Masaryk Univesity
Faculty of Science
Geometric constructions
and correspondences
in action
Habilitation thesis
Vojtˇech ˇZ´adn´ık
Brno, 2019
Abstract
T his thesis is primarily concerned with several geometric notions and constructions that all
have interesting history as well as useful applications. Nowadays, they are well established
and put into the framework of Cartan, respectively parabolic, geometries. After setting the scene
with some general context, we investigate selected problems with notable outcomes. In all the
situations discussed here, the constructions are used or combined in such a way that is in some
respect uncommon. Moreover, although our setting is of a rather functorial nature, we can often
provide concrete down-to-earth interpretations. We consider this symbiosis as an interesting and
not always automatic feature.
Contents
Introduction 1
I Toolkit 5
1 Cartan geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Cartan connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Related notions and connections . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Extension functors and correspondence spaces . . . . . . . . . . . . . . . . . 9
2 Parabolic geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Underlying data, regularity and normalization . . . . . . . . . . . . . . . . 12
2.2 Weyl structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Distinguished curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 BGG sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Concrete geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Reductive geometries, mutations etc. . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Projective geometry and Thomas cone . . . . . . . . . . . . . . . . . . . . . 20
3.3 Flag varieties and correspondence spaces . . . . . . . . . . . . . . . . . . . . 22
3.4 Conformal geometry and ambient metric . . . . . . . . . . . . . . . . . . . . 27
3.5 CR contact geometry and Feﬀerman space . . . . . . . . . . . . . . . . . . . 29
II Usage 35
4 Geometry of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 History and uniﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 The action: curved Cartan extension . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Other parabolic contact structures . . . . . . . . . . . . . . . . . . . . . . . 39
5 Conformal Patterson–Walker metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 History and comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 The action: non-normal Feﬀerman-type construction . . . . . . . . . . . . . 44
5.3 Further action: ambient constructions . . . . . . . . . . . . . . . . . . . . . 47
6 Conformal theory of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.1 History and initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 The action: tractor Frenet construction . . . . . . . . . . . . . . . . . . . . 53
6.3 Further action: r-null curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.4 Comparisons and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
References 58
Contents
Attachments 63
On the geometry of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A projective-to-conformal Feﬀerman-type construction . . . . . . . . . . . . . . . . . . . 99
Feﬀerman–Graham ambient metrics of Patterson–Walker metrics . . . . . . . . . . . . . 133
Conformal theory of curves with tractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Introduction
T ransforming a given problem into one of a diﬀerent character is an inﬂuential approach, which
has dozens of incarnations across the mathematics. This opens new perspectives that help
both to solve the initial problem and, importantly, to investigate new interrelations. The success
of respective techniques depends on the control of the transition between the two settings. The
constructions discussed below exhibit very satisfactory level of control which is then demonstrated
in applications that follow. Moreover, in these applications we are able to follow the transition in
various modes.
It is a common feature of all notions and constructions discussed in this treatise that they can
be dealt within the framework of Cartan, respectively parabolic, geometries. This ﬁeld is nowadays
well developed and provides a wide range of conceptual approaches and instruments.
Background
The background material is collected in chapter I. The very general principles we are going to use
are introduced in section 1. The extension and correspondence space constructions belong among
the most prominent notions, which are later combined in diverse ways.
The typical and omnipresent features of parabolic geometries—the main area of our interest—
are concentrated in section 2. These include the canonical normalization condition, underlying
Weyl structures, distinguished families of curves and basics of the BGG industry.
The previous abstract notions are concretized in section 3, where particular geometric structures
are discussed. In this context, two constructions of ambient character related to projective
and conformal geometries are presented.
Most of the discussed generalities as well as many of their concrete realizations are thoroughly
described by ˇCap and Slov´ak in the monograph [15]. Therefore this book is used as the main
reference throughout this thesis.
Our contributions
Selected applications of previous general concepts, containing our original contributions, are presented
in chapter II. In section 4 we discuss the geometry of chains of parabolic contact structures.
These are distinguished curves determined by the geometric structure which are, as unparametrized
curves, uniquely given by a tangent direction in one point. In this respect they behave as geodesics
of an aﬃne, respectively projective, structure (in other respects, however, they behave very diﬀerently).
As such they form a path geometry on the base manifold, which can also be described in
terms of parabolic geometries.
The most interesting results are obtained for CR and Lagrangean contact structures. It is
the main contribution of our attached article [17] that the correspondence between the initial
2 Introduction
geometric structure and the associated path geometry of chains can be described directly, i.e.,
without any prolongation, via a functorial extension construction provided that the initial structure
is integrable. In these cases, the construction preserves the normality of parabolic geometries
attached to the structures in question, hence allows an immediate use. The point is that we
control fully the transition—as well as a backward reconstruction of the initial structure—also in
non-ﬂat cases. This represents an unusual usage of this kind of construction.
In section 5, an application of the construction of Feﬀerman type is discussed, which induces
a conformal structure of split signature on the (weighted) cotangent bundle of a projective manifold.
The construction preserves normality of induced parabolic geometries only in very limited
situations (in dimension two or in ﬂat case). It might therefore seem surprising how much can be
said, including a full characterization of the induced structure, despite this unfriendly setting. This
is indeed an unexpected case, which is also emphasized as the main achievement of the attached
article [43]. In fact the construction can be seen as a composition of two sub-constructions with a
natural intermediate Lagrangean contact structure. This turns to be half-integrable which implies
the resulting conformal structure is somehow ‘half-normal’.
It is also an interesting issue that our construction can be interpreted as a conformal analogue
of the classical Patterson–Walker metric construction. This point of view is elaborated in our
associated article [42]. As an icing on the cake, we can show that such conformal structures admit
a global and explicit ambient metric, which is a rather rare phenomenon. The ambient metric
can further be described as the true Patterson–Walker metric of the Thomas ambient connection
associated to the very starting projective structure. This construction is the main subject of the
next attached article [44].
In section 6 we consider the problem of describing a generating set of absolute conformal invariants
of curves on conformal manifolds. There is a very classical solution for curves in Euclidean
spaces, the Frenet frame construction, which is based on expressing the inﬁnitesimal change of a
natural frame associated to a given curve. There are several attempts to apply these ideas to other
geometries, both in homogeneous and curved setting. The main complication lies in detecting reasonable
invariant tools to treat the problem, which is why any careful study proceeds only case
by case. Our approach in the last attached article [62] is based on a transition to the conformal
standard tractor bundle, where a natural linear connection and a parallel bundle metric are ready
to use. With these instruments, the original Frenet frame construction is easily adapted for general
curves in conformal manifolds of any signature, including null curves. As a side result we get a
new characterization of conformal circles, the distinguished curves of conformal structures.
Conception
The conception of this text is subordinated to its aim, which is introducing the results of the four
attached articles. They are, of course, not chosen randomly, but rather for their common features
which were already mentioned above. To give a reasonable exposition of the results, it is important
to describe those common features in a wider context. That is why we spend a relatively large
space with processing this background knowledge in an individual chapter I, which might help to
follow the subsequent presentation more smoothly.
Also in chapter II, where the results are presented, we take the liberty to introduce each subject
more widely than is common in journal articles. We always start with a brisk historical survey,
including some comparisons, after which the key parts follow. We help the rushing readers by
directing them to the key parts of these sections by using the label ‘action’ in the section title.
Many of the topics discussed are still subjects of a ongoing research, which is usually commented
in accompanying remarks.
Throughout the text many things are explained and many are not: we tend to explain relevant
notions from Cartan geometry, whereas we do not bother with the notions and ideas from standard
diﬀerential geometry on the level of classical textbooks, e.g., [48, 49]. Besides the notions of smooth
manifolds, connections etc., we rely on Lie groups and their actions, principal bundles and their
reductions, associated bundles and induced connections and the like. Some basics on Lie algebras
and their representations are involved as well.
Introduction 3
Acknowledgements
I thank many colleagues for many discussions that helped to orient my thoughts and research. Of
course, the most intensive exchange of ideas were the ones with the coauthors, namely, Andreas
ˇCap, Boris Doubrov, Matthias Hammerl, Martin Pan´ak, Katja Sagerschnig, Josef ˇSilhan, Jan
Slov´ak, Arman Taghavi-Chabert and Lenka Zalabov´a. I also thank my chieftain, Helena Durnov´a,
for a professional support and many language corrections. But the most thanks go to my wife and
daughters for their patience, fun and inspiration.
CHAPTER I
Toolkit
T hroughout this thesis we enjoy the opportunity to remain in the safe haven of Cartan geometries.
This is our preferred point of view and this is also why we open the current chapter by
introducing general concepts that will be repeatedly applied below. After focusing on the subclass
of parabolic geometries, we present concrete geometric structures and other related notions that
are further employed in chapter II. Individual sections are meant to be independent, although
there are many corss-references. The standard reference covering most of the material collected
here is [15]; generally, if we do not refer anywhere, details can be found in that book.
There is nothing really original in this chapter except for some remarks on Lagrangean contact
structures in section 3.5, which are useful for later considerations. Although all those observations
are rather direct and natural, we are not aware they have been published anywhere.
1 Cartan geometries
The concept of Cartan connections was developed by ´E. Cartan hand in hand with the method
of moving frame. These techniques proved to constitute an eﬃcient approach to the equivalence
problem, associating a natural parallelism (connection) and essential invariants (curvatures) to a
particular geometric problem. However, learning about Cartan connections from original sources
may be somewhat troubled. One of main reasons is “the diﬃculty Cartan faced in trying to express
notions for which there was no truly suitable language”. This concerns primarily the conception
of principal and associated bundles, the terms in which Cartan’s ideas are nowadays usually
presented. As a gentle introduction to the topic we refer the reader to the monograph [58] by
Sharpe, probably the ﬁrst comprehensive textbook on Cartan geometries, from which also the
quotation above is taken.
The driving idea to what we now call Cartan geometries is indicated in the following diagram:
Cartan geometry (of a particular type) generalizes the corresponding Klein geometry (on a homogeneous
space of a particular Lie group) in an analogous way as the Riemannian geometry
generalizes the Euclidean one.
Riemannian GG Cartan
Euclidean
yy
GG Klein
yy
6 I Toolkit
In other words, Cartan geometry is regarded as a ‘curved’ geometry modeled over a homogeneous
space of a particular Lie group. We employ both this terminology and this point of view
consistently below.
In the following subsections we deﬁne Cartan connection and describe some related notions and
alternative approaches, among which the tractor bundles and connections are emphasized. The
next topic concerns a construction relating Cartan geometries, respectively connections, of diﬀerent
types. The motivation comes from the characterization of homogeneous Cartan connections, i.e.,
(non-ﬂat) Cartan connections that are invariant under a group action. However, constructions of
this type apply more generally and they may have many guises as is demonstrated in what follows.
1.1 Cartan connections
With the previous motivation, we may roughly interpret Cartan’s original approach as follows
(incorporating, however, some non-original terminology to be used below). Considering a particular
geometric structure, take its homogeneous model, i.e., the most symmetric case, ﬁrst. Let the model
space be the coset space G/P of a pair of principal Lie group G and its structure subgroup P ⊂ G.
In this situation, the absolute parallelism on G, the ﬂat Cartan connection, is provided by the
Maurer–Cartan form ω : TG → g, i.e., the left-invariant 1-form on G with values in the Lie algebra
g of G. The ﬂatness of this connection means that the structure equation is satisﬁed, dω+ω∧ω = 0.
In order to emphasize the role of the Lie algebra g, the same is rather written as
dω + [ω, ω] = 0. (1.1)
Instead of the 1-form ω : TG → g one may consider its pull-backs θ := s∗
ω : TU → g, the
so-called Cartan gauges, with respect to local sections s : U → G of the canonical projection
p : G → G/P, where U ⊆ G/P. For a choice of basis of g, any gauge θ is represented by a bunch of
ordinary 1-forms on U. The structure equation (1.1) then expands to a couple of equations relating
the constituents of θ and their exterior diﬀerentials. For sections that are somehow adapted to
the geometric structure in question, these equations simplify. In a more general setting, one starts
with a local adapted coframe and other 1-forms that are consecutively adjusted in a sequence
of steps (which may often seem mysterious) so that one ends as close to the previous structure
equations as possible. This yields a ‘natural’ Cartan gauge, respectively Cartan connection, and
the invalidity of the structure equation is interpreted as the contribution of its curvature. Also,
among the pieces that obstruct the validity of the structure equation, one may distinguish the
essential ones whose vanishing imply the vanishing of all others.
Reading this overview, several natural questions have to arise; in particular, it is not clear what
are the ‘natural’ choices and restrictions to follow. In order to make the previous sketch more
serious (and ready to use), one needs a number of illustrative examples (plus a bit of Cartan’s
insight) or a general theory. It is a good news that many seemingly unrelated cases can be put on
a common foundation, which is the result of decades of later investigations. This brings especially
the notion of normalization into the game, which we exemplify in section 2. Anyhow, the existence
and description of a ‘natural’ Cartan connection associated to a particular problem is typically
not an obvious statement. For the time being, we consider Cartan connection as the given data
and focus on some related notions.
Basic deﬁnitions
We keep in mind the model Cartan geometry of type (G, P) consists of the principal P-bundle
p : G → G/P with the Maurer–Cartan form ω : TG → g which, among other things, deﬁnes an
absolute parallelism on G and resonates with the principal (right) action of the structure group P
so that it reproduces its inﬁnitesimal generators (fundamental vector ﬁelds) and is equivariant. A
general Cartan geometry of type (G, P) on a smooth manifold M consists of a principal P-bundle
p : G → M with Cartan connection, which is a g-valued 1-form ω : TG → g satisfying the previous
three properties, i.e.,
1 Cartan geometries 7
• ωu : TuG → g is a linear isomorphism, for each u ∈ G,
• ω d
dt 0
rexp tX(u) = X, for each u ∈ G and X ∈ p,
• (rh)∗
ω = Adh−1 ◦ ω, for each h ∈ P,
where p ⊂ g is the Lie algebra of the subgroup P ⊂ G and rh : G → G, respectively Adh : g → g,
denotes the principal right action, respectively the adjoint action, by an element h ∈ P. Remember
that Cartan connections are not principal connections (which are p-valued 1-forms satisfying the
last two conditions).1
A morphism of two Cartan geometries of the same type is a morphism of the corresponding
principal bundles that preserves Cartan connection.
The curvature of Cartan connection ω is the g-valued 2-form on G given by
κ := dω + [ω, ω]. (1.2)
Note that κ is strictly horizontal, i.e., it vanishes under insertion of any vertical vector. Cartan
curvature vanishes identically if and only if the Cartan geometry is locally isomorphic to its
homogeneous model.
The composition of ω : TG → g with the quotient projection g → g/p is the so-called solder
form. For any u ∈ G, this yields a linear isomorphism TxM → g/p, where x = p(u). For a diﬀerent
element over x, say rh(u) ∈ p−1
(x) with h ∈ P, this isomorphism is aﬄicted by the action of
Adh−1 . Hence the identiﬁcation TG ∼= G × g provided by ω descends to the identiﬁcation of TM
with G ×P (g/p), the associate bundle to G with respect to the indicated P-action.
TG
ω GG
T p

g

TM GG g/p
Under this identiﬁcation, various geometric objects on M are reﬂected as P-invariant objects in
g/p. Furthermore, the previous identiﬁcation can be read so that a vector ﬁeld on M is represented
by a P-equivariant map G → g/p, the so-called frame form. The same logic applies to any tensor
ﬁeld on M, more generally, to sections of any associated bundle to G. In particular, the frame form
of the Cartan curvature κ is a P-equivariant function
G → Λ2
(g/p)∗
⊗ g. (1.3)
Composing this with the quotient projection g → g/p, we obtain a P-equivariant map G →
Λ2
(g/p)∗
⊗(g/p), which represents a tensor ﬁeld of type Λ2
T∗
M ⊗TM, which is called the torsion
of Cartan connection ω.
Remarks
Cartan gauge is deﬁned alike to the ﬂat case, i.e., θ := s∗
ω : TU → g, where s : U → G is a local
section of the projection p : G → M. Under change of section s = rh ◦ s, for h : U → P, the gauge
changes so that
θ = Adh−1 θ + δh, (1.4)
where δh : TU → p is the left logarithmic derivative of h, i.e., the pull-back of the Maurer–
Cartan form on P. Cartan geometry can be dealt in terms of (atlases of) Cartan gauges with the
equivalence relation given by (1.4) (on intersections of domains).
1Cartan connection determines a principal connection on G only in very speciﬁc cases of so-called reductive
geometries, see section 3.1.
8 I Toolkit
1.2 Related notions and connections
Since Cartan connection ω : TG → g is not a principal connection, it does not induce connection
neither on TM nor on any associated bundle. It, however, determines a principal connection
ω : TG → g on the principal G-bundle G := G ×P G via the equivariant extension of ω from
the image of the canonical inclusion G → G ×P G (mapping u ∈ G to the equivalence class
represented by u and the unit element of G). The principal connection ω induces connections on
all associated bundles to G → M. Hence the Cartan connection ω gives rise to a connection on
any associated bundle to G → M provided that the standard ﬁbre is actually a G-module (and
not only P-module).
Cartan space
As a typical instance we have to mention the Cartan space, S := G ×P (G/P) = G ×G (G/P) with
the obvious G-action on G/P, which is important for several reasons. There is a canonical section
of the projection S → M (mapping x ∈ M to the equivalence class represented by any u ∈ Gx and
the origin in G/P), along which the vertical subbundle is naturally identiﬁed with G ×P (g/p),
i.e., the tangent bundle TM. This observation makes precise the intuitive idea of ‘osculating’ M
at each point by the model space G/P, the idea that may be spotted in Cartan’s work.
The induced connection on S, respectively the parallel displacement, allows to deﬁne the development
of a curve on M into the homogeneous space G/P. More concretely, for a curve c : I → M
and its canonical lift ˆc : I → S (given by the canonical section of the projection S → M), one may
transport parallelly the points ˆc(t) into a chosen ﬁber over x = c(t0). Locally, this draws curves
in Sx
∼= G/P passing through the origin. The previous identiﬁcation depends on u ∈ Gx so that,
for rh(u) with h ∈ P, the resulting curve is shifted by the left action of h−1
on G/P. However,
ﬁxing u ∈ Gx, this construction provides a bijection between germs of curves through x in M and
germs of curves through origin in G/P preserving the order of contact. This is an important tool
for studying curves on M via their counterparts in G/P. In particular, the curves developing to
homogeneous curves in G/P form a distinguished family of curves on M, cf. section 2.3.
Note also that sections of S → M are in bijective correspondence with reductions of G → M to
the subgroup P ⊂ G (in particular, the canonical section corresponds to the canonical reduction
G → G ×P G above). Hence, taking the principal connection on G as the prior object, Cartan
connection on the reduced bundle is given by its pull-back. In this vein, the holonomy of Cartan
connection is deﬁned to be the holonomy of the corresponding principal connection on G and as
such it is a subgroup of G.
Tractors
Another prominent family of associated bundles with natural induced connections are the tractor
bundles, which are the vector bundles with standard ﬁber being a G-representation. Among these
bundles there is a particularly important one, namely, the adjoint tractor bundle A := G ×P g =
G ×G g given by the adjoint representation Ad : G → GL(g). Among other things, this bundle is
equipped with a natural algebraic bracket { , } : A × A → A, given by the Lie bracket on g, and
a bundle projection Π : A → TM, given by the quotient projection g → g/p and the identiﬁcation
TM ∼= G ×P (g/p). Using the Cartan connection on G, sections of A → M are identiﬁed with
vector ﬁelds on G that are invariant with respect to the principal P-action.
For a P-representation ρ : P → GL(V ), let V := G×P V be the associated bundle corresponding
to ρ. The diﬀerentiation of any P-equivariant map G → V in the direction of a P-invariant vector
ﬁeld is again P-equivariant, which yields a natural diﬀerential operator
D : Γ(A) × Γ(V) → Γ(V), (1.5)
the so-called fundamental derivative.
1 Cartan geometries 9
If V is a tractor bundle, i.e., ρ is the restriction of a homomorphism G → GL(V ) and V =
G ×P V = G ×G V , then we also have the tractor connection,
: Γ(TM) × Γ(V) → Γ(V), (1.6)
i.e., the linear connection induced by ω. It follows that, for any s ∈ Γ(A) and t ∈ Γ(V), these two
operations are related by
Π(s)t = Dst + s • t, (1.7)
where the bundle map • : A × V → V is given by the derivative of the deﬁning representation,
g × V → V , and Π : A → TM is as above. (In the case V = A, the map • coincides with the
bracket { , } above.) Notice an important feature behind the equality (1.7), namely, that the result
does not depend on the section s ∈ Γ(A) covering ξ = Π(s) ∈ Γ(TM).
One of often used aspects is that the curvature κ of the Cartan connection ω is naturally
interpreted as a section of Λ2
T∗
M ⊗A, cf. (1.3). Denoting the curvature of the tractor connection
by R, which is a section of Λ2
T∗
M ⊗ End(V), then these two objects are related by
R(ξ, η)(t) = κ(ξ, η) • t,
for any ξ, η ∈ Γ(TM) and t ∈ Γ(V). In particular, the holonomy of the tractor connection
coincides with the image of the holonomy group of the principal connection ω under the deﬁning
representation G → GL(V ).
tractor (V, )
ÖÖ
principal (G, ω)

UU
Cartan (G, ω)
B
UU
55
M
Previous reasonings indicate how to deal with Cartan geometry in terms of induced tractorial
objects. Among all conceivable tractor bundles, other popular candidates (besides the adjoint ones)
are the standard tractor bundles corresponding to the standard representation of the principal
group G. It is interesting that these entities, in the case of projective and conformal geometries,
have roots in the work of T.Y. Thomas from the similar period in which Cartan developed notions
leading to Cartan connections. Despite this coincidence, the full understanding of all relationships
is of much later date, see [2, 11].
1.3 Extension functors and correspondence spaces
The previous extension of the Cartan connection ω : TG → g of type (G, P) to the principal
connection ω : TG → g was simply driven by the inclusion P ⊂ G. This notion can be generalized
as follows.
General principal extension
In the homogeneous case, i.e., when G = G and ω = Maurer–Cartan form, the bundle G = G×P G ∼=
(G/P)×G is a homogeneous principal bundle and the principal connection ω is invariant under the
action of G. More generally, any homomorphism of Lie groups i : P → G deﬁnes a homogeneous
principal G-bundle G := G ×P G. For p ⊂ g and ˆg being the Lie algebras of respective Lie groups,
it follows that G-invariant principal connections on G are in bijective correspondence with linear
maps α : g → ˆg that are compatible with i and P-equivariant, i.e., such that
10 I Toolkit
• α|p = i , the derivative of i,
• α ◦ Adh = Adi(h) ◦ α, for each h ∈ P.
It also follows that, for a given Cartan geometry of type (G, P), any such compatible pair (i, α)
determines a principal G-bundle with a principal connection ω and this construction is functorial.
One just substitutes G for G and a general Cartan connection ω for the Maurer–Cartan form.
More concretely, ω extends equivariantly the form α ◦ ω : TG → ˆg from the image of the canonical
inclusion j : G → G ×P G, i.e., it is determined by j∗
ω = α ◦ ω. In this view, the construction at
the beginning of section 1.2 corresponds to G = G so that i is the inclusion P ⊂ G and α is the
identity map g = ˆg.
General Cartan extension
In a similar manner, one may characterize invariant Cartan connections and apply the corresponding
functor generally. In addition to the previous setting, there is a new element entering the game
in the place of the structure group of the extended principal bundle. To be more precise, let P ⊂ G
and P ⊂ G be the pairs of Lie groups and their closed subgroups and let p ⊂ g and ˜p ⊂ ˜g be
the corresponding Lie algebras. A Lie group homomorphism i : P → P deﬁnes a homogeneous
principal P-bundle G := G×P P and it follows that G-invariant Cartan connections of type (G, P)
on G are in bijective correspondence with linear maps α : g → ˜g such that
• α|p = i , the derivative of i,
• α ◦ Adh = Adi(h) ◦ α, for each h ∈ P,
• the induced map α : g/p → ˜g/˜p is a linear isomorphism.
It also follows that, for a given Cartan geometry (G → M, ω) of type (G, P), any such compatible
pair (i, α) determines a Cartan geometry (G ×P P → M, ω) of type (G, P) and this construction
is functorial. Again, the induced connection is determined by
j∗
ω = α ◦ ω
and its required P-equivariance, where j : G → G ×P P is the canonical inclusion. The inner action
of P, respectively its adjoint action on ˜g, deﬁnes an equivalence relation on the pairs of compatible
maps (i, α) so that two equivalent pairs induce isomorphic functors.
The eﬀect of this construction on the Cartan curvature follows easily from deﬁnitions. In the
invariant setting, i.e., starting with ﬂat ω, the curvature κ of the induced Cartan connection ω is
fully given just by α : g → ˜g so that the frame form of κ is constant along the image of j with
value the P-invariant element Ψα ∈ Λ2
(g/p)∗
⊗ ˜g given by
(X + p, Y + p) → [α(X), α(Y )] − α([X, Y ]). (1.8)
In particular, the induced Cartan connection is ﬂat if and only if α is a homomorphism of Lie
algebras.
To describe the general case, we use the same symbol Ψα also for the corresponding section
of Λ2
T∗
M ⊗ A, where A is the adjoint tractor bundle to the induced Cartan geometry. Similarly,
since α is P-equivariant, it gives rise to a bundle map α : A → A, where A is the adjoint tractor
bundle to the initial Cartan geometry. Now, if κ ∈ Γ(Λ2
T∗
M ⊗ A) is a curvature of the initial
Cartan connection, then the curvature κ ∈ Γ(Λ2
T∗
M ⊗ A) of the induced one is given by
κ = α ◦ κ + Ψα. (1.9)
Correspondence space
Until now we considered various bundles and connections over a common base manifold. Factorizing
a given bundle, respectively connection, via an intermediate manifold leads to the notion of
correspondence spaces. For a Cartan geometry (G → M, ω) of type (G, P), let Q ⊂ P ⊂ G be
2 Cartan geometries 11
another closed subgroup and let M := G/Q be the corresponding quotient space. This is called
the correspondence space of M induced by the inclusion Q ⊂ P.
Easily, M is the total space of the ﬁber bundle over M with the standard ﬁber P/Q, the
initial Cartan geometry factorizes to the Cartan geometry (G → M, ω) of type (G, Q) and this
construction is functorial. It is also clear that the initial Cartan geometry over M is ﬂat if and
only if the induced Cartan geometry over M ﬂat.
(G, ω)
77

M
yy
M
By its horizontality, the curvature κ of the Cartan connection ω vanishes under insertion of
any vertical vector of the projection M → M. Under the identiﬁcation TM ∼= G ×Q (g/q), the
vertical subbundle is V ∼= G ×Q (p/q) and we highlight the previous condition as
ιvκ = 0, for any v ∈ Γ(V ). (1.10)
It is both interesting and important result that this condition locally characterizes Cartan geometries
obtained by the correspondence space construction.
Remarks and outlook
Combinations of the previous constructions provide an eﬃcient approach for constructing (or
better understanding of already existing) relations between Cartan geometries of diﬀerent types.
Below in this text we meet several concrete realizations belonging to this framework.
(G, ω)
P extension

(G, ω) correspondence +
Q
88
P

(G, ω)
Q
88
C
VV
M
xx
M
M
The correspondence space construction is induced by the inclusion Q ⊂ P, the Cartan extension
by the pair of maps i : Q → P and α : g → ˜g satisfying the compatibility conditions listed on the
previous page (but substituting Q in the place of P). Let us highlight several cases:
• Correspondence spaces themselves (no further extension) appear in sections 3.2, 3.3 and
other places.
• Mutations are the constructions that correspond to i being an equality Q = P and α being
a (Q-equivariant) linear isomorphism, see section 3.1.
• Constructions of Feﬀerman type are induced by an embedding φ : G → G of Lie groups so
that i = φ|Q and α = φ , see sections 3.5 and 5.2.
• General Cartan extension is discussed in section 4.2.
12 I Toolkit
2 Parabolic geometries
Parabolic geometries are Cartan geometries of type (G, P) where G is a semisimple Lie group and
P ⊂ G is its parabolic subgroup. This terminology reﬂects the one for corresponding Lie algebras:
Lie algebra g is semisimple if it has no non-zero solvable ideal, equivalently, it is a direct sum of
simple Lie algebras, equivalently, the Killing form is non-degenerate (the Cartan’s criterion), etc.
Subalgebra p ⊂ g is parabolic if it contains the Borel subalgebra (the maximal solvable subalgebra
of g) and the ortho-complement of p in g with respect to the Killing form is the nilradical of p.
Previous explanations does not sound too convincing in the front of geometric audience. To
make a better impression, one usually adds that parabolic geometries involve such important and
well studied structures as projective, conformal, CR contact and many others. These relations
are get closer in section 3. Here we focus on several uniform features of parabolic geometries,
among which the conceptual and canonical normalization condition is the prominent one. Also,
when establishing the correspondence between concrete and more abstract notions, the relations
to ordinary G-structures with subordinate compatible connections can be dealt in generality. Next
topics to be introduced here are distinguished curves and BGG sequences.
Basics
In fact, parabolic subalgebras are related to gradings of semisimple Lie algebras, which is a more
appropriate starting position. Let
g = g−k ⊕ · · · ⊕ g0 ⊕ · · · ⊕ gk
be a |k|-grading of a semisimple Lie algebra g, i.e., [gi, gj] ⊆ gi+j (where we consider gl = 0, for
|l| > k), and let
g = g−k
⊃ · · · ⊃ g0
⊃ · · · ⊃ gk
= gk
be the corresponding ﬁltration, where gi
:= gi⊕· · ·⊕gk. This turns g into a ﬁltered Lie algebra, i.e.,
[gi
, gj
] ⊆ gi+j
. Let the grading be such that the subalgebra p+ := g1
= g1 ⊕ · · · ⊕ gk is generated
by g1 or, equivalently, g− := g−k ⊕ · · · ⊕ g−1 is generated by g−1. Then p := g0
= g0 ⊕ p+ is a
parabolic subalgebra with nilradical p+.
Any parabolic subalgebra can be obtained this way, in particular, their classiﬁcation corresponds
to the classiﬁcation of gradings of g. It follows that this is equivalent to distinguishing
subsets of simple roots of g (which is pictured by crossing nodes of the corresponding Satake diagram).
The number of distinguished elements corresponds to the dimension of the center of the
reductive subalgebra g0.
Passing to the group level, there is a freedom in choosing a Lie group G and its subgroup P
whose Lie algebras are g and p, respectively. Fixing G, the extreme choices for P ⊂ G are the
normalizer of p in G and its connected component of the identity. Under the restricted adjoint
action, the subgroup P preserves the ﬁltration of g. Given a parabolic subgroup P ⊂ G, let G0 ⊂ P
be the subgroup preserving the gradation of g (the Lie algebra of G0 is obviously g0). This inclusion
induces an isomorphism G0
∼= P/P+, where P+ denotes the nilpotent normal subgroup exp p+.
Note that the Killing form on g provides a natural isomorphism g ∼= g∗
compatible with the
ﬁltration, respectively gradation. This leads to a natural isomorphism of P-modules p+
∼= (g/p)∗
,
respectively G0-modules p+
∼= g∗
−, which we often use below without explicit mentioning. Note
also that for many considerations that follow, it is an important assumption that no simple ideal
of g is contained in g0. This is a vacuous condition for simple g and we will not emphasize it below.
2.1 Underlying data, regularity and normalization
For a parabolic geometry (G → M, ω) of type (G, P), we now describe the induced underlying
data on M. We comment the conditions under which this association can be made bijective: the
regularity and the normality.
2 Parabolic geometries 13
Underlying data, regularity
The subgroup P+ ⊂ P acts freely on G, let us denote by G0 := G/P+ the orbit space. The ﬁbration
G → G0, respectively G0 → M, is a principal bundle with structure group P+, respectively G0.
The P-invariant ﬁltration of g gives rise to ﬁltrations of the tangent bundles TG, TG0, respectively
TM (which are of length 2k + 1, k + 1, respectively k). Finally, the Cartan connection ω descends
to a couple of partially deﬁned G0-equivariant 1-forms (ωi) on G0 with values in gi (where i =
−k, . . . , −1).
(G, ω)
ww

(G0, ωi)
99
M
In the case of trivial ﬁltration, k = 1, we get the classical picture of G-structures on M, with
the structure group G0 ⊆ GL(g−1), where ω−1 : TG0 → g−1 plays the role of the solder form. In
general, there is an important notion of regularity tightening the relation between the ﬁltration
of TM and the reduction to G0. The parabolic geometry (respectively its underlying structure) is
called regular if the ﬁltration of TM is compatible with the Lie bracket of vector ﬁelds and the
corresponding Levi bracket
L : gr(TM) × gr(TM) → gr(TM), (2.1)
i.e., the induced algebraic bracket on the associated graded bundle, coincides with the one given
by the Lie bracket in the Lie algebra g− via the identiﬁcation gr(TM) ∼= G0 ×G0
g−. Hence the
notion of symbol algebra etc.
The regularity condition may be equivalently formulated in terms of the curvature κ of the
Cartan connection ω as follows. The ﬁltration of g gives rise also to a ﬁltration of the adjoint tractor
bundle A, which is compatible with the algebraic bracket { , } : A×A → A and the corresponding
objects on TM via the projection Π : A → TM. This leads to the notion of homogeneity degree of
sections of Λ2
T∗
M ⊗ A (and similar bundles). It follows that the parabolic geometry is regular if
and only if its curvature κ is of positive homogeneity. Clearly, |1|-graded or torsion-free parabolic
geometries are automatically regular.
Normalization
It is an important achievement that, up to two exceptional cases2
, the correspondence between
(isomorphic classes of) regular parabolic geometries and induced underlying data can be made
bijective imposing a normalization condition. The whole story involves an abstraction of the underlying
data (the notion of inﬁnitesimal ﬂag structures), the construction of a bigger P-bundle
with a Cartan connection of type (G, P) (via a choice of compatible principal G0-connection and
an application of a convenient extension functor discussed above) and a possible normalization
of the resulting Cartan connection (to be discussed below). This process should be seen as a
ﬁltration adapted generalization of the standard prolongation procedure for G-structures. In the
case of |1|-graded parabolic geometries, these can indeed be related so that g1
∼= g∗
−1 coincides
with the ﬁrst prolongation of g0 and its second prolongation vanishes.
Now we focus on the natural normalization condition. First of all, one has to ﬁnd a reasonable
description of the fact that two Cartan connections induce the same underlying data and, in such
case, a description of the diﬀerence of their curvatures. This is done in terms of the diﬀerence
map of the two connections (which is seen due to its horizontality as a 1-form Φ : TM → A), and
2These are the projective and projective contact structures, in which cases a class of connections on the base
manifold has to be added as a part of underlying data.
14 I Toolkit
the bundle map gr(T∗
M ⊗A) → gr(Λ2
T∗
M ⊗A) induced by the standard diﬀerential of the chain
complex
· · · −→ g∗
− ⊗ g
∂
−→ Λ2
g∗
− ⊗ g −→ . . . (2.2)
computing the Lie algebra cohomology of g− with coeﬃcients in g (which is indeed G0-equivariant).
A normalization condition may be given by a subbundle of Λ2
T∗
M ⊗ A, respectively by a
P-invariant subspace of Λ2
(g/p)∗
⊗ g, whose associated graded is complementary to im ∂. For any
parabolic geometry, there is a canonical choice given by the kernel of a natural adjoint map, the
Kostant codiﬀerential, which is the codiﬀerential of the complex
· · · ←− p+ ⊗ g
∂∗
←− Λ2
p+ ⊗ g ←− . . . . (2.3)
This map is indeed P-equivariant etc.; the corresponding bundle map is denoted by the same
symbol. The Cartan connection, respectively the parabolic geometry, is normal if its curvature is
a section of ker ∂∗
⊂ Λ2
T∗
M ⊗ A.
For a given Cartan connection, its potential normalization proceeds inductively according to
the homogeneity degree. A rough sketch of the process (with concrete outcomes in the case of
conformal structures) is described in section 5.2.
Harmonic curvature
For normal parabolic geometries, the composition of the curvature function κ with the quotient
projection to the cohomology space ker ∂∗
→ ker ∂∗
/ im ∂∗
yields the so-called harmonic curvature,
denoted by κH. In fact, P+ acts trivially on this space, which means the corresponding associated
bundle allows an interpretation in underlying terms. The adjointness of ∂ and ∂∗
gives rise to the
G0-equivariant self-adjoint endomorphism
:= ∂ ◦ ∂∗
+ ∂∗
◦ ∂,
the Kostant Laplacian. This determines a G0-invariant Hodge decomposition of the chain complex,
in particular, the kernel of this operator is isomorphic to the cohomology space in each degree.
For regular and normal parabolic geometries, it follows that the lowest non-zero homogeneous
component of κ has values in ker , i.e., it coincides with the corresponding homogeneous component
of κH. In particular, κ vanishes identically if and only if κH does. In fact, the full curvature
can be recovered from the harmonic one via the BGG splitting operator, see section 2.4. Finally,
note that ker is algorithmically computable as a g0-representation and often consists only of few
components, see [59].
Previous results allow to reduce many treatments involving curvature just to its harmonic
counterpart. This is also behind number of conclusions discussed below.
2.2 Weyl structures
Weyl structures provide a useful tool in transition between the two descriptions of parabolic
geometries (Cartan geometries vs. underlying data). In particular, they induce a canonical class
of compatible aﬃne connections on the base manifold. All notions that follow have well known
preimages in conformal geometry, from which also a lot of terminology is taken.
Reductions, splittings, connections
Let (G → M, ω) be a parabolic geometry of type G/P and let G0 = G/P+ → M be the underlying
principal G0-bundle as above. A Weyl structure is a global G0-equivariant section σ : G0 → G of
the P+-projection G → G0, i.e., a reduction of the P-principal bundle G → M to the subgroup
G0 ⊂ P. The pull-back σ∗
ω : TG0 → g of the Cartan connection decomposes into G0-equivariant
pieces in accord with the grading of g. The negative component σ∗
ω− : TG0 → g− provides
an identiﬁcation of the ﬁltered tangent bundle TM with the associated graded gr(TM), hence
2 Parabolic geometries 15
a splitting of the ﬁltration of TM. The zeroth component σ∗
ω0 : TG0 → g0 deﬁnes a principal
G0-connection, hence also an induced connection on any associated bundle; they are jointly called
the Weyl connections. The positive component σ∗
ω+ : TG0 → p+ can be interpreted as a 1-form
TM → T∗
M, respectively a section of T∗
M ⊗ T∗
M, and is called the Rho or Schouten tensor.
(G, ω)

ww
(G0, σ∗
ωj)
σ HH
@@
M
The splitting of the ﬁltration of the tangent, respectively cotangent, bundle caused by a choice
of Weyl structure applies similarly to any associated bundle corresponding to a P-representation
V that is completely reducible as a G0-representation. There is a P-invariant ﬁltration of V with
the property that the action of P+ shifts each subspace to the next smaller one. This yields a
ﬁltration of V = G ×P V ,
V ⊃ V0
⊃ V1
⊃ . . . , (2.4)
and a choice of a Weyl structure provides an identiﬁcation of V with its associated graded gr(V) ∼=
G0 ×G0
V . The ﬁltration of V may be described directly via a natural bundle map • : T∗
M ×V → V
induced by the derived action of p+ on V and the identiﬁcation T∗
M ∼= G ×P p+.
In the case when V is a tractor bundle, this map is the restriction of the characteristic map
• : A × V → V (as in (1.7)) under the natural inclusion T∗
M ⊂ A (given by p+ ⊂ g), which
is independent of a choice of Weyl structure. On the other hand, since g− ⊂ g as a G0-module,
a choice of Weyl structure yields an inclusion of bundles TM ⊂ A, respectively their associated
graded. In this context we can express the tractor connection , according to (1.6), as follows. For
a Weyl structure σ, let D and P be the corresponding Weyl connection on V and the Rho tensor,
respectively. A comparison of the fundamental derivative D with these objects yields that
ξt = Dξt + P(ξ) • t + ξ • t, (2.5)
for each ξ ∈ Γ(TM) and t ∈ Γ(V). This can further be expanded with respect to the induced
splitting of A and V, which yields the concrete formulas that appear below.
Although the constituents of (2.5) depend on a choice of Weyl structure, the resulting tractor
connection does not. In the just indicated approach this follows for free, while a direct check of
this fact can be a rather tedious exercise. This would involve the knowledge of the transformation
rules of respective objects according to a change of Weyl structure.
Changes of Weyl structures
Two Weyl structures σ and ˆσ diﬀer by a G0-equivariant map Υ : G0 → p+ so that
ˆσ(u) = rexp Υ(u)(σ(u)), (2.6)
for each u ∈ G0. Such Υ corresponds to a 1-form on M, more precisely, to a section of the associated
graded bundle gr(T∗
M). Weyl structures thus form an aﬃne bundle over M modeled on this vector
bundle. The bundle of Weyl structures can also be described as W := G ×P P/G0 → M; indeed,
sections of this bundle are in bijective correspondence with Weyl structures σ : G0 → G. The
Cartan connection on G induces a surprisingly rich geometric structure on the total space of W,
which is discovered and employed only recently, see [45, 14].
The eﬀect of a change of Weyl structure on various associated quantities can be described in
terms of corresponding 1-forms. Below we meet explicit formulas only for |1|-graded geometries,
in which cases the uniform background is as follows. For Weyl structures σ and ˆσ diﬀering by
Υ ∈ Γ(T∗
M) as in (2.6), let (v0, v1, v2, . . . ) and (ˆv0, ˆv1, ˆv2, . . . ) be the corresponding splittings of
an element of a natural bundle V = G ×P V , labeled so that the zeroth gradation component is
16 I Toolkit
the projecting part, i.e., a quotient of the whole V, cf. (2.4). Then, expanding the action of exp Υ,
it easily follows that
ˆv0 = v0, ˆv1 = v1 − Υ • v0, ˆv2 = v2 − Υ • v1 + 1
2 Υ • Υ • v0, . . . . (2.7)
Further, the corresponding Weyl connections D and ˆD on V are related as
ˆDξv = Dξv − {Υ, ξ} • v (2.8)
and the corresponding Rho tensors P and ˆP diﬀer as
ˆP(ξ) = P(ξ) + DξΥ + 1
2 {Υ, {Υ, ξ}}, (2.9)
where ξ ∈ Γ(TM) and v ∈ Γ(V).
Scales
Scales are sections of scale bundles, which are oriented line bundles convenient to a given parabolic
geometry. The point is that sections of a scale bundle determine special Weyl structures, called
exact, whereas general Weyl structures correspond to principal connections on them. Note that
the terminology reﬂects the fact that the 1-form Υ measuring the diﬀerence between two exact
Weyl structures is exact.
As associated bundles to G0 → M, scale bundles are given by a representation G0 → R+.
As such they can be classiﬁed by certain elements from the center of g0 (describing the derived
representation). Any scale bundle is identiﬁed with the quotient bundle G0/G0 → M, where G0
is the kernel of the deﬁning representation. In particular, sections of this bundle correspond to
reductions of the principal G0-bundle G0 → M to the structure group G0. Thus, exact Weyl
connections have reduced holonomies, corresponding to the parallelity of the geometric quantity
related to a choice of scale.
For |1|-graded geometries, the center of g0 is 1-dimensional (consisting of multiples of the
grading element), hence G0 is just the semisimple part of the reductive group G0.
Curvature invariants
Likewise to the decomposition of the pullback of the Cartan connection with respect to a Weyl
structure, one may decompose the pullback of its curvature. These data allow an interpretation
of the Cartan, respectively harmonic, curvature in underlying terms as well as an alternative
treatment of parabolic geometries themselves.
Bellow we need some details only for |1|-graded geometries, in which case the general discussion
simpliﬁes considerably as follows. For a normal Cartan connection ω and a Weyl structure σ, let
D be the corresponding Weyl connection on TM and let R be its curvature tensor. Let κ be the
curvature of ω and let σ∗
κ =: T + W + Y be the decomposition of its pullback to G0 according
to the grading g = g−1 ⊕ g0 ⊕ g1. The individual components, interpreted as 2-forms on M with
values in TM, End0(TM) and T∗
M, are the torsion, Weyl curvature and Cotton–York tensor of
D, respectively. It follows the last two tensors may be expressed in previous terms as
W = R + ∂P, Y = dD
P,
where ∂ is the bundle map induced by the diﬀerential (2.2) and dD
is the covariant exterior
derivative determined by D. Note that the ﬁrst non-zero tensor in the sequence (T, W, Y ) is an
invariant of the structure, i.e., is independent of the choice of Weyl structure, as it corresponds to
the lowest non-zero homogeneous component of the harmonic curvature κH.
2 Parabolic geometries 17
2.3 Distinguished curves
As for any Cartan geometry (G → M, ω), the notion of development of curves described in section
1.2 allows to distinguish (families of) curves on M using their counterparts in the corresponding
homogeneous space G/P. The prominent curves there are the exponential curves, respectively
orbits of 1-parameter subgroups of G. Within the framework of parabolic geometries we usually
restrict to such curves whose generators lie in g− ⊂ g. The corresponding curves on M are called
the distinguished curves.3
Besides the development, there are other ways to distinguished curves in parabolic geometry.
For instance, for a curve c : I → M, the following properties are equivalent:
• The curve c admits a development into a curve L(t) = exp(tX)/P in G/P, where X ∈ g−.
• The curve c admits a lift ˆc : I → G such that ω(ˆc (t)) ∈ symL, for all t, where symL ⊂ g is
the symmetry algebra of L.
• The curve c admits a lift ˆc : I → G such that ω(ˆc (t)) = X, for all t.
• The curve c admits a lift ˆc : I → A such that Dˆcc = 0, where D is the fundamental
derivative.
• There is a Weyl structure such that Dc c = 0 and P(c ) = 0, where D and P is the corresponding
Weyl connection and Rho tensor, respectively.
In the second condition, the symmetry algebra symL ⊂ g consists of those inﬁnitesimal automorphisms
of G/P that are everywhere tangent to L. The third condition can be rephrased so
that the curve is the projection to M of the ﬂow of a constant vector ﬁeld in G. In the fourth
condition, the lift to the adjoint tractor bundle A is the same as a P-invariant vector ﬁeld in G
over the curve, hence a lift to G whose tangent ﬁeld is extended as required. The relation to the
last condition follows from relations between D, P and D as is behind (2.5).
Remarks
It is worth emphasizing that, for diﬀerent distinguished curves with the same trajectory, the
freedom in possible reparametrizations is not arbitrary. It turns out that the only eventualities are
either projective of aﬃne ones.
For a ﬁner discussion of possible types of distinguished curves and their properties, we further
restrict the set of possible generators in g− ⊂ g. It follows that these types can be classiﬁed according
to G0-invariant subsets A ⊂ g−. Diﬀerent subsets contained in a common P-hull correspond
to diﬀerent types of distinguished curves sharing the same tangent vectors (this may happen only
for more graded parabolic geometries).
Among other approaches to distinguished curves, let us mention the one from [45] employing
a natural connection on the bundle of Weyl structures. Below in section 6 we also formulate other
characterizations of distinguished curves in conformal setting (where they are named conformal
circles) using tractor calculus. Some of those ideas are easily, but not uniformly, transformable to
other parabolic geometries, cf. section 6.4.
2.4 BGG sequences
BGG sequences form a really powerful instrument of parabolic geometry. Many important invariant
operators, equations, respectively problems, can be interpreted in this framework. The general
theory was introduced by ˇCap, Slov´ak and Souˇcek in [16], later simpliﬁed by Calderbank and
Diemer in [9]. The sequences are intrinsically constructed for any tractor bundle of given parabolic
geometry. The key player is the so-called splitting operator, let us introduce the basics.
3Even in this generality, distinguished curves can be found under various nicknames in the literature: canonical
curves, parabolic geodesics, generalized geodesics, Cartan circles and the like.
18 I Toolkit
For a given tractor bundle V = G ×P V , the starting ingredient is the exterior covariant
derivative given by the normal tractor connection ,
0 −→ Γ(V)
d
−→ Γ(T∗
M ⊗ V)
d
−→ Γ(Λ2
T∗
M ⊗ V)
d
−→ . . .
In the opposite direction it operates ∂∗
, the bundle map given by the Kostant codiﬀerential as in
(2.3), where only the coeﬃcients are now taken in the g-module V rather than in g. This deﬁnes
the cohomology bundles Hi := ker ∂∗
/ im ∂∗
and projections Πi : ker ∂∗
→ Hi, both labeled by
the corresponding form degree. The crucial fact is the existence of a unique diﬀerential operator
Li : Γ(Hi) → Γ(ker ∂∗
), for any i, splitting the projection Πi and satisfying
∂∗
(d (Li(τ))) = 0,
for all τ ∈ Γ(Hi). These are the splitting operators. The displayed property allows the composition
Θi = Πi+1 ◦ d ◦ Li, which yields the sequence of BGG operators,
0 −→ Γ(H0)
Θ0
−→ Γ(H1)
Θ1
−→ Γ(H2)
Θ2
−→ . . .
Note that his sequence forms a complex if and only if the parabolic geometry is ﬂat.
Some applications and remarks
One of many use of BGG techniques is the reconstruction of the curvature κ ∈ Γ(Λ2
T∗
M ⊗ V) of
normal Cartan connection from its harmonic counterpart κH = Π2(κ). As a consequence of the
diﬀerential Bianchi identity, it follows that κ = L2(κH).
Another noteworthy application concerns the characterization of correspondence spaces discussed
in section 1.3, which allows a much stronger formulation. With the notation as before, it
follows that the condition
ιvκH = 0, for any v ∈ Γ(V ), (2.10)
implies the one in (1.10), hence a local characterization of Cartan connection obtained by the
correspondence space construction. This was shown by ˇCap in [10].
Below we deal with ﬁrst splitting, respectively ﬁrst BGG, operators. Note that the ﬁrst cohomology
bundle H0 is simply the quotient V/V0
, where V0
is the largest ﬁltration component in V,
cf. (2.4). In particular, the ﬁrst splitting operator maps sections of the projecting part of gr(V) to
sections of V.
3 Concrete geometries
In this section we discuss mostly those structures and constructions that appear also in chapter II.
It is a preparatory section, but it was not meant to be just a collection of conventions and notations.
References to the previous text should indicate how to see this material within the given context.
Besides concrete incarnations of the general constructions above, we are going to introduce two
ambient constructions for projective and conformal structures. Except the structures in section
3.1, all others belong among parabolic geometries.
3.1 Reductive geometries, mutations etc.
Riemannian and aﬃne geometries are not parabolic geometries, but they are particular examples of
so-called reductive geometries, which form a subclass of Cartan geometries. It is worth reminding
some general features of these geometries.
3 Concrete geometries 19
Reductive geometries
Reductive geometry is a Cartan geometry of type (G, H) such that the corresponding Lie algebra
admits an H-invariant decomposition g = n ⊕ h (with respect to the adjoint action).4
This is a
very strong condition with many restrictive implications.
The Cartan connection ω : TG → g of a reductive Cartan geometry splits according to the
decomposition of g into well deﬁned H-equivariant pieces θ : TG → n and γ : TG → h. The solder
form θ provides an identiﬁcation TM ∼= G ×H n. If, in addition, the homomorphism H → GL(n)
induced by the adjoint action is (inﬁnitesimally) injective, then we get an identiﬁcation of G with
a (covering of) a subbundle of the ﬁrst order frame bundle of M, i.e., an ordinary G-structure
with structure group H. The form γ is the principal connection, which induces structure adapted
connections on all bundles associated to G. These induced connections coincide with the restriction
of the fundamental derivative (1.5), where the restriction (of the ﬁrst argument from A to TM)
indeed makes sense only in reductive cases (when TM is a subbundle of A).
The reductivity inﬂuences also the behavior of distinguished curves. Considering the curves
corresponding to n ⊂ g, it easily follows that all of them seem like ordinary geodesics: as unparametrized
curves, they are uniquely given by a tangent direction in one point and each of them
admits an aﬃne class of distinguished parametrizations.
In the discussion above, we met reductive geometries in section 2.2: any Weyl structure σ of
a parabolic geometry (G → M, ω) of type (G, P) induces a reductive Cartan geometry (G0 →
M, σ∗
(ω− ⊕ ω0)) of type (Pop
, G0), where Pop
⊂ G is the opposite parabolic subalgebra to P.
In particular, Pop
∩ P = G0 and the corresponding Lie algebra is pop
= g− ⊕ g0, which is a
G0-invariant decomposition. Two particular examples, which might be seen in this scheme, follow.
Aﬃne structures
An aﬃne structure on a smooth manifold M is given by an aﬃne connection, i.e., a linear connection
on TM, which brings the notion of parallel displacement of vectors along curves. The
model is the aﬃne space Rm
with the absolute parallelism given by the translations. This is the
homogeneous space of the group of aﬃne motions, the semidirect product of the abelian group of
translations and the general linear group, Aff(m, R) = Rm
GL(m, R). Under the identiﬁcation
Rm
X → ( 1
X ) ∈ Rm+1
, the action of this group corresponds to the standard action of
G =
1 0
Y A
: Y ∈ Rm
, A ∈ GL(m, R) ⊂ GL(m + 1, R). (3.1)
The stabilizer of the origin in Rm
, which corresponds to the ﬁrst vector from the standard basis
of Rm+1
, is H = GL(n, R), which is given by Y = 0 in the previous description.
The natural decomposition g = n ⊕ h, with n = Rm
and h = gl(m, R) in accord with the block
decomposition above, is H-invariant. Thus the Cartan geometry is reductive, but the corresponding
G-structure is the full frame bundle (no reduction) and the only underlying object is just the aﬃne
connection D. Since n ⊂ g is abelian, the torsion and the curvature of D corresponds to
T = dθ + [γ, θ] and R = dγ + [γ, γ],
respectively, where ω = θ ⊕ γ is the decomposition of the Cartan connection as above.
Riemannian structures
A Riemannian structure on a manifold M is given by a Riemannian metric g ∈ Γ(S2
T∗
M), which
brings the notion of distance and parallel displacement provided by the Levi-Civita connection,
the unique metric connection with vanishing torsion. The model is the Euclidean space, i.e., the
aﬃne space Rm
with the standard inner product. This is the homogeneous space of the group
of rigid motions, the semidirect product of translations and the orthogonal group preserving the
4Warning: this notion has nothing to do with the reductivity of the Lie algebra g in algebraic sense.
20 I Toolkit
inner product, Euc(m) = Rm
O(m), where H = O(m) is that stabilizer of the origin. According
to (3.1), the group Euc(m) ⊂ Aff(m, R) is described by restricting the values of A to O(m).
The current Cartan geometry is reductive, the corresponding G-structure is the reduction to
the orthonormal frame bundle and the Levi-Civita connection corresponds to the unique ‘normal’
Cartan connection: the normalization condition in this case is trivial (vanishing torsion) and the
fact that the ﬁrst prolongation of h = o(m) is also trivial pins down the connection uniquely.
A transition from positive deﬁnite metrics to indeﬁnite ones is reﬂected by a transition from
orthogonal structure groups O(m) to pseudo-orthogonal ones O(p, q), where p + q = m, with the
model space being the pseudo-Euclidean aﬃne space of signature (p, q).
Mutations
As we know from the very deﬁnition of Cartan geometry of type (G, H), the crucial role is played
by the structure group H and the Lie algebra g of the principal group G, whereas the group
itself enters mainly the model description, respectively extensions. In fact, in the deﬁnition of the
Cartan connection just the H-module structure of g is addressed, its Lie algebraic structure enters
the curvature.
In this sense, Riemannian structures may be viewed as Cartan geometries modeled over any
space form and not just the Euclidean space Rm ∼= Euc(m)/O(m). Indeed, the round sphere,
respectively the hyperbolic space, is the homogeneous space Sm ∼= O(m + 1)/O(m), respectively
Hm ∼= O(m, 1)/O(m), and euc(m, R), o(m + 1) and o(m, 1) are mutually isomorphic as O(m)modules,
but not as Lie algebras. With respect to their standard block realizations, the isomorphisms
o(m + 1) ← euc(m, R) → o(m + 1, 1) are given as
0 −kY t
kY A
←
0 0
Y A
→
0 lY t
lY A
,
where k, l ∈ R are arbitrary constants.
These are concrete examples of model mutations leading to a functorial construction described
in section 1.3. The curvature of a mutated Cartan geometry is controlled by the (constant) contribution
(1.8) so that (1.9) holds.
3.2 Projective geometry and Thomas cone
Projective geometries are one of the two exceptional types of parabolic geometries, for which the
underlying data described in section 2.1 are vacuous and the structure is rather given by a class
of aﬃne connections.
Projective structures
A projective structure on a manifold M is given by a class of torsion-free projectively equivalent
aﬃne connections, i.e., connections having the same unparametrized geodesics. The model is the
real projective space RPm
, the projectivization of Rm+1
, with projective lines as the distinguished
family of curves. This is the homogeneous space of the group of collineations, Proj(m, R), which
coincides with the projectivized general linear group G = PGL(m+1, R), the quotient of GL(m+
1, R) by its center. The stabilizer of the origin in RPm
, respectively the ﬁrst vector from the
standard basis of Rm+1
, is the subgroup P ⊂ G represented by block upper-triangular matrices
according to the block decomposition indicated in (3.1).
The corresponding Lie algebras are simple g = sl(m + 1, R) and parabolic p = g0 ⊕ g1 with the
corresponding grading indicated in the block form as
g =
g0 g1
g−1 g0
,
with blocks of sizes 1 and m. In particular, g is |1|-graded and we identify g−1
∼= Rm
, g0
∼= gl(m, R)
and g1
∼= Rm∗
. The semisimple part of g0 is g0
∼= sl(m, R).
3 Concrete geometries 21
The underlying structure of the corresponding Cartan geometry consists just of family of Weyl
connections (there is no reduction of the principal frame bundle since the Levi factor G0 coincides
with GL(m, R)). The general formula (2.8) for the change of Weyl connection according to a 1-form
Υ expands in this case to
ˆDξη = Dξη + Υ(ξ)η + Υ(η)ξ,
for ξ, η ∈ Γ(TM), which indeed recovers the well known description of projectively equivalent
connections. Moreover all these connections have the same torsion, which vanishes in the normal
case. The ﬁrst (and only) non-trivial harmonic curvature component is the Cotton–York tensor,
for m = 2, respectively the Weyl tensor, for m > 2. These are the obstructions to the ﬂatness of
the projective structure.
Thomas cone
Another choice of principal Lie group to the Lie algebra g = sl(m+1, R) clearly is G = SL(m+1, R).
This group acts transitively not only on lines, but also on rays in Rm+1
. Hence both the projective
space RPm
and the sphere Sm
, its 2-fold cover, can be considered as the homogeneous spaces of
SL(m+1, R) (only the action on RPm
is not eﬀective). In the latter case, the family of distinguished
curves is formed by great circles. The corresponding Cartan geometry modeled on Sn
deﬁnes the
oriented projective structures, i.e., projective structures on oriented manifolds.
Great circles on Sm
are geodesics of the standard round metric, whose Levi-Civita connection
deﬁnes the projective class of connections. Among these connections, there is a subclass of special
ones, each of which preserves a volume form (these are the exact Weyl connections in the terminology
of section 2.2). Volume forms on Sm
correspond to sections of the canonical R+-bundle
q : Rm+1
\ {0} → Sm
. It is an inspiring observation that any special connection on Sm
can be described
as a ‘pull-back’ of the ﬂat ambient connection on Rm+1
with respect to the corresponding
section of q. This is a (simpliﬁed) model interpretation of the cone, respectively ambient, description
of projective structures, which allows a straight generalization. Before doing so, let us denote
P ⊂ G the stabilizer of a ray in Rm+1
and Q ⊂ P the stabilizer of its generating vector. Thus,
Sm ∼= G/P, Rm+1
\ {0} ∼= G/Q and P/Q ∼= R+.
Let (M, [D]) be an oriented projective structure, let (G → M, ω) be the corresponding normal
Cartan geometry of type (G, P) and let M := G/Q be the correspondence space given by the
inclusion Q ⊂ P. As usually, the tangent bundle is identiﬁed as TM ∼= G ×Q (g/q), where q is the
Lie algebra of Q. As unusually, the Q-module g/q is isomorphic to the restriction of the standard
representation of G = SL(m + 1, R) on Rm+1
. Hence TM is a tractor bundle over M and as such
it carries an induced linear connection , which is called the Thomas or ambient connection. This
connection has many characteristic properties: easily, it is torsion-free and preserves the canonical
volume form (given by the standard G-invariant volume on Rm+1
). It is also compatible with the
principal R+-action on M and any section of the projection q : M → M determines a special
connection from the projective class [D]. Finally, it is Ricci ﬂat, which is a consequence of the
normality of ω.
(G, ω)
99

(M, )
ww
Thomas
(M, [D]) projective
It follows that the initial projective structure on M is fully encoded in the Thomas ambient
connection on M. Concrete coordinate expressions and direct relations of corresponding curvature
quantities can be found in Thomas’ original work, cf. [64, chapter III]. Let us notice that this is
the place where the so-called Thomas projective parameters, the projectively invariant bunch of
22 I Toolkit
functions with suspicious transformation laws, appear. For later reference, they are given by
Π C
A B := Γ C
A B −
2
m + 1
δC
(AΓ D
B) D , (3.2)
where Γ C
A B are the Christoﬀel symbols of any connection from the projective class.
Relation to standard tractors
Since g/q is identiﬁed with the standard representation of the principal group G, there is a close
relation between the tangent bundle TM → M with its ambient connection and the standard
tractor bundle T = G ×P Rm+1
→ M with its tractor connection . It follows that the latter data
can be obtained by factorizing the former ones with respect to the R+-action (which naturally
extends to TM). In particular, sections of the standard tractor bundle T are in bijective correspondence
with vector ﬁelds on M which are of homogeneity −1 with respect to the R+-action
and the normality of is equivalent to the Ricci ﬂatness of .
The standard tractor bundle is ﬁltered, T ⊃ T 0
, so that T 0
is the tautological line bundle,
which is conventionally labeled as E(−1), the density bundle of projective weight −1. Under the
previous identiﬁcation, T 0
corresponds to the vertical subbundle of the cone projection M → M.
With respect to a choice of Weyl structure, the standard tractor bundle splits as
T = TM(−1) ⊕ E(−1),
where TM(−1) = TM ⊗ E(−1) corresponds to the projecting part T /T 0
.
There are many variants of cone constructions, but we do not know any direct analogy to the
previous one among |1|-graded parabolic geometries. Very similar approach, however, applies to
projective contact structures, see [38]. Note that the ambient construction for conformal structures
(discussed in sections 3.4 and 5.3) is signiﬁcantly more complicated, though the model observations
are of similar nature.
3.3 Flag varieties and correspondence spaces
Projective, Grassmannian and Lagrangean contact manifolds are particular examples of parabolic
geometries whose homogeneous models are ﬂag varieties, i.e., algebraic varieties of linear subspaces
of particular type in a real vector space. In fact, all parabolic homogeneous spaces G/P can
be realized (after a possible complexiﬁcation) this way, which is why they are sometimes called
generalized ﬂag varieties. Previously mentioned examples correspond to the most ordinary ﬂags,
with simple Lie algebras g = sl(−, R) in the background. In this context we just add some details.
Real projective space RPm
is the space of 1-dimensional subspaces in vector space Rm+1
. It
is the homogeneous space of G = PGL(m + 1, R), respectively SL(m + 1, R), with P ⊂ G the
stabilizer of a line. This is an example of Grassmannian, respectively ﬂag, variety of type (1, m+1).
General Grassmannian of type (p, N) is the variety of p-dimensional subspaces in vector space RN
.
General ﬂag variety of type (p1, p2, . . . , N) is the variety of ﬂags of subspaces V 1
⊂ V 2
⊂ · · · ⊂ RN
of dimensions p1 < p2 < · · · < N. It is a homogeneous spaces with G as above and P ⊂ G the
stabilizer of a particular ﬂag.
Mapping any ﬂag V 1
⊂ V 2
⊂ · · · ⊂ RN
to the subspace of a particular dimension (say V i
)
yields a ﬁbration of ﬂag manifolds (whose ﬁber over V i
consists of all ﬂags of the current type
containing V i
). As all these manifolds are homogeneous spaces of the same Lie group, the previous
ﬁbrations are model examples of correspondences discussed in section 1.3. In particular, for ﬂags
of depths two, we have two such ﬁbrations:
3 Concrete geometries 23
G = PGL(N, R)

ÖÖ $$
Flags (p1, p2, N)
ww 99
Flags (p1, N) Flags (p2, N)
Lagrangean contact structures
Let us consider parabolic geometries modelled on ﬂag varieties of lines in hyperplanes, i.e., of type
(1, N − 1, N). Counting dimensions, one observes that the base manifold is odd dimensional, say
m = 2n + 1, in which case N = n + 2. Let P ⊂ G be the stabilizer of the ﬂag determined by
the ﬁrst and the ﬁrst n − 1 vectors from the standard basis. The corresponding grading of the Lie
algebra g = sl(n + 2, R) is indicated in the following block form
g =


g0 gL
1 g2
gL
−1 g0 gR
1
g−2 gR
−1 g0


with blocks of sizes 1, n and 1. In particular, g is |2|-graded with g−2
∼= R, gL
−1
∼= Rn
, gR
−1
∼= Rn∗
etc. The decomposition g−1 = gL
−1 ⊕ gR
−1 is g0-invariant, the Lie bracket g−1 × g−1 → g−2 is nondegenerate
and, with respect to this map, the subspaces gL
−1 and gR
−1 are isotropic (aka Lagrangean,
if we interpret the bracket as a symplectic form on g−1). In particular, we have a contact grading
of g.
Lagrangean contact structure is the structure underlying the current parabolic geometry. On
a manifold M of dimension m = 2n + 1 it consists of a contact distribution H ⊂ TM and a
decomposition H = L ⊕ R into two Lagrangean subspaces with respect to the Levi form L :
H × H → TM/H, the only non-trivial part of the Levi bracket (2.1).
Normal parabolic geometries of this type are always regular, the harmonic curvature decomposes
according to the dimension of M as follows. For m = 3, there are two components, both
of homogeneity 4; in particular, the geometry is torsion-free. An interpretation of these object in
underlying terms yields two tensors of Cotton–York type, namely,
CL ∈ Γ(L∗
∧ (TM/H)∗
⊗ L∗
), CR ∈ Γ(R∗
∧ (TM/H)∗
⊗ R∗
). (3.3)
For m > 3, there are two components of homogeneity 1 and one component of homogeneity 2; the
corresponding underlying objects are two torsions and one curvature of Weyl type,
TL ∈ Γ(L∗
∧ L∗
⊗ R), TR ∈ Γ(R∗
∧ R∗
⊗ L), W ∈ Γ(L∗
∧ R∗
⊗ End(H)). (3.4)
It follows that vanishing of TL and TR is equivalent to the integrability of the distribution L and
R, respectively.
Projective to Lagrangean correspondence
The model correspondence of ﬂag varieties described above reduces in the current setting to
Flags (1, n + 1, n + 2)
vv AA
Flags (1, n + 2) ∼= RPn+1
Flags (n + 1, n + 2) ∼= RP(n+1)∗
In this picture it is easy to identify the top manifold, which is the model Lagrangean contact
manifold of dimension m = 2n + 1, with P(T∗
RPn+1
), the projectivized cotangent bundle of real
24 I Toolkit
projective space of dimension n + 1, so that its canonical contact distribution coincides with the
one spanned by the vertical subbundles of the two projections. Anyway, this is the space of contact
elements of hyperplane type in RPn+1
. We come back to these interpretations in sections 3.5 and
4.3.
According to the general principles from section 1.3, this correspondence translates immediately
to curved cases. Let (G → X, ω) be the normal parabolic geometry associated to a projective
structure on X. For the inclusion of structure groups (the ﬂag stabilizers) as above, the correspondence
space of X is identiﬁed with its projectivized cotangent bundle M := P(T∗
X) so that
the vertical subbundle of the projection M → X is one of the two Lagrangean subspaces of the
contact distribution H ⊂ TM, say R.
(G, ω)

ØØ
(M, L ⊕ R)
ww
Lagrangean
(X, [D]) projective
From the description of the Lagrangean harmonic curvature components, and the fact that the
vertical subbundle is identiﬁed with R, we see that:
• for dim X = 2, i.e., dim M = 3, the induced structure on M satisﬁes CR = 0,
• for dim X > 2, i.e., dim M > 3, the induced structure on M satisﬁes TR = W = 0.
In particular, in the latter case, the induced Lagrangean contact structure is torsion-free if and
only if it is ﬂat, which means the initial projective structure is ﬂat. Anyway, due to the key result
materialized in (2.10), the previous equalities guarantee that, locally, the Lagrangean contact
structure is induced by a projective structure.
Note that a discussion concerning the second ﬁbration of the correspondence above is quite
parallel because of the very symmetry of the whole picture.
Path geometries
Let us consider parabolic geometries modelled on ﬂag varieties of lines in planes, i.e., of type
(1, 2, N). Counting dimensions, one observes that the base manifold is odd dimensional, say m =
2n + 1, in which case N = n + 2. Note that, for n = 1, we just recover the Lagrangean contact
case in the lowest dimension, hence we assume n > 1 hereafter.
Let P ⊂ G be the stabilizer of the ﬂag determined by the ﬁrst and the ﬁrst two vectors from
the standard basis. The corresponding grading of the Lie algebra g = sl(n + 2, R) is indicated in
the following block form
g =


g0 gE
1 g2
gE
−1 g0 gV
1
g−2 gV
−1 g0


with blocks of sizes 1, 1 and n. In particular, g is |2|-graded with g−2
∼= Rn
, gE
−1
∼= R, gV
−1
∼= Rn
etc. The decomposition g−1 = gE
−1 ⊕gV
−1 is g0-invariant, the Lie bracket g−1 ×g−1 → g−2 is trivial
on gV
−1 × gV
−1 and induces a linear isomorphism gE
−1 ⊗ gV
−1 → g−2. Since the bracket is also trivial
on gE
−1 × gE
−1 (because of dim gE
−1 = 1), it is fully determined by the previous two bits.
Generalized path geometry is the structure underlying a regular Cartan geometry of the current
type. On a manifold M of dimension m = 2n + 1 it consists of two transversal subbundles E, V ⊂
TM such that, for H := E ⊕ V , the ﬁltration H ⊂ TM is compatible with the vector ﬁeld
bracket and the corresponding Levi bracket (2.1) vanishes on V × V and induces an isomorphism
E ⊗ V → TM/H.
3 Concrete geometries 25
The terminology abstracts a natural presence of such structures on the projectivized tangent
bundle of a manifold, Y , endowed with a system of paths, i.e., unparametrized curves, with the
property that through each point in each direction there passes exactly one path from the system.
The tangent directional ﬁelds of paths from the system foliate M := P(TY ) and tangent spaces
of this foliation give a line subbundle E ⊂ TM. This E is obviously contained in the tautological
subbundle H ⊂ TM and complementary to the vertical subbundle V ⊂ H of the projection
M → Y , which are there by nature. The decomposition H = E ⊕ V satisﬁes the above declared
properties for generalized path geometries, hence system of paths on Y determines this fancy
geometric structure on M. This way, geometric properties of systems of second order ODEs,
modulo point transformations, may be studied.
In particular, geodesics of an aﬃne connection on Y , respectively its projectively equivalent
class, give rise to a (generalized) path geometry, which has to be somehow special. Before its full
characterization, we need to learn about the harmonic curvatures.
Generally, i.e., for any n ≥ 2, there are three harmonic curvature components, two of which are
independent of n: they are of homogeneity 2 and 3 and they correspond to torsion and curvature
of Weyl type, namely,
TE ∈ Γ(E∗
∧ (TM/H)∗
⊗ V ), W ∈ Γ(V ∗
∧ (TM/H)∗
⊗ End(V )).
The third component is a torsion whose type depends on n: for n = 2, respectively n > 2, it is of
homogeneity 1, respectively 0,
TV ∈ Γ(V ∗
∧ V ∗
⊗ E), respectively TV ∈ Γ(V ∗
∧ V ∗
⊗ TM/H). (3.5)
Note that, for generalized path geometries in dimension m = 2n + 1 > 5, the bad torsion TV
vanishes by the assumption of regularity.
Projective to path geometry correspondence
The model correspondence in the current setting is
Flags (1, 2, n + 2)
ww 77
Flags (1, n + 2) ∼= RPn+1
Flags (2, n + 2)
The bottom right object is the Grassmannian of planes in Rn+2
, whose dimension is 2n. The
top object, which is the model of generalized path geometry of dimension m = 2n + 1, is indeed
identiﬁed with P(TRPn+1
) so that its tautological subbundle coincides with the one spanned by
the vertical subbundles of the two projections. The curved version of this correspondence is already
outlined in the previous paragraph: unparametrized geodesics of a projective structure on Y form
a system of paths, which determines E, and hence a generalized path geometry on M := P(TY ).
This is covered by the normal parabolic geometry (G → Y, ω) so that we have
(G, ω)

ØØ
(M, E ⊕ V )
ww
path
(Y, [D]) projective
From the description of harmonic curvature components, and the fact that the vertical subbundle
is identiﬁed with V , we see that:
26 I Toolkit
• the induced structure on M satisﬁes TV = W = 0.
In particular, the induced generalized path geometry is torsion-free if and only if it is ﬂat, which
means the initial projective structure is ﬂat. Due to the harmonic curvature characterization
of correspondence spaces, the previous equalities guarantee that, locally, the generalized path
geometry is induced by a projective structure or, in other words, the paths on the local leaf space
are unparametrized geodesics of an aﬃne connection.
For dim M > 5, the condition TV = 0 is satisﬁed by regularity, in which cases M is always
identiﬁed with the projectivized tangent bundle of a local leaf space. Hence only W = 0 is relevant
in these dimensions.
Grassmannian to path geometry correspondence
In a similar manner, one may discuss the correspondence concerning the second ﬁbration from the
picture above, which relates generalized path geometries and almost Grassmannian structures. An
almost Grassmannian structure on manifold Z is given by an isomorphism TZ ∼= K ⊗ L of the
tangent bundle with the tensor product of auxiliary vector bundles K and L of ranks 2 and n,
respectively. Starting with the normal parabolic geometry (G → Z, ω) to the almost Grassmannian
structure and the inclusion of parabolics given by the model description, the correspondence space
M is identiﬁed with P(K), the projectivization of the smaller auxiliary bundle over Z.
(G, ω)

%%
path (M, E ⊕ V )
@@
Grassmanian (Z, K ⊗ L)
Almost Grassmannian structures of the current type are speciﬁc for n = 2, in particular, they
are torsion-free, i.e., integrable. For n > 2, it follows that the torsion of the almost Grassmannian
structure ﬁts the torsion (3.5) of the induced Cartan geometry. Thus, such structure gives rise to
a regular geometry, i.e., to a generalized path geometry, just in the integrable case. Anyhow,
• the induced structure on M satisﬁes TE = 0.
Conversely, this condition for generalized path geometries allows to descend to a local leaf space,
interpreted as the space of paths, with an (integrable) Grassmannian structure. Such path geometries,
respectively systems of ODEs, are of particular importance as they generically allow to
construct enough ﬁrst integrals (from the Cartan curvature) to describe the paths, respectively to
solve the system, see [40].
Remarks
In the previous treatment, we observed speciﬁcities in two particular dimensions. At ﬁrst, for n = 1,
the 3-dimensional Lagrangean contact structures coincide with 3-dimensional generalized path
geometries as both are given by two complementary 1-dimensional subbundles in a 2-dimensional
non-integrable distribution H ⊂ TM. Let us comment some aspects of one of correspondences
above. Let us consider a second order ODE,
y = Q(x, y, y ), (3.6)
whose graphs of solutions are considered as the system of paths in the plane, Y , with local coordinates
(x, y). Let (x, y, p = y ) be the local coordinates of the space of contact elements M := P(TY )
(which coincides canonically with P(T∗
Y ) in this dimension). The geometry of the ODE is fully
3 Concrete geometries 27
encoded in the generalized path geometry on M, whose determining subbundles E and V are given
by
dp − Q dx = 0, and dy − p dx = 0,
respectively. The two fundamental invariants, which correspond to the harmonic curvatures from
(3.3), can be expressed in terms of Q and its partial derivatives with respect to y and p. These
invariants were known and used ages before their current interpretations by Lie, Tresse and others.
The one, whose vanishing guarantees that (3.6) is equivalent to a geodesic equation, turns out to
be Qpppp. In such case, the equation has the form
y = A0 + A1y + A2(y )2
+ A3(y )3
, (3.7)
where Ai are functions only of x and y. This equation, and the associated projective Cartan
connection, appears in Catran’s projective papers, see, e.g., [22, chapter V]. We come back to
these issues in section 5.1.
Another speciﬁc behaviour appears for generalized path geometries in dimension ﬁve, i.e., for
n = 2. In this case, the related Grassmannian structure on Z is equivalent to a conformal structure
of split signature so that its two harmonic curvatures correspond to self-dual and anti-self-dual
part of the conformal Weyl tensor. In particular, the model correspondence between 3-dimensional
projective space and 4-dimensional conformal quadric (via 5-dimensional ﬂag variety) reveals the
classical Klein–Pl¨ucker correspondence. Note also that this picture (but complexiﬁed and primarily
read backwards) is the starting position of the far reaching Penrose twistor program, see, e.g., the
survey article [1].
3.4 Conformal geometry and ambient metric
The conformal class of the ﬂat metric on the Euclidean space Rm
deﬁnes a ﬂat conformal structure.
But this is not an accurate conformal model since, e.g., the inversions does not act globally on
Rm
. One rather takes the standard sphere Sm
⊂ Rm+1
, the compactiﬁcation of Rm
with the
stereographic projection, as the right model. This is actually ﬁne, but to describe satisfactorily
the group of (global) conformal automorphisms, one would further pass to the ‘celestical’ sphere
in a bigger Minkowski space Rm+1,1
, i.e., to the space of its null rays. These two spheres can,
of course, be identiﬁed, but not canonically. This leads to the notion of scales etc. The current
viewpoint generalizes straightly to an arbitrary signature.
Conformal structures
Conformal structure of signature (p, q) on a manifold M of dimension m = p+q is given by a class
of conformally equivalent pseudo-Riemannian metrics of this signature, i.e., a class of metrics that
diﬀer just by multiples of everywhere positive functions. The conformal class of metrics is viewed
as a R+-bundle over M, which is a subbundle on S2
T∗
M.
The model structure arises as follows. Let Rp+1,q+1
be a vector space with an inner product of
signature (p+1, q+1) and let N ⊂ Rp+1,q+1
be the cone of non-zero null vectors. The space of rays
in N, i.e., the oriented projectivization of N, is called the M¨obius sphere and denoted Sp,q
. Hence
we have the projection q : N → Sp,q
= N/R+. The M¨obius sphere carries a natural conformal
structure of signature (p, q), whose each representative metric is given by pulling back the induced
metric from the image of a smooth section of the projection q. This conformal structure is obviously
invariant under the action of simple Lie group G = O(p+1, q+1), which acts transitively both on N
and Sp,q
. Let Q ⊂ P ⊂ G be the nested subgroups stabilizing a null vector and the corresponding
ray, respectively. Thus, N ∼= G/Q, Sp,q ∼= G/P and P/Q ∼= R+. The subgroup P ⊂ G is parabolic,
the so-called Poincar´e group.
For concrete matrix realizations we need some choices. Let the inner product on Rp+1,q+1
be
given in the standard basis by the Gram matrix


0 0 1
0 Ep,q 0
1 0 0

 with Ep,q =
Ep 0
0 −Eq
, (3.8)
28 I Toolkit
where Ei is the rank i identity matrix, and let P be the stabilizer of the ﬁrst vector form the basis.
Then the corresponding grading of the Lie algebra g = o(p + 1, q + 1) is indicated as
g =


g0 g1 0
g−1 g0 g1
0 g−1 g0

 ,
with blocks of sizes 1, n and 1. In particular, g is |1|-graded with g−1
∼= Rm
, g0
∼= R ⊕ o(p, q)
and g1
∼= Rm∗
(of course, the blocks that are symmetric with respect to the anti-diagonal are
determined by each other).
Underlying data of the parabolic geometry of the current type are G-structures with structure
group CO(m) = R+ ×O(p, q), the conformal orthogonal group. The Weyl connections are the ones
which preserve the conformal class of metrics, Levi-Civita connections of representative metrics
are the exact ones. The general formula (2.8) describing the change of Weyl connections according
to parametrizing 1-forms Υ expands in this case to
ˆDξη = Dξη + Υ(ξ)η + Υ(η)ξ − g(ξ, η)Υ , (3.9)
for ξ, η ∈ Γ(TM), where g is any metric from the conformal class (and Υ ∈ Γ(TM) is the image of
Υ ∈ Γ(T∗
M) under the corresponding isomorphism T∗
M → TM). Moreover all these connections
have the same torsion, which vanishes in the normal case. The ﬁrst non-trivial harmonic curvature
component is the Cotton–York tensor, for m = 3, respectively the Weyl tensor, for m > 3 (which
splits into two components, for m = 4). These are the obstructions to the ﬂatness of the conformal
structure.
Note that diﬀerent choices of the principal Lie group to the Lie algebra g = o(p + 1, q + 1) lead
to variations on the current structure, some of which are subjects of later interests. Our choice
here is motivated by similar reasons as in section 3.2.
Note also that the full projectivization of the null cone N (rather than the oriented one) yields
a hyperquadric in RPn+1
as the conformal homogeneous model. In order to distinguish these two
cases, one speaks about the M¨obius space rather that the M¨obius sphere (which is its 2-fold cover).
The group O(p + 1, q + 1) does not act eﬀectively on the M¨obius space and it may be substituted
by PO(p + 1, q + 1), the quotient by center (which is Z2).
Feﬀerman–Graham ambient metric
The model interpretations generalize easily so that the cone of conformal metrics N ⊂ S2
T∗
M,
which forms a R+-bundle over M, is the correspondence space induced by the inclusion Q ⊂ P.
But, since the Q-action on g/q is not a restriction of a G-action, there are no easily induced
ambient objects as in section 3.2. Still, the ambient description of the model conformal structure
can be mimicked also in the general setting. Following the approach of [33] by Feﬀerman and
Graham, one extends the natural R+-bundle N → M by one more dimension and constructs an
ambient metric g on this bigger manifold M so that it determines the individual metrics on M
from the conformal class in the very same way as above. For this task one actually needs to control
the behaviour along the cone N ⊂ M. Besides the obvious requirement that g is homogeneous of
degree 2 with respect to the R+-action, one imposes its Ricci ﬂatness as a natural normalization
condition. The procedure is quite subtle and typically describes the ambient metric only in terms
of power series expansion.
To proceed, let I ⊂ R be an interval containing 0, let M = N × I be an ambient space and let
t, (xi
) and ρ be local coordinates on R+, M and I, respectively. To describe the ambient metric,
one may start with the prescription
g = t2
gij(x, ρ) dxi
dxj
+ 2ρ dt dt + 2t dt dρ, (3.10)
where gij(x, 0) corresponds to a representative metric in the conformal class [g] on M. Now,
iteratively, one determines gij(x, ρ) as a Taylor series in ρ so that the Ricci ﬂatness condition is
3 Concrete geometries 29
satisﬁed along N, i.e., for ρ = 0. Necessarily, the ﬁrst order expansion is
gij(x, ρ) = gij(x) + 2Pij(x)ρ + O(ρ2
), (3.11)
where P is the Schouten tensor of the representative metric g on M.
It turns out that, for general conformal structures in odd dimension, there is a (essentially
unique) solution of the problem. But, in even dimension, the construction is obstructed at a ﬁnite
order by the so-called Feﬀerman–Graham tensor, a conformally invariant tensor (which coincides
with the Bach tensor in dimension 4). There are not too many examples of conformal structures
for which the ambient metric is Ricci ﬂat globally and not just asymptotically along N. We are
going to extend this rare family by a new class of members in section 5.3.
Conformal standard tractors and relation to ambient data
The conformal standard tractor bundle T is the tractor bundle corresponding to the standard
representation Rp+1,q+1
of the principal group G = O(p + 1, q + 1). The G-invariant inner product
on Rp+1,q+1
gives rise to a bundle metric that is preserved by the tractor connection . The
standard tractor bundle is ﬁltered, T ⊃ T 0
⊃ T 1
, so that the line bundle T 1
is identiﬁed with
E[−1], the density bundle of conformal weight −1, and T 0
is its ortho-complement with respect to
the bundle metric. With respect to a choice of Weyl structure, the standard tractor bundle splits
as
T = E[1] ⊕ TM[−1] ⊕ E[−1], (3.12)
where TM[−1] = TM ⊗ E[−1] and the projecting part is E[1] ∼= T /T 0
. General formulas (2.5),
(2.7) etc. expand in this case to well known expressions that we employ repeatedly below.
Possible relation of the standard tractor bundle T with its bundle metric (parallel with respect
to the tractor connection) and the ambient space M with the ambient metric (parallel with respect
to its Levi-Civita connection) is not clear at all. In particular, the previous description of the
general ambient space might seem slightly artiﬁcial. However, this can be sorted out so that the
similar factorization as in section 3.2 is possible provided that we restrict to the cone N ⊂ M.
This is proved by ˇCap and Gover in [12], where even much weaker normalization condition on the
ambient metric is considered.
TM

i∗
TM

tt
9 ×
RR
T

(M, g) ambient
N
tt
9 ×
i RR
(M, [g]) conformal
In particular, sections of the standard tractor bundle T → M are in bijective correspondence
with ambient vector ﬁelds along N (more precisely, sections of i∗
TM → N) which are of homogeneity
−1 with respect to the R+-action. Under this correspondence, the line subbundle T 1
⊂ T
corresponds to the vertical subbundle of the cone projection N → M.
3.5 CR contact geometry and Feﬀerman space
Prototypical CR structures are induced on real submanifolds in complex spaces. Submanifolds of
the same real codimension need not be mutually equivalent under biholomorphic transformations,
hence their local geometry may be studied and invariants constructed. This was shown already
30 I Toolkit
by Poincar´e in [55] for real hypersurfaces in C2
, which are the simplest examples of CR contact
structures. Popular and the most symmetric ones are the hyperquadrics given by
|z1|2
+ |z2|2
= 1 and Im z2 = |z1|2
, (3.13)
(z1, z2) ∈ C2
, which are the boundaries of the unit ball and the Siegel domain, respectively.
It follows that they are globally biholomorphically equivalent up to one point under a complex
projective map. Thus, they may be seen as diﬀerent aﬃne realizations of the same hyperquadric
in CP2
, i.e., the complex projectivization of the null cone in C3
with respect to a Hermitean inner
product of signature (2, 1). This is the proper homogeneous model of 3-dimensional CR contact
structures.
Almost CR contact structures
Generalizing according to dimension and signature, let Cp+1,q+1
be a complex vector space with a
Hermitean inner product of signature (p + 1, q + 1) and let N ⊂ Cp+1,q+1
be the cone of non-zero
null vectors. The complex projectivization of N yields a real hyperquadric Cp,q
⊂ CPn+1
, where
n = p + q, which is the model CR contact structure of signature (p, q). It is a homogeneous space
of the simple Lie group G = SU(p + 1, q + 1), respectively PSU(p + 1, q + 1), and the stabilizer
P ⊂ G of a point of Cp,q
, i.e., a complex line in N, is a parabolic subgroup. Thus, Cp,q ∼= G/P.
Let the Hermitean inner product on Cp+1,q+1
be represented by the same matrix as in (3.8)
and let P be the stabilizer of the ﬁrst vector from the standard basis. Then the corresponding
grading of the Lie algebra g = su(p + 1, q + 1) has the form
g =


g0 g1 g2
g−1 g0 g1
g−2 g−1 g0

 ,
with blocks of sizes 1, n and 1. In particular, g is |2|-graded with g−2
∼= R, g−1
∼= Cp,q
, g0
∼=
C ⊕ u(p, q) etc. (of course, the blocks that are symmetric with respect to the anti-diagonal are
determined by each other). The Lie bracket g−1 ×g−1 → g−2 is (a multiple) of the imaginary part
of the standard Hermitean product on g−1, in particular, it is non-degenerate and of type (1, 1)
with respect to the standard complex structure.
The underlying geometric structure of the current parabolic geometry on a manifold M (of
dimension m = 2n + 1) consists of a contact distribution H ⊂ TM with an almost complex
structure J on H such that the Levi form L : H × H → TM/H is of type (1, 1) with respect
to J. Thus L is the imaginary part of a non-degenerate Hermitean form on H whose signature,
(p, q) with p + q = n, is the signature of the structure. In other words, the signature in encoded
in the map L(−, J(−)). Such structures are known as non-degenerate partially integrable almost
CR structures of hypersurface type or, shortly, almost CR contact structures. The structure is
integrable if the almost complex structure J is integrable, i.e., the corresponding Nijenhuis tensor,
which is the map H × H → H of type (0, 2), vanishes.
The notion of partial integrability refers to the compatibility of the Levi form L with the almost
complex structure J. Both this and the stronger condition of integrability can be equivalently
stated in terms of complexiﬁcations: for H1,0
⊕H0,1
being the decomposition of H ⊗C ⊂ TM ⊗C
into the holomorphic and anti-holomorphic part, the almost CR structure is partially integrable
if and only if [H0,1
, H0,1
] ⊂ H ⊗ C, the structure is integrable if and only if [H0,1
, H0,1
] ⊂ H0,1
.
In this view one also relates Lagrangean contact structures and almost CR contact structures as
two real forms of a complex parabolic contact structure: indeed, sl(n + 2, R) and su(p + 1, q + 1),
with p + q = n, are distinct real forms of complex Lie algebra sl(n + 2, C).
This remark concerns also the interpretation of the corresponding cohomologies and harmonic
curvatures: the two components in the Lagrangean case that are labeled CL and CR in (3.3),
respectively TL and TR in (3.4), correspond to only one component in the almost CR case. In
particular, for n = 1 (i.e., dim M = 3), there is only one harmonic curvature of Cotton–York type,
for n > 1, there is one torsion and one curvature of Weyl type (that all are somehow compatible
3 Concrete geometries 31
with the almost complex structure J). It follows that the torsion part coincides with the Nijenhuis
tensor of J up to a multiple, hence its vanishing is equivalent to the integrability of the almost
CR structure.
Feﬀerman space
Feﬀerman’s original construction in [32] concerns hypersurfaces in complex spaces, the boundaries
of strictly pseudoconvex domains D ⊂ Cn+1
. It associates a natural Lorentzian metric on ∂D ×S1
so that its conformal class is invariant under biholomorphic transformations of D, respectively CR
automorphisms of ∂D. This way, the conformal geometry is used to tackle many problems of the
original complex, respectively CR, nature. The construction was later generalized for abstract CR
contact structures by several authors until it was realized it may be considered for almost CR contact
structures in any signature and, moreover, it allows a neat Cartan geometric reinterpretation,
see [13].
As usual, one may start with the homogeneous model and then consider generalizations. With
the usual identiﬁcation of the multiplicative group C×
with S1
×R+, we may factorize the projection
N → Cp,q
= N/C×
above via N/R+, which gives a S1
-ﬁbration over the model CR quadric
Cp,q
.
N

99
N/R+ = S∗,∗
ww
N/C×
= C∗,∗
To interpret this picture, we just forget about the complex structure on Cp+1,q+1
and consider
N as the cone of non-zero null vectors in R2p+2,2q+2
with respect to the inner product given by
the real part of the Hermitean inner product on Cp+1,q+1
. Hence, N/R+ is the M¨obius sphere
S2p+1,2q+1
with its ﬂat (oriented) conformal structure as the Feﬀerman conformal structure of the
model CR quadric. The inner product on R2p+2,2q+2
is by its deﬁnition G-invariant, which yields
an embedding of Lie groups
φ : G = SU(p + 1, q + 1) → SO(2p + 2, 2q + 2) = G. (3.14)
As a homogeneous space, S2p+1,2q+1 ∼= G/P, where P ⊂ G is the stabilizer of a ray in N ⊂
R2p+2,2q+2
. Since also the group G acts transitively on N, we may identify S2p+1,2q+1 ∼= G/Q,
where Q ⊂ G is the preimage of P ⊂ G with respect to φ; roughly we write Q = G∩P. Obviously,
Q ⊂ P, where P ⊂ G is the stabilizer of the complex line containing the ray stabilized by Q.
But this is exactly the setting of section 1.3: the M¨obius sphere is the correspondence space of
the model CR quadric and the Cartan connection encoding the (ﬂat) conformal structure on the
former space can be seen as the equivariant extension of the Maurer–Cartan form on G encoding
the (ﬂat) CR structure on the latter space.
(G, ω)

(G, ω)
@Ø
φ
SS
AA

S∗,∗ ∼= G/Q ∼= G/P
uu
conformal
C∗,∗ ∼= G/P CR contact
32 I Toolkit
In other words, the G-invariant conformal geometry on the model Feﬀerman space is controlled
by the pair of maps i = φ|Q : Q → P and α = φ : g → ˜g via the corresponding extension
functor. The natural—and for applications also important—question is how this functor behaves
in generality, in particular, when it is compatible with the normality condition. In contrast to
previous correspondence space constructions (where there were no extension), there is no general
answer and it demands a careful analysis of curvature interrelations given by (1.9). The only
bonus in the current situation is that there is no contribution of Ψα, since α is a Lie algebra
homomorphism. It is shown in [13] that the current construction preserves the normality condition
just for integrable CR contact structures. More precisely, for (G → M, ω) being the normal Cartan
geometry describing an almost CR contact structure and (G → M, ω) being the Cartan geometry
with an underlying conformal structure constructed via the extension given by the pair (i, α) as
above,
• the Cartan connection ω is normal if and only if ω is torsion-free.
Remarks
For the description of the model Feﬀerman construction, it is sometimes useful to switch the point
of view: instead of considering a complex vector space with a Hermitean inner product and then
forgetting the complex structure, one may start with a real vector space with a pseudo-Euclidean
inner product and then imposing a compatible complex structure. The compatibility means that
the complex structure is skew with respect to the inner product. This allows to reconstruct the
Hermitean inner product which is behind the embedding (3.14). In particular, elements from
its image commute with the complex structure. Note that this embedding admits a lift to the
corresponding Spin group.
The previous factorization of the complex projectivization N → Cp,q
= N/C×
via N/R+ led
to the model interpretation of the Feﬀerman construction. Naturally, one may ask what interesting
issues appear when factorization via N/S1
. It easily follows that the ﬁbration N/S1
→ Cp,q
is
just a scale bundle that can be directly identiﬁed with the bundle of (positive) contact forms.
As we already noticed above, Lagrangean and almost CR contact structure correspond to different
real forms of the contact grading of sl(n + 2, C). In fact, any complex simple Lie algebra
allows a unique (up to inner automorphism) contact grading. Up to several exceptions (including
mainly the compact forms), real simple Lie algebras also admit unique contact gradings; the corresponding
parabolic geometries are collectively called contact parabolic geometries. Quick comments
on these structures, in association with chains, can be found in section 4.3. A slightly extended
comment on Lagrangean contact structures follows.
Lagrangean contact structures revised
Lagrangean contact structures are deﬁned in section 3.3 in terms of decomposition of the contact
distribution into Lagrangean subspaces, H = L⊕R, with the model being a ﬂag variety of speciﬁc
type. Almost CR structures are deﬁned earlier in this section in terms of an almost complex
structure J on H (i.e., an endomorphism such that J2
= − id |H) compatible with the Levi
bracket, with the model being a real hyperquadric in complex projective space. The relationship
between these two structures can be emphasized by interpreting the Lagrangean decomposition
H = L⊕R as the eigenspace decomposition of an almost para-complex structure K on H (i.e., an
endomorphism such that K2
= id |H) that is compatible with the Levi bracket. This can indeed
be precised so that, in particular, vanishing of the Nijenhuis tensor of K is equivalent to the
integrability of both L and R.
Similar aﬃnity should, of course, be perceptible also on the level of homogeneous models. This
is indeed possible: starting with the CR description and imposing the ‘para-’ preﬁx everywhere, we
claim that the homogeneous model of the Lagrangean contact structure of dimension m = 2n + 1
can be naturally identiﬁed with the ‘para-complex projectivization’ of the cone of non-zero null
vectors of a para-Hermitean inner product in a ‘para-complex vector space’ of real dimension
2n + 4. The quote marks indicate that one has to be careful about typical para-issues, i.e., that
3 Concrete geometries 33
para-complex numbers do not form a ﬁeld, hence ‘para-complex vector spaces’ are actually modules
etc. To be on the safe side, let us formulate things in real terms.
Let V be a real vector space of dimension 2n + 4 with an inner product h of signature (2p +
2, 2q + 2), where p + q = n, and N ⊂ V be the cone of non-zero null vectors of h. Let us consider
K to be a para-complex structure on V that is skew with respect to h. This compatibility forces
the eigenspaces V+ and V− of K to be isotropic. In particular, h must have split signature and V+
and V− are dual one another with respect to h. For X ∈ V , the para-complex hull X, K(X) is
2-dimensional if and only if X belongs to the open subset V0 := V \{V+ ∪V−}; such subspaces are
called para-complex lines. Any para-complex line X, K(X) intersects V+ and V− in 1-dimensional
subspaces spanned by X+ := X +K(X) and X− := X −K(X), respectively. Since V− is identiﬁed
with the dual space of V+, we may interpret the latter subspace as a hyperplane in V+. Since
the identiﬁcation is provided by h, the pair X+ and X− represents a ﬂag in V+ if and only if
h(X+, X−) = 0. This is also equivalent to the fact that the para-complex line is null, i.e., contained
in N or, more precisely, in N0 := N \ {V+ ∪ V−}. Altogether, we have
Proposition 3.1. The space of para-complex null lines in V = Rn+2,n+2
is naturally identiﬁed
with the ﬂag variety of type (1, n + 1, n + 2).
Thus, the ‘para-complex projectivization’ of N0 recovers the model Lagrangean ﬂag variety
together with the double ﬁbration as displayed on page 23.
Considering the real projectivization of N, which is the M¨obius space Sn+1,n+1
, it has three
components in accord with the decomposition
N = V+ ∪ N0 ∪ V−. (3.15)
The group G = SO(n + 2, n + 2) acts transitively both on N and Sn+1,n+1
, however, the previous
decomposition corresponds to the orbit decomposition of the subgroup of elements commuting
with K. Such elements are represented by linear isomorphisms of V+ (and their duals on V−),
hence the subgroup is isomorphic to GL(n + 2, R). Restricting only to isomorphisms preserving a
volume form on V+, we have an embedding
φ : G = SL(n + 2, R) → SO(n + 2, n + 2) = G. (3.16)
The compatibility of K and h allows to construct a para-Hermitean inner product on V that is
preserved by G and whose real part is h. In this context we may see G as a special ‘para-unitary’
group.
Note that map (3.16) admits a lift to the corresponding Spin group. All these observations
are employed later in section 5.2 in connection with a (closest possible) analogue of the Feﬀerman
construction.
CHAPTER II
Usage
H ere we collect our concrete contributions to the generalities described in the previous chapter.
The exposition is based on the attached articles, extended by motivations and commentaries
for which, usually, there is not enough space in journal publications. Although the individual
sections concern independent stories, one can easily spot many interrelations. The ordering of the
material was driven also by this aspect.
A quick summary of the results obtained is given already in chapter . The core of each of the
following sections follows the introductory historical survey. In the explanations of the results we
suppress all technical details and rather stress the key points and remarkable relations.
At this place we face a slight problem concerning the notation and terminology: as we refer
both to the attached articles and the previous preparatory text, we hope the reader will understand
that we cannot guarantee an absolute consistency.
4 Geometry of chains
Chains are non-contact distinguished curves of parabolic contact structures. Certainly, they are
best known and studied in the CR contact setting, in which case they were introduced by Cartan,
and later generalized by Chern and Moser, in the course of solution to the equivalence problem.
Despite their innocent deﬁnition and model realization, they form a rather intricate family of
curves. This is probably why chains and their behaviour became a subject of independent study.
In the introductory section we make several remarks on this account.
In the main part we focus on the conceptual description of the associated path geometry
following our article [17]. This can be attained by a general extension of Cartan geometries as
described in section 1.3. In the CR and Lagrangean contact case, this construction preserves the
normality condition for an important subclass of integrable (torsion-free) structures, in which cases
we can add further non-trivial observations. This, in particular, concerns the reconstruction of the
initial structure from the path geometry of chains.
Considering a similar adventure for other parabolic contact structures, one observes some
limitations: in most cases the whole circle of ideas works only in the ﬂat case, for projective
contact structures the construction reduces to a construction of Feﬀerman type. These situations
are commented in the concluding section.
36 II Usage
4.1 History and uniﬁcation
Cartan, Chern and Moser
The notion of chains (chaˆınes) was introduced in the work of Cartan on real hypersurfaces in C2
,
[21, 20]. On hyperquadrics (3.13), they are just the intersections with complex lines, which turn
to be either real circles or straight lines. Passing to the model projective hyperquadric in CP2
,
which is the homogeneous space G/P with G = PSU(2, 1) and P ⊂ G the parabolic subgroup as in
section 3.5, it follows that chains are the homogeneous curves with generators in the 1-dimensional
subspace g−2 ⊂ g. Easily, all chains are mutually equivalent, any two points determine a unique
chain and any chain is everywhere transverse to the contact distribution. Thus, each chain (as an
unparametrized curve) is uniquely determined by a tangent direction in one point, provided it is
transverse to the complex contact subspace.
Cartan’s generalization to general real hypersurfaces M ⊂ C2
can be phrased in our terms as
follows. For G → M being the associated P-bundle with the normal Cartan connection ω, chains
are the projections of integral submanifolds of a closed diﬀerential ideal on G determined by ω.
This somehow vague description may be interpreted so that the g±1-parts of a suitable Cartan
gauge along each chain vanish.
Chains are further considered on real hypersurfaces in complex spaces of general dimension
by Chern and Moser in [25]. Firstly, chains play a key role in the construction of normal forms,
where they (and certain framings along them) are related to an external coordinate system under
biholomorphic mappings. This way, chains (and the associated framings) satisfy a system of differential
equations which is invariantly given by the hypersurface. Secondly, chains are described
intrinsically on a general (integrable) CR manifold of hypersurface type in terms of the associated
(normal) Cartan connection generalizing the above mentioned Cartan’s description.
Beside the previous views, there is another one by Feﬀerman via his construction discussed
in section 3.5: chains are understood as projections of null geodesics of the conformal Feﬀerman
space.
Uniﬁcation and remarks
It is not immediately clear how all these concepts are related, if ever. Detailed comparisons of Cartan
and Chern–Moser approach can be found in [47], many illuminating remarks are also scattered
in [4]. It follows that, in the Cartan geometric interpretation, chains coincide with projections to
the base manifold of ﬂows of constant vector ﬁelds on G that corresponds to elements from g−2 ⊂ g.
This ﬁts nicely to the concept of distinguished curves introduced in section 2.3. In particular, the
subalgebra of g with vanishing g±1-parts is the symmetry algebra of a model chain. Also, via
the notion of development of curves, the above listed properties of model chains remain valid in
general. And, ﬁnally, the notion of chains can be instantly transferred to any parabolic contact
geometry.
Before we fully focus on the previous standpoint, let us remark several interesting features of
chains that were studied by many authors, cf. [32, 46, 50, 51]: It holds that, any two nearby points
on strictly pseudoconvex CR contact manifolds may be connected by a chain. But, this statement
fails in general signature. Furthermore, it may happen that chains spiral into a point. Another
result, which was proved in generality by Cheng [23], is that chain preserving diﬀeomorphisms
between CR contact manifolds are either CR automorphisms or conjugate CR automorphisms.
We are going to recover this result in the next section.
4.2 The action: curved Cartan extension
The fact that chains (as unparametrized curves) are determined by a tangent direction at one
point means that they form a system of paths in the sense of section 3.3, restricted, however, to
the non-contact directions. Since this structure also allows a parabolic geometric description, it is
natural to look for a direct relation of the two structures via some extension construction described
in previous chapter. Experiments with the corresponding system of ODEs in homogeneous model
4 Geometry of chains 37
give some clue on the character of such relation; see [27] for concrete outcomes in small dimensions.
The most interesting discussions concern the CR and Lagrangean contact structures, which we
reproduce here following our article [17] stimulated by [26]. According to the current exposition,
we focus on the CR case and only comment the Lagrangean one below (although the structure of
[17] is right opposite).
Let M be a smooth manifold of dimension m = 2n + 1 endowed with an almost CR contact
structure (H, J) of signature (p, q), p+q = n. Let G = PSU(p+1, q +1) and P ⊂ G be the pair of
simple Lie group and its parabolic subgroup, let p ⊂ g be their Lie algebras and let (G → M, ω) be
the corresponding regular normal parabolic geometry as in section 3.5. Let further M ⊂ PTM be
the open subset of the projectivized tangent bundle of M consisting of the non-contact directions
and let E ⊕ V be the decomposition of the tautological subbundle of TM determined by the path
geometry of chains. Let G = PGL(2n + 2, R) and P ⊂ G be the pair of simple Lie group and
its parabolic subgroup, let ˜p ⊂ ˜g be their Lie algebras and let (G → M, ω) be the corresponding
regular normal parabolic geometry as in section 3.3.
The construction
Searching for a direct relation of the two Cartan geometries, one has to investigate the homogeneous
model ﬁrst. For M = G/P, the ﬁrst observation is that G acts transitively on M, the space of noncontact
lines in TM. The stabilizer Q ⊂ G of an appropriate element contains G0 and is contained
in P so that its Lie algebra is q = g0 ⊕ g2. In particular, M is the homogeneous space G/Q and
the associated path geometry as well as the regular normal parabolic geometry (G → M, ω) are
homogeneous with respect to G. From the generalities in section 1.3 we know that such Cartan
geometry can be described by a pair of compatible extension maps i : Q → P and α : g → ˜g. It
follows that, in the current setting, such pair is actually unique up to the natural equivalence, see
[17, Theorem 3.4].
(G, ω)
P

(G, ω)
@Ø
SS
Q
AA
P

(M, E ⊕ V )
uu
path
(M, H, J) CR contact
To prescribe the pair (i, α) in concrete, one may ﬁx as much as possible to satisfy all the compatibility
conditions and eliminate the remaining freedom so that the induced parabolic geometry
is normal, i.e., that ∂∗
Ψα = 0, for Ψα as in (1.8). For instance, according to the block decompositions
as before, the following choice guarantees that α induces a linear isomorphism g/q → ˜g/˜p
and g−2 ⊂ g hits ˜gE
−1 ⊕ ˜p ⊂ ˜g,


· Z ·
X · −Ep,qZ∗
ix −X∗
Ep,q ·

 α
−→




· · · ·
x · · ·
Re(X) Im(Ep,qZ∗
) · ·
Im(X) − Re(Ep,qZ∗
) · ·



 ,
where Ep,q is as in (3.8) and ∗
denotes the conjugate transposition. This setup can be completed
and integrated to i : Q → P so that the remaining compatibility and normality conditions hold,
which determines the pair (i, α) uniquely. The full matrix realization is given in [17, section 5.2].
It follows that the corresponding Ψα is non-trivial, hence we deal with general extension functor.
38 II Usage
Normality and applications
The ﬁrst serious task concerns answering the fundamental question under which conditions the
extension functor yields a regular normal parabolic geometry. In order to tackle this problem,
a careful analysis of the Cartan curvatures in accord with (1.9) is inevitable. With a detailed
knowledge of harmonic curvatures and the properties of the map α, it turns out the direct relation
between the regular normal parabolic geometries associated to an almost CR contact structure
and its path geometry of chains is available exactly for the subclass of integrable structures:
Theorem 4.1 ([17, Theorem 5.2]). Let (M, H, J) be an almost CR contact structure and let
(G → M, ω) be the corresponding regular normal parabolic geometry of type (G, P). Then the
parabolic geometry (G → M, ω) constructed using the extension functor associated to the pair
(i, α) as above is regular and normal if and only if ω is torsion-free, i.e. the almost CR structure
is integrable.
In particular, this is automatically satisﬁed when dim M = 3, i.e., n = 1. With such direct relation
we can control some properties of chains—for integrable CR contact structures—in relatively
easy way:
Firstly, the general principle that the induced Cartan curvature κ is fully given by the CR
Cartan curvature κ and the constant contribution Ψα may be nicely speciﬁed in our setting. It
follows that the two ingredients, respectively their equivariant extensions, are well separable from
the whole curvature κ in terms of certain projections. Note that this is not an obvious task since
this has to be done without the prior knowledge of the bundle extension. In particular, these two
parts can never eliminate each other, the one corresponding to Ψα is never vanishing and the
one corresponding to κ supports just the torsion of κ. These facts and the interpretation of the
vanishing of individual curvature components as in section 3.3 lead to the following results:
Theorem 4.2 ([17, Theorem 5.3]). Let (M, H, J) be a CR contact structure.
• There is no linear connection on TM which has the chains among its geodesics.
• The path geometry of chains is torsion-free if and only if the CR structure is ﬂat.
Secondly, the previous separation can be pushed even further so that the equivariant extension
of Ψα, respectively κ, coincides up to a non-zero multiple with the component of the harmonic
curvature of homogeneity 3, respectively 2 (torsion), of the induced Cartan connection. This and
the explicit knowledge of Ψα, which is just an algebraic map given by (1.8), allows us to interpret
the corresponding harmonic curvature component via the complete symmetrization of the map
(ξ, η, ζ) → L(ξ, J(η))J(ζ), (4.1)
for ξ, η, ζ ∈ Γ(H), where L : H ×H → TM/H is the Levi bracket. This shows an intimate relation
between the initial CR contact structure and the associated path geometry of chains on the very
underlying level. Now, it is an algebraic exercise to show the following:
Theorem 4.3 ([17, Theorem 5.3]). The almost complex structure J can be reconstructed up to
sign from the harmonic curvature of the associated path geometry of chains.
In the proof one has to consider the complexiﬁcation of (4.1) to H ⊗ C, which then allows
to reconstruct the subset H1,0
∪ H0,1
. This determines the almost complex structure on H up
to the sign, hence the ambiguity. Notice that particularly the signature of the CR structure is
incorporated in the path geometry of chains and the previous reasoning indicates how.
Finally, as an easy consequence, we recover the result of [23] on chain preserving transforma-
tions:
Corollary 4.4 ([17, Corollary 5.3]). A contact diﬀeomorphism between two CR contact manifolds
which maps chains to chains is either a CR isomorphism or a CR anti-isomorphism.
4 Geometry of chains 39
Remarks
From the viewpoint of path geometry, chains form a complicated system of curves even in the
homogeneous model since the corresponding Cartan geometry is non-ﬂat. However, it is torsionfree
and has a relatively large automorphism group. From the previous we know this group is
G = PSU(p + 1, q + 1), whose dimension is n2
+ 4n + 3, where n = p + q. A natural question in
similar situations is whether this realizes the submaximal case or not. It follows from [52, section
5.3] that this is not the case, since the maximal automorphism group of a non-ﬂat path geometry
has dimension 4n2
+ 4n + 6.
Since the path geometry of chains in the homogeneous model is torsion-free, the twistor space
of chains carries a canonical Grassmannian structure of type (2, 2n) (equivalently, Segre or paraquaternionic
structure). Again, this is an example of a non-ﬂat Grassmannian structure with
relatively large (but not submaximal) automorphism group. Moreover, this is an example of a
non-ﬂat Grassmannian (globally) symmetric space, cf. [69].
Although chains are not geodesics of any linear connection on M, there is a variational approach
to their description and study. This has been recently shown in [24] via a Finsler-like metric on
M, the so-called Kropina metric.
4.3 Other parabolic contact structures
The just described story of chains for CR contact structures has analogies for other parabolic
contact structures. The Lagrangean contact ones are, of course, the closest.
Lagrangean contact structures
In this case, the development of section 4.2 is very parallel with analogous results, which are
also described in [17]: the extension functor preserves the normality condition just in torsion-free
cases, chains are not geodesics of any linear connection and the path geometry of chains is torsionfree
just for ﬂat Lagrangean contact structures. In the reconstruction, the harmonic curvature
component corresponding to Ψα allows to recreate the subset L ∪ R ⊂ H, i.e., the para-complex
structure on H up to sign. Hence the chain preserving diﬀeomorphisms are either Lagrangean
automorphism or anti-automorphism, i.e., they either preserve or swap the Lagrangean subspaces
from the decomposition L ⊕ R = H.
Concerning the homogeneous model, it allows several descriptions as discussed in sections
3.3 and 3.5. Anyhow, it is the correspondence space of RPn
which is identiﬁed with the space
of contact elements in RPn
of hyperplane type. Consequently, chains of the model Lagrangean
contact structure should be described in the underlying projective terms. Indeed, seeing them as
1-parameter families of contact elements, it easily follows that each chain consists of a pencil of
hyperplanes in RPn
with a common projective line whose base points also form a projective line. In
particular, chains project to projective geodesics and this fact holds generally for any Lagrangean
contact structure induced by projective structure.
Projective contact structures
Projective contact structure (in a narrower sense) is the underlying structure of a regular normal
parabolic geometry corresponding to the contact grading of simple Lie algebra g = sp(2n + 2, R),
where the dimension of base manifold is 2n + 1. Roughly, the structure is given by a class of
‘contact projectively equivalent’ aﬃne connections with vanishing ‘contact torsion’, i.e., basically,
by a family of ‘contact geodesics’. In a broader sense, the condition of vanishing contact torsion may
be suppressed, which still allows a parabolic geometric description with a generalized normalization
condition. This is considered by Fox in [38], where also a subordinated projective structure on the
same manifold is described. The subordinateness means that the contact geodesics of the former
structure are among the geodesics of the latter one.
40 II Usage
The approach of [38] is via the ambient connections—the Thomas one for projective structures
and its direct analogue for projective contact structures. Considering the structure with vanishing
contact torsion, the subordinated projective structure may be interpreted as the result of a
construction of Feﬀerman type. In such case, geodesics in non-contact directions are exactly the
chains of the projective contact structure, hence the associated path geometry is covered just by the
correspondence space construction. This is indeed unusual situation compared to other parabolic
contact structures. Anyhow, the chain preserving transformations turn out to be automorphisms
of projective contact structures. All details can be found in our article [18].
Lie contact structures
Other parabolic contact geometries are less studied than the ones mentioned above. Among them,
probably the Lie contact structures are the most prominent as they are related to the classical Lie
sphere geometry. Lie contact structures correspond to contact gradings of simple Lie algebras g =
so(p+2, q+2), where the dimension of base manifold is 2p+2q+1. The underlying structure allows
several equivalent descriptions, among which the shortest is via a para-quaternionic structure on
the contact distribution compatible with the Levi bracket (the role of the signature (p, q) is rather
involved in this setting).
According to the development of section 4.2 one can follow the key steps including the reconstruction
of the structure from the harmonic curvature of the associated path geometry, provided
that the Lie contact geometry is torsion-free. The only defect on the neck is that, for dimensions
grater or equal to 7, this restriction implies the Lie contact structure is ﬂat and in such cases the
used machinery seems a bit heavy. Anyhow, in the homogeneous case the path geometry of chains
is again non-ﬂat, torsion-free, with relatively large automorphism group, see [68] for more.
Furthermore, the model Lie contact structure can be identiﬁed on the projectivized cotangent
bundle of the M¨obius sphere, and this observation can be covered by a construction of Feﬀerman
type, see [67]. In the case of positive deﬁnite signature, interpreting chains as 1-parameter families
of contact elements, one can show that the base points of each chain form a conformal circle
which intersect the hyperplanes from the family in constant angle. In particular, chains project to
conformal circles and this fact holds for any Lie contact structure induced by conformal structure.
In low dimensions, Lie contact structures are basically equivalent to other, better studied,
parabolic contact structures: the Lie algebra isomorphisms so(4, 2) ∼= su(4, 2) and so(3, 3) ∼=
sl(4, R) indicate the equivalence to almost CR contact structures of split signature and Lagrangean
contact structures, respectively, in dimension 5. Note that the latter identiﬁcation (respectively,
its complexiﬁcation) is also behind the famous line-sphere correspondence by Lie.
Remaining cases
There is also an isomorphism so(6, 2) ∼= so∗
(8), which is an isomorphism of real forms of the
same complex Lie algebra. Parabolic contact structures corresponding to contact gradings of so∗
algebras are almost unknown. However, it can be shown that the underlying structure consists of
a quaternionic structure on the contact distribution that is compatible with the Levi bracket. The
behaviour of chains is not studied in detail for these structures, but it is expected to be parallel
to the Lie contact case.
One may also list the structures corresponding to contact gradings of exotic simple Lie algebras.
Similarly to previous two cases, the common feature is that their harmonic curvatures consists
only of torsions, hence the same restrictions as above are expected.
5 Conformal Patterson–Walker metrics
Walker manifolds are pseudo-Riemannian manifolds admitting a parallel isotropic distribution. In
the case of split signature, there is an interesting subclass of Patterson–Walker metrics, which
are the Walker metrics induced naturally on the cotangent bundle of a manifold with an aﬃne
connection. If the base manifold has dimension two, the construction was modiﬁed in [30] in a
5 Conformal Patterson–Walker metrics 41
projectively invariant manner. We can grasp this construction as a construction of Feﬀerman type
and consider its generalization to any dimension.
In the following we recall some basics following our three main sources of inspiration, [54, 30,
53]. Then we compare various viewpoints, which should serve as a useful background for understanding
the general Feﬀerman-type construction relating the associated parabolic geometries. As
emphasized already in the introductory chapter, this construction is typically non-normal, but it
is in a sense ‘half-normal’, which allows its surprisingly satisfactory treatment. This is described
in section 5.2 following our article [43].
It follows that many of the abstract functorial notions have very concrete counterparts. From
the latter perspective, new interrelations between the two geometric structures were researched
in our next article [42]. This is not treated systematically in the current exposition, but here and
there we mention concerning particular results. As a nice mixture of abstract and concrete, we are
able to draw also a very favourable ambient picture. This is described in section 5.3 following our
last article [44].
5.1 History and comparisons
Patterson and Walker
The concept of the Riemann extension appeared for the ﬁrst time in the article [54] by Patterson
and Walker. This is a pseudo-Riemannian metric of split signature (n, n) with a parallel isotropic
distribution of rank n induced by a geometric data on an n-dimensional manifold. Already in that
reference, there are considered several variants of such construction, which are further modiﬁed
and generalized by many authors. What we mean by the Patterson–Walker metric is the Riemann
extension induced by a torsion-free aﬃne connection as follows. Let (xA
) be local coordinates on
M, let Γ C
A B = Γ C
B A be the Christoﬀel symbols of a torsion-free aﬃne connection D on M and let
(pA) be the extended coordinates on the bigger manifold.1
Then the associated Patterson–Walker
metric is given by
g := 2 dxA
dpA − 2 Γ C
A B pC dxA
dxB
. (5.1)
It is easy to see that the distribution spanned by ∂pA
is isotropic and parallel with respect to
the Levi-Civita connection of g. Besides concrete local expressions and relations, it is indicated
already in [54] how to interpret the metric g in large, namely, as living on the total space of the
cotangent bundle T∗
M. We come back to this issue later in this section.
Right after this initiation, many related contributions appeared, including a local characterization
of Patterson–Walker metrics and reﬂections of various speciﬁc properties. For instance, we
list several typical statements:
• D is ﬂat if and only if g is ﬂat,
• D is Ricci ﬂat if and only if g is Ricci ﬂat,
• D is projectively ﬂat if and only if g is conformally ﬂat.
These and similar results can be found in papers by Patterson, Walker, Aﬁﬁ and others, see, e.g.,
[66, chapter VII, §15] for a quick introduction and many references.
Dunajski and Tod
In the 2-dimensional case, the previous construction was modiﬁed by Dunajski and Tod in [30] so
that single aﬃne connection is substituted by a whole projective class. This was done using the
Thomas projective parameters in the place of Christoﬀel symbols,
¯g := 2 dxA
dpA − 2 Π C
A B pC dxA
dxB
, (5.2)
where the symbols Π C
A B are deﬁned by (3.2).
1We use the notation that ﬁts best to our later development, in particular, uppercase indices range from 1 to
n = dim M, while lowercase ones will range from 1 to 2n.
42 II Usage
It follows that the conformal structure represented by ¯g is anti-self-dual and admits a lightlike
conformal Killing ﬁeld, namely, k := pA∂pA
. A close relation between the two structures is
demonstrated in [30, Theorem 4.1] which states that the initial projective structure is metrizable
if and only if the induced conformal structure contains K¨ahler or para-K¨ahler metric.
A proper interpretation of the extended manifold, an alternative description of the induced
conformal class and its characterization is commented below.
Nurowski and Sparling
It is a useful observation and one of our research motivations that the previous construction can be
directly related to the one by Nurowski and Sparling in [53], which is demonstrably a construction
of Feﬀerman type. In that article, authors are concerned with the geometry of 2-order ODE,
y = Q(x, y, y ), (5.3)
in the plane (x, y) modulo point transformations; cf. remarks around (3.6). Since, in this dimension,
the corresponding path geometry is equivalent to a Lagrangean contact structure, the article
is designed in full analogy with the Cartan’s original treatment of 3-dimensional CR contact
structures. In particular, the equivalence method yields the normal Cartan connection ω associated
to the ODE together with the fundamental curvature invariants.
Moreover, this approach allows to deﬁne an analogue of the Feﬀerman conformal metric on a 4dimensional
quotient of the Cartan’s principal bundle. A representative metric from the conformal
class is then explicitly given by
g := (dp − Q dx) dx − (dy − p dx) (2
3 i dφ + 2
3 Qp dx + 1
6 Qpp(dy − p dx)), (5.4)
where (x, y, p = y , φ) are local extended coordinates on the Feﬀerman space and i is a non-zero
real constant; see [53, formula (31)].
It is shown in [53, Proposition 1] that the conformal structure [g] is half-ﬂat (i.e., self-dual or
anti-self-dual) if and only if the Cartan connection ω is half-ﬂat (i.e., one of the two fundamental
invariants vanishes). In particular, this happens for ODEs which are point-equivalent to a geodesic
equation or, in other words, for Lagrangean contact structures which are induced by a projective
structure. In such case, the resulting conformal structure is the result of composition of two functorial
construction, namely, the correspondence space construction (from projective to Lagrangean
contact) and the Feﬀerman-type construction (from Lagrangean contact to conformal).
Direct comparison
In order to compare the current approach with the setting of Dunajski and Tod, we just need to
expand the condition that the ODE (5.3) is point-equivalent to a geodesic equation. From (3.7)
we know that this is equivalent to
Q = A0 + A1p + A2p2
+ A3p3
, (5.5)
where Ai are functions only of x and y and p = y . Computing partial derivatives Qp and Qpp and
substituting into (5.4) yields
g = dp dx + 2i
3 p dφ dx − 2i
3 dy dφ−
− (A0 + 1
3 A1p) dx dx − 2
3 (A1 + A2p) dx dy − (1
3 A2 + A3p) dy dy.
(5.6)
To compare this metric with the one from (5.2), we need some relations among the coeﬃcients in
(5.5), the Christoﬀel symbols of a representative connection and the Thomas projective parameters.
Considering the representative connection is torsion-free, it easily follows the relations are
A0 = −Γ2
11, A1 = Γ1
11 − 2Γ2
12, A2 = 2Γ1
12 − Γ2
22, A3 = Γ1
22,
Π2
11 = −A0, Π1
11 = −Π2
21 = −Π2
12 = 1
3 A1, Π1
12 = Π1
21 = −Π2
22 = 1
3 A2, Π1
22 = A3,
5 Conformal Patterson–Walker metrics 43
cf., e.g., [6]. This gives the following expression of the metric from (5.2),
¯g = dp1 dx1
+ dp2 dx2
−
− 1
3 A1p1 dx1
dx1
− 2
3 A2p1 dx1
dx2
− A3p1 dx2
dx2
+
+ A0p2 dx1
dx1
+ 2
3 A1p2 dx1
dx2
+ 1
3 A2p2 dx2
dx2
.
(5.7)
Now, employing a coordinate transformation
x = x1
, y = x2
, p = −p1
p2
, φ = 3
2i ln |p2|,
it is a straightforward exercise to show that
Proposition 5.1. Metrics (5.6) and (5.7) are conformally equivalent.
Conceptual comparison
An alternative way to realize the whole story is to interpret, and then modify, appropriately the
original Patterson–Walker construction. The ﬁrst part of this program is scattered in literature,
the second part, including a comparison with previously used Thomas projective parameters,
is novel. This approach, initiated by [29], is developed in [42]. General interpretation and further
investigation of this construction as a construction of Feﬀerman type is the subject of next section.
In what follows we consider a smooth manifold M of general dimension n and a torsionfree
aﬃne connection D. As before, let (xA
) be local coordinates on M and let Γ C
A B be the
Christoﬀel symbols of D. But, now, let (pA) be the dual ﬁbre coordinates of T∗
M, i.e., the
ones so that θ = pA dxA
is the tautological 1-form, and let g be the Patterson–Walker metric
on T∗
M deﬁned by (5.1). The connection D determines (and is determined by) a horizontal
distribution H ⊂ TT∗
M which is complementary to the vertical distribution V of the bundle
projection π : T∗
M → M. Via the tangent map of π, the bundle H is isomorphic to TM, whilst
V is canonically isomorphic to T∗
M. Concretely, the vertical and the horizontal distribution is
spanned by ∂pA
and ∂xA +Γ C
A B pC ∂pB
, respectively. The Patterson–Walker metric g is the unique
one such that both V and H are isotropic with respect to g and the value of g with one entry from
V and another entry from H is given by the natural pairing between V ∼= T∗
M and H ∼= TM.
I.e., for any ξ, η ∈ H and ϕ, ψ ∈ V ,
g(ξ, η) = g(ϕ, ψ) = 0, g(ξ, ϕ) = ϕ(Tπ·ξ).
Note that, extending the previous natural pairing between V and H in anti-symmetric (rather
than symmetric) way, one recovers the canonical symplectic form on T∗
M,
µ = dθ = dpA ∧ dxA
. (5.8)
The relation to the Patterson–Walker metric g is provided by the almost para-complex structure,
K, induced by the decomposition H ⊕V = TT∗
M so that H and V are its +1 and −1 eigenspace,
respectively. It follows that g is para-Hermitean with respect to K, i.e., g(K−, K−) = −g(−, −),
and g(−, −) = µ(K−, −). The vector ﬁeld corresponding to the tautological 1-form with respect
to g, respectively µ, is the Euler vector ﬁeld of the ﬁber vector structure,
k := pA∂pA
. (5.9)
This is an important companion and one of the characteristics of the Patterson–Walker metrics
as it is its homothety, i.e., a conformal Killing ﬁeld with constant divergence.
For the projective-to-conformal variant of the previous construction, let us consider its obvious
modiﬁcation to weighted cotangent bundles T∗
M(w), with general w. In such cases, in order to
trivialize the density bundle, we have to assume that M is orientable and an aﬃne connections
on M are special, i.e., volume preserving. Now, the eﬀect of the change of the underlying aﬃne
connection can be directly spotted with the following outcome:
44 II Usage
Proposition 5.2 ([42, Proposition 3.1]). Let D and D be projectively equivalent special torsionfree
aﬃne connections on M and let g and ˆg be the associated Patterson–Walker metrics on
T∗
M(w). Then g and ˆg are conformally equivalent if and only if w = 2.
Note that the deﬁnition (5.2) using Thomas projective parameters can be considered in any
dimension. It is an instructive exercise to directly compare that approach with the just mentioned
one, see [42, remark 3.3] for the details.
Altogether, the construction we are going to investigate associates to an oriented projective
structure on M a conformal structure on T∗
M(2), which we call the conformal extension or conformal
Patterson–Walker metric. The induced conformal structure is also oriented and, moreover, it
can be lifted to a conformal spin structure. We employ this addition as a convenient interpretative
instrument.
5.2 The action: non-normal Feﬀerman-type construction
Before we launch the Feﬀerman-type construction covering the previous observations, we have
to consult the model situation. Important preparatory observations were done in section 3.5, in
which setting we continue here. In order to be in concordance with our main reference [43], we only
decrease n (which controls dimensions of all the spaces around) by 1. We also consider the spin
group G = Spin(n + 1, n + 1), instead of SO(n + 1, n + 1), as the principal one. The embedding
φ : G = SL(n + 1, R) → Spin(n + 1, n + 1), (5.10)
which lifts the one from (3.16), is described in [43, section 3.2].
The construction
The group G acts transitively on null rays in V = Rn+1,n+1
, whereas G has three orbits in accord
with the decomposition (3.15). Let P ⊂ G be the stabilizer of a ray from N0, i.e., a null ray not
contained in V+ ∪ V−, and let Q = G ∩ P. Obviously Q ⊂ P ⊂ P, where P ⊂ G is the stabilizer
of the para-complex line containing the ray stabilized by Q and P ⊂ G is the stabilizer of the
ray given by the projection to V+
∼= Rn+1
. Hence the homogeneous space G/Q coincides with the
open orbit in G/P ∼= Sn,n
, the M¨obius sphere, G/P coincides with the oriented ‘para-complex
projectivization’ of N0, the model oriented Lagrangean ﬂag variety, and G/P coincides with the
oriented projectivization of V+, the projective n-sphere.
(G, ω)

(G, ω)
9 ×
φ
RR
BB

ÖÖ
P+(N0) ∼= G/Q
tt
G/P ∼= Sn,n
conformal
Flags+(1, n, n + 1) ∼= G/P
tt
Lagrangean
P+(V+) ∼= G/P projective
The central part of this picture reminds the one from section 3.5, the left part is the standard
ﬂag correspondence. In the following, we are primarily interested in the extension from projective
to conformal structures. We might therefore ignore the intermediate Lagrangean step, however,
we will see it is worth remembering it is naturally there.
For the general construction, let M be an oriented manifold of dimension n ≥ 2 endowed with
a projective structure and let (G → M, ω) be the corresponding normal parabolic geometry of
5 Conformal Patterson–Walker metrics 45
type (G, P). Let P ⊂ G and Q = G ∩ P ⊂ P be as above and let ˜p ⊂ ˜g and q ⊂ p ⊂ g be
the corresponding Lie algebras. Let us consider the extension functor of Feﬀerman type given by
the embedding (5.10) and let (G → M, ωind
) be the induced parabolic geometry with underlying
conformal spin structure on the Feﬀerman space M = G/Q. Let us interpret and further investigate
the induced conformal structure:
First interpretations
The Feﬀerman space, as the quotient bundle M = G/Q is identiﬁed with the associated bundle
G ×P (P/Q). Focusing on the typical ﬁbre P/Q, one easily shows that this is isomorphic as a
P-module to the set of non-zero elements of (g/p)∗
(2), the typical ﬁbre of T∗
M(2) seeing as an
associated bundle to G. Thus, the Feﬀerman space M is naturally identiﬁed with the weighted
cotangent bundle of projective weight 2, without the zero section, see [43, Proposition 3.1]. Similar
argument suﬃces to show that among the metrics from the induced conformal class there are
genuine Patterson–Walker metrics, see [43, Proposition 6.2]. (The corresponding scales, i.e., the
ones which correspond to projective scales on M, are called reduced.) Hence we know the current
construction indeed covers the conformal Patterson–Walker extension described by Proposition
5.2.
As we indicated in previous commentaries, the induced conformal structure has number of
special properties. Immediately from the algebraic setup we can easily observe several distinguished
objects, which play a role in later characterizations, see [43, Proposition 3.2]. Namely, they are pure
tractor spinors, denoted as sE and sF , and a compatible involution of the standard tractor bundle,
K, which are all parallel with respect to the induced (not necessarily normal) tractor connection.
The underlying objects are pure spinors, η and χ, and a null vector ﬁeld, k, generating the 1dimensional
intersection of the distributions ker η and ker χ. Of course, one of these distributions
forms the vertical subbundle of the projection M → M and the notation is chosen so that V =
ker χ. Note also that the sum of these two distributions coincides with the ortho-complement of k
in TM.
Normality
As usual, the critical problem is to detect in which cases the Feﬀerman-type construction preserves
the normality condition. This can be discussed either directly or by analysing the two natural steps
which are behind the current construction (cf. the model diagram above): (1) the correspondence
space construction from projective to Lagrangean contact structure and (2) the Feﬀerman-type
construction from Lagrangean contact to conformal structure.
From section 3.3 we know that the ﬁrst construction always preserves the normality. Concerning
the second construction, we know from section 3.5 that the classical Feﬀerman construction
preserves the normality only in the torsion-free (integrable) case. The current Lagrangean setting
can be dealt in full analogy to the CR case with the very same outcome: here the torsion-freeness
means that both Lagrangean subbundles of the contact distribution are integrable. Combining
these results, we conclude with the following.
Proposition 5.3 ([43, Proposition 3.8]). Let (G → M, ω) be a normal projective parabolic geometry
and let G → M, ωind
be the conformal parabolic geometry obtained by the Feﬀerman-type
construction.
• If dim M = 2 then ωind
is normal.
• If dim M > 2 then ωind
is normal if and only if ω is ﬂat.
Moreover, independently of the dimension of M, ωind
is ﬂat if and only if ω is ﬂat.
Hence, the normal conformal Cartan connection in the 2-dimensional case is directly obtained
by the current Feﬀerman-type construction, which recovers the Nurowski–Sparling construction;
see [43, section 3.7] for a quick summary with a full characterization. In general dimension, this is
46 II Usage
not the case. However, from section 3.3 we know that one of the two torsions of the intermediate
Lagrangean contact geometry is trivial and it is just the contribution of the second one which spoils
the normality of the resulting conformal parabolic geometry. Thus, even though this geometry
is generally non-normal, we interpret it as ‘half-normal’ and this is what allows further, and
eventually satisfactory, manipulation.
Normalization and characterization
To start with the normalization, one has to quantify the non-normality of the induced Cartan
connection, i.e., to specify the values of ∂∗
κind
. Of course, the discussion includes a lot of technicalities
which obstructs formulating results in a too relaxed way. The key observation is in [43,
Lemma 4.3], which is enough to follow the normalization process in a reasonable detail allowing
some interpretations. The process typically (for |1|-graded geometries) has two steps,
ω0
:= ωind
, ω1
:= ω0
+ Ψ1
, ω2
:= ω1
+ Ψ2
= ωnor
, (5.11)
where the correction 1-forms Ψ1
and Ψ2
have values in well described submodules of ˜p = ˜g0 ⊕ ˜g1
and ˜p+ = ˜g1 and are controlled by ∂∗
κ0
and ∂∗
κ1
, respectively. An important information about
the curvature of the normalized Cartan connection is stated in [43, Proposition 4.6], involving also
a ﬁrst version of the integrability condition that controls values of κnor
under insertions of vectors
that are vertical with respect to the projection M → M.
Previous results can be phrased so that the tractor spinor sF is parallel with respect to the
normal tractor connection. In particular, the underlying spinor ﬁeld χ is pure, the tractor spinor
sF is its BGG splitting, sF = L0(χ), and the holonomy of the induced conformal structure is
reduced. The other allied tractorial objects, sE and K, need not be parallel but it turns out they
are also BGG splittings of their underlying objects, sE = L0(η) and K = L0(k), and, moreover, k
is a conformal Killing ﬁeld. All these informations are condensed in [43, Proposition 4.8].
Interpreting the integrability condition in terms of the conformal Weyl tensor, W, we have at
this stage the ﬁrst half of the characterization theorem:
Theorem 5.4 ([43, Theorem 4.14]). A split-signature (n, n) conformal spin structure on a manifold
M is (locally) induced by an n-dimensional projective structure via the Feﬀerman-type construction
if and only if the following properties are satisﬁed:
• M admits a nowhere vanishing light-like conformal Killing ﬁeld k such that the corresponding
tractor endomorphism K = L0(k) is an involution, i.e., K2
= idT .
• M admits a pure twistor spinor χ with k ∈ Γ(ker χ) such that the corresponding parallel
tractor spinor sF = L0(χ) is pure.
• K acts by minus the identity on ker sF .
• The following integrability condition holds:
va
wc
Wabcd = 0, for all v, w ∈ Γ(ker χ).
For the converse direction one may use an auxiliary Cartan connection, given by
ω := ωnor
− 1
2 ιkκnor
, (5.12)
and an adapted version of the correspondence space descending theorem. This manoeuvre is described
in [43, section 4.3]. However, a detailed analysis of the Cartan curvatures brings an interesting
reﬁnement, which simpliﬁes the previous discussion and allows us to descend directly. It
follows that the Cartan connection (5.12) coincides with the induced one and that the diﬀerence
between the induced and the normal Cartan connection is, according to the notation from (5.11),
Ψ1
= −1
2 ∂∗
κind
= 1
2 ιkκnor
, Ψ2
= 0,
see [43, Theorem 5.7]. In particular, the above indicated normalization process completes already
in the ﬁrst step. Moreover, the diﬀerence term can be written explicitly using the conformal
Weyl and Cotton–York tensor. These results rely on the use of reduced scales and their intrinsic
characterization in [43, section 5.1].
5 Conformal Patterson–Walker metrics 47
Alternative characterization
In [42] we obtained (by diﬀerent means) a slightly diﬀerent version of the characterization theorem
5.4. This is formulated fully in terms of underlying objects, which is also why we speak about the
conformal Patterson–Walker metric rather than Feﬀerman-type construction:
Theorem 5.5 ([42, Theorem 1], [43, Theorem 6.3]). A split-signature (n, n) conformal spin structure
on a manifold M is (locally) induced by an n-dimensional projective structure as a conformal
Patterson–Walker metric if and only if the following properties are satisﬁed:
• M admits a nowhere vanishing light-like conformal Killing ﬁeld k.
• M admits a pure twistor spinor χ such that ker χ is integrable and k ∈ Γ(ker χ).
• The Lie derivative of χ with respect to the conformal Killing ﬁeld k is
Lkχ = −1
2 (n + 1)χ. (5.13)
• The following integrability condition holds:
vr
ws
Warbs = 0, for all vr
, ws
∈ Γ(ker χ).
It is easy to see that this characterization follows from the one in Theorem 5.4. For the converse
direction, one has to consider the BGG splittings K = L0(k) and sF = L0(χ), translate their
relationship and, in particular, show that K is an involution of the standard tractor bundle. All
this can be found in [43, section 6.2].
In both characterizations we collect just minimal data allowing a reconstruction of the underlying
projective structure. From the previous exposition we know there are more typical features
of such conformal structures. For instance, the subtle condition in (5.13) implies, among other
things, that k is a homothety (and not just a conformal Killing ﬁeld) of a representative metric
for which χ is parallel. This also plays the role of Euler vector ﬁeld of the ﬁber vector structure of
the (local) identiﬁcation M ∼= T∗
M(2), cf. (5.9). Moreover, there is another accompanying object,
namely, the pure spinor η, which pairs non-trivially with χ and whose tractorial counterpart was
denoted as sE. This spinor ﬁeld satisﬁes the twistor spinor equation only in the ﬂat case.
Remarks
So far we concentrated on various interpretations and characterizations of conformal extensions
of projective structures. Concerning further relations of the two structures, there is a great space
to investigate, although a lot is already known, cf. section 5.1. In the article [42] we were able
to fully describe, in underlying projective terms, conformal Killing ﬁelds of conformal extensions
and Einstein metrics contained in the conformal class. This analysis naturally includes also the
original Patterson–Walker construction. For an illustration and later use, let us mention several
facts:
Each conformal, respectively genuine, Killing ﬁeld of a Patterson–Walker metric can be uniquely
written as a sum of several individuals, one of which is a lift of a projective, respectively aﬃne,
inﬁnitesimal symmetry. It follows that the Killing ﬁeld on T∗
M corresponding to an inﬁnitesimal
aﬃne symmetry on M is just its Hamiltonian lift with respect to the canonical symplectic
structure (5.8).
5.3 Further action: ambient constructions
Beside the properties and relations discussed so far, there are other striking features of conformal
Patterson–Walker metrics. Namely, they admit a global Feﬀerman–Graham ambient metric which,
moreover, has a very easy form as it is also the Patterson–Walker metric of the Thomas connection,
the ambient linear connection associated to the underlying projective structure. In particular, the
Feﬀerman–Graham obstruction tensor (which is the Bach tensor in dimension four) vanishes.
Following [44], we exhibit some details.
48 II Usage
Ambient relations
The ambient description for projective and conformal structures is introduced in section 3.2 and
3.4, respectively. In the following diagram we indicate the individual associations: vertically goes
the Thomas cone, respectively the Feﬀerman–Graham ambient, construction, horizontally goes
the Patterson–Walker construction, respectively its conformal version.
aﬃne (M, ) GG (M, g) Riemannian
projective (M, [D])
yy
GG (M, [g]) conformal
yy
It is an easy observation that these constructions nicely commute in the homogeneous setting.
In particular, according to the description in section 3.5, the conformal ambient space M coincides
with V = Rn+1,n+1
= V+ × V−, which is naturally identiﬁed with T∗
V+ for current needs. In the
general setting, it is not a priori clear whether these relations still hold. One can, however, trace
the individual constructions indicated in the diagram and compare the results in the upper right
corner. The key facts—and strong motivations for such an endeavour—are that the normalization
of both Thomas and Feﬀerman–Graham construction is controlled by the Ricci ﬂatness of the
respective ambient objects and this property is preserved by the Patterson–Walker construction.
Elaborating these ideas in detail, we conclude with
Theorem 5.6 ([44, Theorem 2]). Given a projective structure [D] on M, the geometric constructions
indicated in the previous diagram commute. In particular, the induced conformal structure
[g] admits a globally Ricci ﬂat Feﬀerman–Graham ambient metric g which is itself a Patterson–
Walker metric.
In the proof we just compose the concrete expressions of the Thomas ambient connection and
the Patterson–Walker extension and, in order to accomplish the ansatz of (3.10), one innocent
coordinate transformation. This in turn yields the following explicit description of the ambient
metric:
Theorem 5.7 ([44, Theorem 1]). Let D be a torsion-free aﬃne connection on M which preserves
a volume form. Denote local coordinates on M by (xA
) and the induced canonical ﬁbre coordinates
on T∗
M by (pA). Let Γ C
A B and RicAB denote the Christoﬀel symbols and the Ricci curvature of
D, respectively. Let
g = 2 dxA
dpA − 2 Γ C
A B pC dxA
dxB
be the Patterson–Walker metric induced on T∗
M by D. Then
g = 2 ρ dt dt + 2 t dt dρ +
+ t2
2 dxA
dpA − 2 Γ C
A B pC dxA
dxB
+
2ρ
n − 1
RicAB dxA
dxB
is a globally Ricci ﬂat Feﬀerman–Graham ambient metric for the conformal class [g].
Comparing with (3.11) we see that we have the simplest conceivable description: the current
ambient metric is linear in ρ, i.e., following the iterative procedure to obtain this local expression
from a representative metric, it actually stops after the ﬁrst step.
Consequences
As an immediate consequence of the existence of ambient metric, the Feﬀerman–Graham obstruction
tensor of the conformal structure whose representative metric is a Patterson–Walker metric
vanishes. This is the Bach tensor in dimension four. In general dimension, Bach tensor is not
6 Conformal theory of curves 49
conformally invariant and it has nothing to do with the obstructions tensor. However, it follows
that for Patterson–Walker metrics the Bach tensor has to vanish as well.
Beside these restrictions, there is another one that is easy to read oﬀ the current description.
Namely, the Q-curvature of a Patterson–Walker metric vanishes as well. Although it is generally
very diﬃcult to compute the Q-curvature for a given metric, things simplify signiﬁcantly in the
current situation. This is shown in [44, Theorem 3], whose proof is based on the method from [34]
and a quick analysis of the ambient Laplace operator.
Remarks
Note that the other ambient objects, namely, the cone N ⊂ M and the Euler vector ﬁeld E of
the R+-action, are described in previous local coordinates by ρ = 0 and E = t ∂t, respectively.
However, by the very construction, there is a more conceptual description. Firstly, as for any
Patterson–Walker metric, there is a homothety k of g. Secondly, the Euler vector ﬁeld of the R+action
on the Thomas cone M is an inﬁnitesimal aﬃne symmetry of , therefore its Hamiltonian
lift is a Killing ﬁeld H of g. It easily follows, that Euler ﬁeld E is the sum of H and (an appropriate)
multiple of k. Having this, it further follows that g(E, E) is the deﬁning function for N ⊂ M.
6 Conformal theory of curves
There is hardly more classical subject in diﬀerential geometry than the local geometry of curves in
Euclidean spaces. Nowadays, many attempts to pass the study to other geometries, both homogeneous
and curved, are known. In this section we present several variations of the typical construct,
the Frenet–Serret frame, in the conformal geometry.
In the introductory overview we comment on the very original construction and several branches
of its possible generalizations: from Euclidean to (curved) Riemannian, from Euclidean to (homogeneous)
M¨obius and from Riemannian to general conformal setting. In this part we follow two of
three main sources of inspiration, namely, [63, 36].
Riemannian GG conformal
Euclidean
yy
GG M¨obius
yyy
y
y
The third inspiration comes from [2], where the tractorial techniques were used for the description
of distinguished curves (conformal circles) and distinguished parametrizations of general
curves. We can push these ideas forward so that a tractor analogue of the Frenet frame is constructed,
which yields a generating set of absolute conformal invariants of a general curve. This
is described in section 6.2 following our article [62], which is based on [60]. This procedure is
presented as a direct passage from (homogeneous) M¨obius to general conformal setting, which
encompasses other approaches and, after possible expansion of tractorial formulas, allows their
comfortable comparisons.
On the way, we also meet a natural family of relative conformal invariants aﬀording further
applications and generalizations. As expected, these techniques can be adapted to other parabolic
geometries, which is indicated in concluding remarks.
6.1 History and initiation
Frenet, Serret and others
For a general curve in Euclidean space, there are several ways to quantify its curving. The idea
of Frenet and Serret in 3-dimensional case, later generalized by Jordan to any dimension, is based
on the association of a natural orthonormal frame along the curve and the expression of its rate of
50 II Usage
change with respect to the natural parametrization, i.e., the one by the arc-length. The resulting
system is known as Frenet–Serret (or just Frenet, for short) equations, whose coeﬃcients form a
distinguished set of (absolute) metric invariants of the curve.
For the arc-length parametrization of a generic curve, c : I → Rn
, the natural starting object
is the unit tangent vector e1 = c ; the whole Frenet frame is built from its higher derivatives,
c , c , etc., by the Gram–Schmidt process. Denoting the Frenet frame by (e1, . . . , en), the Frenet
equations read as
e1 = κ1e2,
ei = −κi−1ei−1 + κiei+1, for i = 2, . . . , n − 1,
en = −κn−1en−1,
(6.1)
where the functions κ1, . . . , κn−1 are the curvatures of the curve. The obvious symmetries of
the system reﬂect the fact that the frame is orthonormal. For later use, let us remark that the
expressions of initial derived vectors with respect to the Frenet frame looks like
c = e1, c = κ1e2, c = −κ2
1e1 + κ1e2 + κ1κ2e3, etc. (6.2)
Previous concepts were generalized to Riemannian manifolds at latest in the article [5] by
Blaschke. Basically, one has to substitute the Riemannian metric for the inner product and the
corresponding (Levi-Civita) covariant derivative for the ordinary ﬂat one. Beside the immediate
expressions of curvatures involving the frame constituents and their derivatives, κi = ei ·ei+1, there
are expressions in terms of initial derived vectors and volumes of corresponding parallelepipeds.
The latter formulas can be adapted easily with respect to an arbitrary parametrization of the
curve, cf. [41, 61]. This is an easy but important remark because it shows how to express absolute
invariants of curves in terms of relative ones, i.e., those which change by a functional multiple
relative to a concrete parametrization of the curve. Here the natural family of relative invariants
is formed by volumes of the parallelepipeds given by ﬁrst k derived vectors along the curve, where
k = 2, . . . , n.
With the help of relative invariants one may characterize special curves for which the full Frenet
frame is not available. For instance, Euclidean straight lines, respectively Riemannian geodesics,
are the curves for which the simplest relative invariant (corresponding to k = 2 in the previous
description) vanishes.
Cartan
The Frenet frame is a particular instance of a much more general concept, the Cartan’s moving
frame. From this perspective, the Frenet frame along a curve on M is an adapted lift to the bundle
of orthonormal frames G → M (whose total space coincides with the group G = Euc(n) of rigid
motions in the ﬂat case). If ˆc : I → G is such a lift of the curve c : I → M and ω is the Cartan
connection on G with values in the Lie algebra g = Rn
o(n) then, following the conventions as
in section 3.1, the Frenet equations (6.1) are written as
ˆc∗
ω =










1 −κ1
κ1
...
...
−κn−1
κn−1










.
The ﬁrst column in this decomposition just says that c = e1, i.e., the tangent vector of the curve
is the ﬁrst vector from the frame.
The point is that Cartan’s method allows to determine the lift by a successive reduction of the
structure group yielding the invariants with increasing order.
6 Conformal theory of curves 51
Schiemangk and Sulanke
In this manner, submanifolds of M¨obius space Sn
were studied in the series of articles starting
with [56, 63] by Schiemangk and Sulanke, where the study is subordinated to the bigger group
G = O(n + 1, 1) of conformal transformations. In the second reference, a lift to G along a generic
curve c : I → Sn
is constructed, which is, by nature, a pseudo-orthonormal moving frame in the
ambient Minkowski space Rn+1,1
. As a natural starting object (called the normed representation
of the curve) it is taken a curve of null vectors in Rn+1,1
whose tangent vectors have constant
length one.
If b : I → Rn+1,1
denotes such a lift, b , b etc. its derivatives and · the inner product in the
ambient vector space, then the assumptions yield the following couple of relations
b·b = 0, b·b = 0, b·b = −1,
b ·b = 1, b ·b = 0,
(6.3)
with the ﬁrst non-trivial pairing being b ·b . From this and the next derived vector, a conformally
invariant 1-form along the curve is extracted, namely,
ds = 4
b ·b − (b ·b )2 dt, (6.4)
where t is the initial parameter of c. Integration of this 1-form gives a natural parametrization of
the curve (up to additive constant), the so-called conformal arc-length.
At this stage it is easy to complete the full frame and to express its change with respect to the
natural parametrization. Hence a conformal analogue of Frenet equations and n − 1 distinguished
conformal invariants follow.
It has to be noted that there are many particular results predating the just described ones,
which would deserve careful comparisons. The list of related contributions is really huge and we
intentionally ignore most of them here. Satisfactory surveys can be found, e.g., in the introductions
of the articles [8, 7]. As an instance, we remark that the form (6.4) can be expressed in terms of
Euclidean curvatures so that
ds = 4
κ 2
1 + κ2
1κ2
2 dt,
which is a formula that appears repeatedly in older literature, cf. [63, Proposition 1.3]. (Here, t is
the Euclidean arc-length parameter, κ1, κ2 are given by (6.1) and also (6.2) is used.)
Fialkow
To our knowledge, the ﬁrst publications where the conformal invariants of curves are studied in full
generality (in positive deﬁnite signature) are [35, 36] by Fialkow, based on tensorial techniques.
In order to build the conformal Frenet frame and hence the distinguished conformal curvatures of
a curve in analogy to the general Riemannian setting, one has to describe a couple of conformally
invariant objects ﬁrst:
• a natural starting object (in the place of the unit tangent vector),
• a natural derivative along the curve (in the place of the restricted Levi-Civita connection),
• a natural distinguished parametrization of the curve (in the place of the arc-length parame-
ter).
In [36, sections 3–4], these objects are explicitly introduced and named the ﬁrst principal
normal, conformal derivative and conformal arc-length, respectively. Then, for generic curves, the
full frame may be constructed and the conformal curvatures extracted. Notice that there is an
exceptional curvature, completing the system of generating invariants, which does not follow from
the Frenet process and is constructed independently in [36, section 5]. Opening the article, one
quickly realizes that many formulas seem a bit complicated, if not mysterious. We are going to
show how to recover them using tractor calculus, from which—by the way—it should be clear they
cannot be easier.
52 II Usage
Before we continue, let us remark the many observed subtleties follow from a sizable freedom
in the change of covariant derivative with respect to a choice of scale, respectively Weyl structure.
It follows from (3.9) that, for any curve, a metric in the conformal class may be chosen so that the
curve is aﬃnely parametrized geodesic. This, in particular, reveals the problem with conformally
invariant notion of osculating subspaces etc.
Bailey, Eastwood and Gover
Here we introduce the tractor approach to conformal geometry of curves following the article [2]
by Bailey, Eastwood and Gover (and adapting to our notation). For the standard tractor bundle,
T → M, associated to a conformal Riemannian structure on M as in 3.12, there is the natural
insertion E[−1] ∼= T 1
→ T which we denote by X. Considering a regular curve c : I → M, a
natural conformal density along the curve is the length of its tangent vector, u := c ∈ E[1].
Hence we have a lift to the standard tractor bundle, T := u−1
X, which is the natural starting
object in the current setting.
T

I c
GG
T
gg
M
Denoting by U, U etc. the derivatives of T with respect to the standard tractor connection
along the curve and by · the standard tractor metric, the following relations are easily satisﬁed
T · T = 0, T · U = 0, T · U = −1,
U · U = 1, U · U = 0,
(6.5)
with the ﬁrst non-trivial pairing being U · U .
It is shown in [2, Proposition 2.11] that this quantity changes under reparametrizations of the
curve so that
U · U = g −2
U · U − 2S(g) , (6.6)
where the objects decorated with tildes are related to the reparametrization ˜t = g(t) and S denotes
the Schwarzian derivative,
S(g) :=
g
g
−
3
2
g
g
2
.
Consequently, vanishing of U · U determines a preferred family of parametrizations of the curve
with freedom given by the projective group of the line. In this derivation, there is no need to
rummage the tractor entries (basically, the chain rule and the Leibniz rule suﬃce). The same
result was obtained in [3, Proposition 4.1] by a direct manipulation with the expanded equation,
which is seen as direct proof of one of Cartan’s observations in [19]. Any parameter from this
preferred family is called projective.
Next, it is shown in [2, Proposition 2.12] how to characterize conformal circles in terms of
tractors. Conformal circles are distinguished curves of conformal structures in the sense of section
2.3. The name reﬂects the fact that on the M¨obius sphere they are ordinary circles. They can
be described by a system of ODEs of 3-order. Expanding the following tractorial equations and
comparing with the determining system of ODEs, it turns out that the curve is a projectively
parametrized conformal circle if and only if it obeys
U · U = 0 and U = 0. (6.7)
We are going to continue this game in the next section.
6 Conformal theory of curves 53
Remarks
At latest after the comparison of starting relations (6.3) and (6.5), it should be clear that the two
approaches have the same footing. Indeed, the curved version of the former construction means
lifting to the Feﬀerman–Graham ambient space, M, and from section 3.4 we know how such things
are related to standard tractors:
Let the lift to the ambient space be a curve contained in the null cone, N ⊂ M, whose tangent
vector ﬁeld has constant length one. The lift can be interpreted as a choice of scale along the curve
and its tangent vector ﬁeld can be uniquely extended to a vector ﬁeld of homogeneous degree −1
in the preimage of the curve with respect to the projection N → M. After factorization by the
R+-action, this yields a section of the standard tractor bundle along the curve coinciding with T
above.
6.2 The action: tractor Frenet construction
Before constructing the full Frenet frame using tractors, we prefer to start with the notion of
relative invariants and an alternative characterization of conformal circles. The starting assumptions
are as yet, in particular, we consider conformal structures of Riemannian signature, in which
case the standard tractor metric has signature (n + 1, 1), where n = dim M. Remarks on general
indeﬁnite signature follow.
Relative conformal invariants and conformal geodesics
A relative conformal invariant of weight k of a curve is a conformally invariant function, I, along
the curve that transforms under any reparametrization, ˜t = g(t), as I = g −k
I. In particular, vanishing,
respectively non-vanishing, of any relative invariant is independent of reparametrizations.
Conformal invariants of weight 0 are the absolute ones.
The easiest relative invariant one can ﬁnd by hand is the one corresponding to (6.4). Indeed,
it is an exercise to show (by continuing the reasoning behind (6.6)) that the function
U · U − (U · U )2
(6.8)
is a relative conformal invariant of weight 4. However, this is also (up to the sign) the ﬁrst nontrivial
determinant from the sequence of Gram matrices corresponding to T , U, U , U , . . . Let us
denote by ∆i the determinant determined by the ﬁrst i tractors from this sequence (in particular,
(6.8) corresponds to −∆4). It easily follows that, for i = 4, . . . , n + 2, the Gram determinant ∆i is
a relative conformal invariant of weight i(i − 3), see [62, Lemma 2.2]. Moreover, all these function
are non-positive.
Hence we have a bunch of relative invariants to play with. On the one hand, in generic cases,
integrating an appropriate root of any of these functions along the curve yields its natural conformally
invariant parametrization. Of course, we take the simplest one, −∆4, and call the corresponding
parameter conformal arc-length as before. In particular, in this parametrization, −∆4
equals identically to 1.
On the other hand, vanishing of some of these invariants (which implies vanishing of all higher
ones) dictates some degeneracy, respectively speciﬁcity, of the curve. Focusing on the simplest one,
it is not surprising that we recover the conformal circles:
Proposition 6.1 ([62, Proposition 3.3]). The following conditions are equivalent:
• The curve is a conformal circle (with and arbitrary parametrization).
• The invariant U · U − (U · U )2
vanishes identically.
• The rank 3 subbundle T , U, U ⊂ T is parallel along the curve.
This way we extend the characterization from (6.7). Another equivalent conditions are commented
in section 6.4.
54 II Usage
Tractor Frenet frame and absolute conformal invariants
Having enough relative conformal invariants of a generic curve, one may construct the absolute
ones in various ways. Constructions of Frenet type provide a natural and minimal (in the generating
sense) set of such invariants. In our situation we may proceed as follows.
Consider a generic curve with the conformal arc-length parametrization and the sequence of
derived tractors as before. The restriction of the tractor metric to the subbundle T , U, U is
non-degenerate and has signature (2, 1). Thus its orthogonal subbundle in T is complementary and
has positive deﬁnite signature. The initial tractor frame (T , U, U , U , . . . ) can be transformed
into a natural pseudo-orthonormal one (U0, U1, U2; U3, . . . ) in two steps: the ﬁrst three tractors
are taken to be
U0 := T , U1 := U, U2 := −U − 1
2 (U · U ) T , (6.9)
the rest is completed by the standard Gram–Schmidt process.
Now the usual techniques yield the tractor version of conformal Frenet equations which are
gathered in [62, formula (34)]. In particular, there are n−1 distinguished curvatures, K1, . . . , Kn−1,
among which the ﬁrst one is slightly exceptional due to the character of the construction. This is
summarized in [62, Proposition 4.2] together with immediate expressions of the curvatures in terms
of Frenet tractors and their derivatives. At this stage, it is also very easy to say something about
the exceptional curvature K1 (recall the projective family of parametrizations was determined by
vanishing of U · U ):
Proposition 6.2 ([62, Proposition 4.3]). The conformal arc-length parameter belongs to the projective
family of parameters if and only if K1 = 0.
From the conceptual point of view, we are basically done. For concrete applications and comparisons,
it is useful to express the curvatures Ki with respect to an arbitrary parametrization of
the curve. There are several strategies to do so, in [62, section 4.2] we use the relative conformal
invariants ∆j. Of course, the discussion for K1 is diﬀerent from the one for remaining curvatures
(where we just exploit well-known ideas behind the Gram–Schmidt process). Altogether, we
conclude with
Theorem 6.3 ([62, Theorem 4.6 and 4.7]). With respect to an arbitrary parametrization of the
curve, the conformal curvatures can be expressed as
Ki =
−1
2 (−∆4)− 5
2 ∆4
2
U · U − 1
2 ∆4∆4 + 9
16 ∆4
2
, for i = 1,
−(−∆4)− 1
4 (∆i+1∆i+3)
1
2 ∆i+2
−1
, for i = 2, . . . , n − 1.
Remarks on indeﬁnite signature
On a conformal manifold of indeﬁnite signature, a rough local division of curves reﬂects types of
tangent vectors, which may be space-, time- or light-like. The curves of ﬁrst two types can be lifted
to the standard tractor bundle as before (with the obvious sign adjustment in the time-like case),
in the third case we have to consider higher derivatives. In any case, though the basic ideas and
instruments described above remain, some subtleties appear.
Let us consider space-, respectively time-like, curves here (light-like curves are dealt individually
in the next section). The ﬁrst diﬀerence to the previous discussion concerns our favourite relative
invariants: because now the standard tractor bundle allows isotropic subspaces of higher dimension,
vanishing of ∆i does not imply the determining tractors are linearly dependent. In particular, in the
characterization of conformal circles (which are well deﬁned for any space- or time-like direction)
in Proposition 6.1, the second condition is weaker than the others. Also, these invariants may have
various signs which should be taken into account when deﬁning the conformal arc-length. In the
construction of the tractor Frenet frame, both the subbundle T , U, U ⊂ T and its orthogonal
complement may have sundry signatures. Hence the construction has to be adapted with regard
to a ﬁner classiﬁcation of types of curves, which can be done, e.g., via the sequence of signatures
of subbundles generated by the derived tractors of increasing order. This is why the discussion
branches quickly and it is fully manageable only in concrete situations, cf. [62, Remark 4.4(3)].
6 Conformal theory of curves 55
6.3 Further action: r-null curves
For the light-like curves, aka null curves, one has to ﬁnd a substitute for the density u = c ∈ E[1]
that was formerly used for the lift to T and that vanishes now. This leads us to the notion of ‘order
of nullity’, respectively r-null curves, which are the curves c : I → M whose ﬁrst r derived vectors
c , . . . , c(r)
are linearly independent and null, whereas c(r+1)
is not null. Although the individual
higher derivatives depend on the choice of scale according to (3.9), the notion of r-null curves
is well deﬁned. On conformal manifolds of signature (p, q), the highest possible r equals to the
smaller of the numbers p and q.
For an r-null curve, the norm of c(r+1)
provides an appropriate substitute for u ∈ E[1], hence
the lift to T . In this vein, space- and time-like curves can be seen as 0-null curves and with this
perspective one should read the following outcomes. Indeed, most of previous concepts generalize
swimmingly up to some shift in orders related to the value of r.
Projective parametrizations and relative invariants
Concerning the analogue of the starting relations (6.5), the pattern is the same only shifted so
that the ﬁrst non-trivial pairing is U(r+1)
· U(r+1)
. Analysis of the change of this quantity with
regard to a reparametrization of the curve reveals a similar behaviour as in (6.6). Hence vanishing
of this function determines a preferred parametrizations of the curve with the projective freedom:
Proposition 6.4 ([62, Proposition 5.1]). For any admissible r, any r-null curve carries a preferred
family of projective parameters.
Concerning the relative conformal invariants ∆i, the ﬁrst non-trivial one is ∆2r+4. Both their
total amount and particular weights depend on r so that
Lemma 6.5 ([62, Lemma 5.2]). Along any r-null curve, the Gram determinant ∆i, for i =
2r + 4, . . . , n + 2, is a relative conformal invariant of weight i(i − 2r − 3).
As before, the simplest of these invariants can be used to deﬁne a natural parametrization of
the curve, the so-called pseudo-arc-length, provided it is nowhere vanishing. But a simple counts
show that all of them may vanish automatically even for generic r-null curves (this happens for
maximal r in the case of split signature). This is a good reason to search for another source of
relative conformal invariants.
The typical situation with vanishing ∆2r+4 occurs when the tractors T , U, . . . , U(2r+2)
are
linearly dependent. This condition can be phrased as a tractor linear ODE, whose coeﬃcients are
functions expressible in terms of pairings of the initial tractors. To this equation one may associate
the so-called Wilczynski invariants, which form a set of 2r + 1 essential invariants denoted as
Θ3, . . . , Θ2r+3. As a matter of fact, each Θi is a relative conformal invariant of weight i. Expressing
the coeﬃcients of our equation, it follows that the subleading one vanishes, in which case it is easier
to gain concrete formulas for Θi (at least for small i) and they indeed can be found in ancient
literature, cf. [57, 65]. Altogether, for an r-null curve, the ﬁrst three Wilczynski invariants are
Θ3 = 0, Θ4 = − β −
r(r + 3)
10
α +
(r + 3)(2r + 5)(5r + 4)
10(r + 1)(r + 2)(2r + 3)
α2
, Θ5 = 0,
where := U(r)
·U(r)
= ±1, α := U(r+1)
·U(r+1)
and β := U(r+2)
·U(r+2)
, see [62, Proposition 5.5].
Note that, for r = 0 and = 1 (which holds in positive deﬁnite signature), the invariant Θ4
recovers ∆4, i.e., the one in (6.8) up to the sign. For general r, the considered tractor equation is
equivalent to the trivial equation U(2r+2)
= 0 if and only if all its Wilczynski invariants vanish. In
our setting, this is equivalent to vanishing of U(j)
·U(j)
, for any j = r+1, . . . , 2r+2. This innocent
remark serves as a preparation to an appropriate generalization of the notion of conformal circles.
56 II Usage
Conformal null helices
For any space- or time-like curve with an arbitrary parametrization, c : I → M, a metric in the
conformal class may be chosen so that c = 0. For null curves, this condition holds only if the curve
is null geodesic, i.e. the null curve whose acceleration vector c (and consequently all higher order
vectors) is proportional to the tangent one c . These curves, although very important, cannot be
lifted to the standard tractor bundle in the sense considered here. That is why they are excluded
from our considerations.
For an r-null curve, it follows that the vectors c , . . . , c(2r+1)
are linearly independent for any
choice of scale. Therefore we cannot achieve any of the conditions c = 0, . . . , c(2r+1)
= 0 by a
conformal change of metric. The simplest conceivable statement is
Lemma 6.6 ([62, Lemma 5.7]). For an r-null curve with an arbitrary parametrization, a metric
in the conformal class may be chosen so that c(r+1)
·c(r+1)
= ±1 and c(2r+2)
= 0.
Now we are ready to identify the closest relatives of conformal circles among r-null curves. The
simplest conceivable generalization of Proposition 6.1 (and surrounding notions) turns out to be
Theorem 6.7 ([62, Theorem 5.8]). For an r-null curve, the following conditions are equivalent:
• The curve with a projective parametrization (i.e., satisfying U(r+1)
· U(r+1)
= 0) obeys
U(2r+2)
= 0.
• The curve with and arbitrary parametrization obeys that the subbundle T , U, . . . , U(2r+1)
⊂
T is parallel along the curve and all Wilczynski invariants vanish Θ3 = · · · = Θ2r+3 = 0.
• There is a metric in the conformal class such that c(r+1)
·c(r+1)
= ±1, c(2r+2)
= 0 and
P(c , −) = 0.
The parallelity assumption in the second condition implies ∆2r+4 = 0, but the opposite implication
does not hold in general (which should be corrected in the original formulation of the
statement in [62]). In the ﬂat case, the former condition means that the conformal r-null helix is
contained in the conformal image of a (2r + 1)-dimensional Euclidean subspace. With the third
condition, it easily follows that these curves are just the conformal images of so-called null Cartan
helices, cf. [28]. This is why we call the curves satisfying (any of) previous three conditions the
conformal r-null helices. Note that these curves are not distinguished curves in the sense of section
2.3, where they are controlled by (subsets of) g− ⊂ g. However, weakening this restriction, the
currently distinguished curves may be considered in a similar way.
Tractor Frenet frame
Constructing the tractor Frenet frame, we face the very similar problems as described at the end
of section 6.2. In contrast to the case r = 0, there is also a freedom in accomplishing the ﬁrst
step, i.e., in an analogy of the transformation (6.9). Of course, any additional choice inﬂuences
the corresponding tractor Frenet equations and hence the resulting system of absolute invariants.
This is why the construction cannot be declared as canonical.
Anyhow, we count n − r − 1 conformal curvatures of an r-null curve, some of which depend on
the described choices. But the ﬁrst of them is not inﬂuenced by any choice and its interpretation
is analogous to the one in Proposition 6.2.
Instead of a toothless general reasoning, it is more illuminating to analyse some concrete
situations. For the case of Lorentzian signature, this is done in [62, section 5.4], where we refer
for details and inspirations for further investigations.
6.4 Comparisons and remarks
As promised in the introduction, expanding our tractorial formulas, one generates concrete (and
usually ugly) tensorial expressions. These allow to compare the previously studied notions with
our current approach. In particular, we can detect the Fialkow’s objects listed in section 6.1 among
6 Conformal theory of curves 57
our tractors and their individual slots. These, and other comparisons to existing literature, are
discussed in [62, section 4.3]. Also the exceptional curvatures that appear both in Fialkow’s and
our approach can be related satisfactorily, see [62, remark 4.8].
Instead of standard tractor bundle, one may consider other ones. As a natural and prominent
candidate, there is the adjoint tractor bundle. In the conformal case, A ∼= T ∧ T and the natural
lift of curve c : I → M to A : I → A is provided by A := T ∧ U (according to the notation
above). With the corresponding tractor connection and parallel bundle metric, one may mimic the
previous development at least in detecting distinguished parametrizations, constructing families
of relative invariants and characterizing conformal circles, respectively other distinguished curves.
Since the adjoint tractor bundle carries a parallel bundle metric for any parabolic geometry
(induced by the Killing form on the background Lie algebra), we consider this object as an appropriate
candidate to experiment with. It also carries additional structures, whose role to the study
of curves is not yet investigated. Only several experiments have been done in the context of CR
and Lagrangean contact geometries.
Recently, several studies of curves in conformal and other geometries appear that use tractorial
(and related) techniques, see [31, 37, 39].
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Attachments
H ere we attach the articles whose main contributions are described in the previous chapter.
They are [17], [43] together with [44] and [62], which correspond to sections 4, 5 and
6, respectively. For reader’s convenience we collect the tables of contents of individual articles
(according to their own page numberings):
On the geometry of chains
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Parabolic contact structures, chains, and path geometries . . . . . . . . . . . . . . 4
3 Induced Cartan connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Partially integrable almost CR structures . . . . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A Projective-to-Conformal Feﬀerman-type Construction
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Projective and conformal parabolic geometries . . . . . . . . . . . . . . . . . . . . 3
3 The Feﬀerman-type construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Normalisation and characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Reduced scales and explicit normalisation . . . . . . . . . . . . . . . . . . . . . . . 24
6 Comparisons with Patterson–Walker metrics and alternative characterisation . . . 28
A Explicit matrix realisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Feﬀerman–Graham ambient metrics of Patterson–Walker metrics
1 Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Geometric construction of the ambient metric . . . . . . . . . . . . . . . . . . . . . 3
3 Vanishing Q-curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
64 Attachments
Conformal theory of curves with tractors
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2 Relative conformal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3 Conformal circles and conserved quantities . . . . . . . . . . . . . . . . . . . . . . . 119
4 Absolute conformal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 Null curves in general signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
j. differential geometry
82 (2009) 1-33
ON THE GEOMETRY OF CHAINS
Andreas ˇCap & Vojtˇech ˇZ´adn´ık
Abstract
The chains studied in this paper generalize Chern–Moser chains
for CR structures. They form a distinguished family of one dimensional
submanifolds in manifolds endowed with a parabolic contact
structure. Both the parabolic contact structure and the system of
chains can be equivalently encoded as Cartan geometries (of diﬀerent
types). The aim of this paper is to study the relation between
these two Cartan geometries for Lagrangean contact structures
and partially integrable almost CR structures.
We develop a general method for extending Cartan geometries
which generalizes the Cartan geometry interpretation of Feﬀerman’s
construction of a conformal structure associated to a CR
structure. For the two structures in question, we show that the
Cartan geometry associated to the family of chains can be obtained
in that way if and only if the original parabolic contact
structure is torsion free. In particular, the procedure works exactly
on the subclass of (integrable) CR structures.
This tight relation between the two Cartan geometries leads to
an explicit description of the Cartan curvature associated to the
family of chains. On the one hand, this shows that the homogeneous
models for the two parabolic contact structures give rise
to examples of non–ﬂat path geometries with large automorphism
groups. On the other hand, we show that one may (almost) reconstruct
the underlying torsion free parabolic contact structure
from the Cartan curvature associated to the chains. In particular,
this leads to a very conceptual proof of the fact that chain preserving
contact diﬀeomorphisms are either isomorphisms or anti–
isomorphisms of parabolic contact structures.
First author supported at diﬀerent times by projects P15747–N05 and P19500–
N13 of the “Fonds zur F¨orderung der wissenschaftlichen Forschung” (FWF). Second
author supported at diﬀerent times by the Junior Fellows program of the Erwin
Schr¨odinger Institute (ESI) and by grant 201/05/2117 of the Czech Science Foundation
(GAˇCR).
Received 05/26/2005.
1
2 A. ˇCAP & V. ˇZ´ADN´IK
1. Introduction
Parabolic contact structures are a class of geometric structures having
an underlying contact structure. They admit a canonical normal
Cartan connection corresponding to a contact grading of a simple Lie algebra.
The best known examples of such structures are non–degenerate
partially integrable almost CR structures of hypersurface type. The
construction of the canonical Cartan connection is due to Chern and
Moser ([11]) for the subclass of CR structures, and to Tanaka ([20]) in
general.
In the approach of Chern and Moser, a central role is played by a
canonical class of unparametrized curves called chains. For each point x
and each direction ξ at x, which is transverse to the contact distribution,
there is a unique chain through x in direction ξ. In addition, each
chain comes with a projective class of distinguished parametrizations.
The notion of chains easily generalizes to arbitrary parabolic contact
structures, and the chains are easy to describe in terms of the Cartan
connection.
A path geometry on a smooth manifold M is given by a smooth family
of unparametrized curves on M such that for each x ∈ M and each
direction ξ at x there is a unique curve through x in direction ξ. The
best way to encode this structure is to pass to the projectivized tangent
bundle PTM, the space of all lines in TM. Then a path geometry is
given by a line subbundle in the tangent bundle of PTM with certain
properties, see [12] and [14] for a modern presentation. It turns out
that these structures are equivalent to regular normal Cartan geometries
of a certain type, which fall under the general concept of parabolic
geometries, see Section 4.7 of [4].
In the description as a Cartan geometry, path geometries immediately
generalize to open subsets of the projectivized tangent bundle.
In particular, given a manifold M endowed with a parabolic contact
structure, the chains give rise to a path geometry on the open subset
P0TM ⊂ PTM formed by all lines transversal to the contact subbundle.
The general question addressed in this paper is how to describe the
resulting Cartan geometry on P0TM in terms of the original Cartan geometry
on M. We study this in detail in the case of Lagrangean contact
structures and, in the end, brieﬂy indicate how to deal with partially
integrable almost CR structures, which can be viewed as a diﬀerent real
form of the same complex geometric structure. Q in the structure group
P The ﬁrst observation is that P0TM can be obtained as a quotient of
the Cartan bundle G → M obtained from the parabolic contact structure.
More precisely, there is a subgroup Q in the structure group P
such that P0TM ∼= G/Q. In particular, G is a principal Q–bundle over
P0TM and the canonical Cartan connection ω ∈ Ω1(G, g) associated to
ON THE GEOMETRY OF CHAINS 3
the parabolic contact structure can be also viewed as a Cartan connection
on G → P0TM. The question then is whether the canonical Cartan
geometry ( ˜G → P0TM, ˜ω) determined by the path geometry of chains
can be constructed directly from (G → P0TM, ω).
To attack this problem, we study a class of extension functors mapping
Cartan geometries of some type (G, Q) to Cartan geometries of
another type ( ˜G, ˜P). These functors have the property that there is a
homomorphism between the two Cartan bundles, which relates the two
Cartan connections. We show that in order to obtain such a functor,
one needs a homomorphism i : Q → ˜P (which we assume to be inﬁnitesimally
injective) and a linear map α : g → ˜g which satisfy certain
compatibility conditions. There is a simple notion of equivalence for
such pairs and equivalent pairs lead to naturally isomorphic extension
functors.
There is a particular simple source of pairs (i, α) leading to extension
functors as above. Namely, one may start from an inﬁnitesimally
injective homomorphism ϕ : G → ˜G, deﬁne i to be its restriction to Q
and α to be the induced homomorphism ϕ′ of Lie algebras. In a special
case, this leads to the Cartan geometry interpretation of Feﬀerman’s
construction of a canonical conformal structure on a circle bundle over
a CR manifold.
One can completely describe the eﬀect of the extension functor associated
to a pair (i, α) on the curvature of the Cartan geometries. Apart
from the curvature of the original geometry, also the deviation from α
being a homomorphism of Lie algebras enters into the curvature of the
extended Cartan geometry.
An important feature of the special choice for ( ˜G, ˜P) that we are
concerned with, is a uniqueness result for such extension functors. We
show (see Theorem 3.4) that if the extension functor associated to a
pair (i, α) maps locally ﬂat geometries of type (G, Q) to regular normal
geometries of type ( ˜G, ˜P), then the pair (i, α) is already determined
uniquely up to equivalence. For the two parabolic contact structures
studied in this paper, we show that there exist appropriate pairs (i, α)
in 3.5 and 5.2.
In both cases, the resulting extension functor does not produce the
canonical Cartan geometry associated to the path geometry of chains in
general. We show that the canonical Cartan connection is obtained if
and only if the original parabolic contact geometry is torsion free. For
a Lagrangean contact structure this means that the two Lagrangean
subbundles are integrable, while it is the usual integrability condition
for CR structures. This ties in nicely with the Feﬀerman construction,
4 A. ˇCAP & V. ˇZ´ADN´IK
where one obtains a conformal structure for arbitrary partially integrable
almost CR structures, but the normal Cartan connection is obtained
by equivariant extension if and only if the structure is integrable
(and hence CR), see [5] and [6].
Finally, we discuss applications of our construction, which are based
on an analysis of the curvature of the canonical Cartan connection associated
to the path geometry of chains. We show that chains never are
geodesics of a linear connection, and they give rise to a torsion free path
geometry if and only if the original parabolic contact structure is locally
ﬂat. Then we show that the underlying parabolic contact structure can
be almost reconstructed from the harmonic curvature of the path geometry
of chains. In particular, this leads to a very conceptual proof
of the fact that a contact diﬀeomorphism which maps chains to chains
must (essentially) preserve the original torsion free parabolic contact
structure.
Acknowledgements. Discussions with Boris Doubrov have been very
helpful.
2. Parabolic contact structures, chains, and path geometries
In this section, we will discuss the concepts of chains and the associated
path geometry for a parabolic contact structure, focusing on the
example of Lagrangean contact structures. We only brieﬂy indicate the
changes needed to deal with general parabolic contact structures.
2.1. Lagrangean contact structures. The starting point to deﬁne
a parabolic contact structure is a simple Lie algebra g endowed with
a contact grading, i.e. a vector space decomposition g = g−2 ⊕ g−1 ⊕
g0 ⊕ g1 ⊕ g2 such that [gi, gj] ⊂ gi+j, g−2 has real dimension one, and
the bracket g−1 × g−1 → g−2 is non–degenerate. It is known that
such a grading is unique up to an inner automorphism and it exists for
each non–compact non–complex real simple Lie algebra except sl(n, H),
so(n, 1), sp(p, q), one real form of E6 and one of E7, see Section 4.2 of
[23].
Here we will mainly be concerned with the contact grading of g =
sl(n + 2, R), corresponding to the following block decomposition with
blocks of size 1, n, and 1:



g0 gL
1 g2
gL
−1 g0 gR
1
g−2 gR
−1 g0


 .
We have indicated the splittings g−1 = gL
−1 ⊕ gR
−1 respectively g1 =
gL
1 ⊕ gR
1 , which are immediately seen to be g0–invariant. Further, the
subspaces gL
−1 and gR
−1 of g−1 are isotropic for [ , ] : g−1 × g−1 → g−2.
ON THE GEOMETRY OF CHAINS 5
Put G := PGL(n + 2, R), the quotient of GL(n + 2, R) by its center.
We will view G as the quotient of the group of matrices whose determinant
has modulus one by the two–element subgroup generated by ± id
and work with representative matrices. The group G always has Lie
algebra g. For odd n, one can identify G with SL(n + 2, R). For even
n, G has two connected components, and the component containing the
identity can be identiﬁed with PSL(n + 2, R).
By G0 ⊂ P ⊂ G we denote the subgroups formed by matrices which
are block diagonal respectively block upper triangular with block sizes
1, n, and 1. Then the Lie algebras of G0 and P are g0 respectively
p := g0 ⊕ g1 ⊕ g2. For g ∈ G0, the map Ad(g) : g → g preserves
the grading while for g ∈ P one obtains Ad(g)(gi) ∈ gi ⊕ · · · ⊕ g2 for
i = −1, . . . , 2. This can be used as an alternative characterization of the
two subgroups. The reason for the choice of the speciﬁc group G with
Lie algebra g is that the adjoint action identiﬁes G0 with the group of
all automorphisms of the graded Lie algebra g−2 ⊕g−1 which in addition
preserve the decomposition g−1 = gL
−1 ⊕ gR
−1.
Let M be a smooth manifold of dimension 2n + 1 and let H ⊂ TM
be a subbundle of corank one. The Lie bracket of vector ﬁelds induces
a tensorial map L : Λ2H → TM/H, and H is called a contact structure
on M if this map is non–degenerate. A Lagrangean contact structure on
M is a contact structure H ⊂ TM together with a ﬁxed decomposition
H = L ⊕ R such that each of the subbundles is isotropic with respect
to L. This forces the two bundles to be of rank n, and L induces
isomorphisms R ∼= L∗ ⊗ (TM/H) and L ∼= R∗ ⊗ (TM/H).
In view of the description of G0 above, the following result is a special
case of general prolongation procedures [21, 17, 7], see [19] and Section
4.1 of [4] for more information on this speciﬁc case.
Theorem 2.1. Let H = L ⊕ R be a Lagrangean contact structure
on a manifold M of dimension 2n + 1. Then there exists a principal
P–bundle p : G → M endowed with a Cartan connection ω ∈ Ω1(G, g)
such that L = Tp(ω−1(gL
−1 ⊕ p)) and R = Tp(ω−1(gR
−1 ⊕ p)). The pair
(G, ω) is uniquely determined up to isomorphism provided that one in
addition requires the curvature of ω to satisfy a normalization condition
discussed in 3.6.
Similarly, for any contact grading of a simple Lie algebra g and a
choice of a Lie group G with Lie algebra g, one deﬁnes a subgroup
P ⊂ G with Lie algebra g0 ⊕ g1 ⊕ g2. One then obtains an equivalence
of categories between regular normal parabolic geometries of type
(G, P) and underlying geometric structures, which in particular include
a contact structure.
The second case of such structures we will be concerned with in this
paper, is partially integrable almost CR structures of hypersurface type,
see Section 5.
6 A. ˇCAP & V. ˇZ´ADN´IK
2.2. Chains. Let (p : G → M, ω) be the canonical Cartan geometry
determined by a parabolic contact structure. Then one obtains an
isomorphism TM ∼= G ×P (g/p) such that H ⊂ TM corresponds to
(g−1 ⊕ p)/p ⊂ g/p. Of course, we may identify g−2 ⊕ g−1 as a vector
space with g/p and use this to carry over the natural P–action to
g−2 ⊕ g−1. Let Q ⊂ P be the stabilizer of the line g−2 under this action.
By deﬁnition, this is a closed subgroup of P. Let us denote by
G0 ⊂ P the closed subgroup consisting of all elements whose adjoint
action respects the grading of g. Then G0 has Lie algebra g0 and by
Proposition 2.10 of [7], any element g ∈ P can be uniquely written in
the form g0 exp(Z1) exp(Z2) for g0 ∈ G0, Z1 ∈ g1, and Z2 ∈ g2.
Lemma 2.2. (1) An element g = g0 exp(Z1) exp(Z2) ∈ P lies in the
subgroup Q ⊂ P if and only if Z1 = 0. In particular, q = g0 ⊕ g2 and
for g ∈ Q we have Ad(g)(g−2) ⊂ g−2 ⊕ q.
(2) Let (p : G → M, ω) be the canonical Cartan geometry determined by
a parabolic contact structure. Let x ∈ M be a point and ξ ∈ TxM \ Hx
a tangent vector transverse to the contact subbundle.
Then there is a point u ∈ p−1(x) ⊂ G and a unique lift ˜ξ ∈ TuG of ξ
such that ω(u)(˜ξ) ∈ g−2. The point u is unique up to the principal right
action of an element g ∈ Q ⊂ P.
Proof. (1) We ﬁrst observe that for a nonzero element X ∈ g−2, the
map Z → [Z, X] is a bijection g1 → g−1. This is easy to verify directly
for the examples discussed in 2.1 and 5.1. For general contact gradings
it follows from the fact that [g−2, g2] consists of all multiples of the
grading element, see Section 4.2 of [23].
By deﬁnition, g ∈ Q if and only if Ad(g)(g−2) ⊂ g−2 ⊕ p. Now from
the expression g−1 = exp(−Z2) exp(−Z1)g−1
0 one immediately concludes
that Ad(g−1)(X) is congruent to −[Z1, X] ∈ g−1 modulo g−2 ⊕p. Hence
we see that g ∈ Q if and only if Z1 = 0, and the rest of (1) evidently
follows.
(2) Choose any point v ∈ p−1(x). Since the vertical bundle of G → M
equals ω−1(p), there is a unique lift η ∈ TvG of ξ such that ω(v)(η) ∈
g−2 ⊕ g−1. The assumption that ξ is transverse to Hx means that
ω(v)(η) /∈ g−1. For an element g ∈ P we can consider v ·g and Tvrg ·η ∈
Tv·gG, where v · g = rg(v) denotes the principal right action of g on v.
Evidently, Tvrg · η is again a lift of ξ and equivariancy of ω implies that
ω(v · g)(Tvrg · η) = Ad(g−1)(ω(v)(η)).
Writing ω(v)(η) = X−2 + X−1 we have X−2 = 0, so from above we
see that there is an element Z ∈ g1 such that [Z, X−2] = X−1. Putting
g = exp(Z) ∈ P we conclude that ω(v · g)(Tvrg · η) ∈ g−2 ⊕ p. Hence
putting u = v · g and subtracting an appropriate vertical vector from
Tvrg · η, we have found a couple (u, ˜ξ) as required.
ON THE GEOMETRY OF CHAINS 7
Any other choice of a preimage of x has the form u·g for some g ∈ P.
Any lift of ξ in Tu·gG is of the form Turg · ˜ξ + ζ for some vertical vector
ζ. Clearly, there is a choice for ζ such that ω(Turg · ˜ξ + ζ) ∈ g−2 if and
only if ω(Turg · ˜ξ) ∈ g−2 ⊕ p and equivariancy of ω implies that this is
equivalent to g ∈ Q. q.e.d.
This lemma immediately leads us to chains. Fix a nonzero element
X ∈ g−2. For a point x ∈ M and a line ℓ in TxM which is transverse
to Hx, we can ﬁnd a point u ∈ G such that Tup · ω−1
u (X) ∈ ℓ. Denoting
by ˜X the “constant vector ﬁeld” ω−1(X) we can consider the ﬂow of ˜X
through u and project it onto M to obtain a (locally deﬁned) smooth
curve through x whose tangent space at x is ℓ. In Section 4 of [9]
it has been shown that, as an unparametrized curve, this is uniquely
determined by x and ℓ, and it comes with a distinguished projective
family of parametrizations.
The lemma also leads us to a nice description of the space of all transverse
directions. For a point u ∈ G, we obtain a line in Tp(u)M which is
transverse to Hp(u), namely Tp(ω−1
u (g−2)). This deﬁnes a smooth map
G → PTM, where PTM denotes the projectivized tangent bundle of
M. Since P acts freely on G so does Q and hence G/Q is a smooth
manifold. By the lemma, we obtain a diﬀeomorphism from G/Q to the
open subset P0TM ⊂ PTM formed by all lines which are transverse to
the contact distribution H.
2.3. Path geometries. Classically, path geometries are associated to
certain families of unparametrized curves in a smooth manifold. Suppose
that in a manifold Z we have a smooth family of curves such that
through each point of Z there is exactly one curve in each direction. Let
PTZ be the projectivized tangent bundle of Z, i.e. the space of all lines
through the origin in tangent spaces of Z. Given a line ℓ in TxZ, we can
choose the unique curve in the family which goes through x in direction
ℓ. Choosing a local regular parametrization c : I → Z of this curve
we obtain a lift ˜c : I → PTZ by deﬁning ˜c(t) to be the line in Tc(t)Z
generated by c′(t). Choosing a diﬀerent regular parametrization, we
just obtain a reparametrization of ˜c, so the submanifold ˜c(I) ⊂ PTZ is
independent of all choices. These curves foliate PTZ, and their tangent
spaces give rise to a line subbundle E ⊂ TPTZ.
This subbundle has a special property. Similarly to the tautological
line bundle on a projective space, a projectivized tangent bundle carries
a tautological subbundle Ξ ⊂ TPTZ of rank dim(Z). By deﬁnition,
given a line ℓ ⊂ TzZ, a tangent vector ξ ∈ TℓPTZ lies in Ξℓ if and
only if its image under the tangent map of the projection PTZ → Z
lies in the line ℓ. By construction, the line subbundle E associated to a
family of curves as above always is contained in Ξ and is transverse to
the vertical subbundle V of PTZ → Z. Hence we see that Ξ = E ⊕ V .
8 A. ˇCAP & V. ˇZ´ADN´IK
Conversely, having given a decomposition Ξ = E ⊕V of the tautological
bundle, we can project the leaves of the foliation of PTZ deﬁned
by E to the manifold Z to obtain a smooth family of curves in Z with
exactly one curve through each point in each direction. Hence one may
use the decomposition Ξ = E ⊕ V as an alternative deﬁnition of such a
family of curves, and this decomposition is usually referred to as a path
geometry on Z. It is easy to verify that the Lie bracket of vector ﬁelds
induces an isomorphism E ⊗ V → TPTZ/Ξ.
It turns out that path geometries also admit an equivalent description
as regular normal parabolic geometries. Putting m := dim(Z) − 1 we
consider the Lie algebra ˜g := sl(m+2, R) with the |2|–grading obtained
by a block decomposition



˜g0 ˜gE
1 ˜g2
˜gE
−1 ˜g0 ˜gV
1
˜g−2 ˜gV
−1 ˜g0


 ,
as in 2.1, but this time with blocks of size 1, 1, and m. Hence ˜gE
±1
has dimension 1 while ˜gV
±1 and ˜g±2 are all m–dimensional. Put ˜G :=
PGL(m + 2, R) and let ˜G0 ⊂ ˜P ⊂ ˜G be the subgroups formed by
matrices which are block diagonal respectively block upper triangular
with block sizes 1, 1, and m. Then ˜G0 and ˜P have Lie algebras ˜g0
respectively ˜p := ˜g0 ⊕ ˜g1 ⊕ ˜g2, where ˜g1 = ˜gE
1 ⊕ ˜gV
1 .
The adjoint action identiﬁes ˜G0 with the group of automorphisms
of the graded Lie algebra ˜g−2 ⊕ ˜g−1 which in addition preserve the
decomposition ˜g−1 = ˜gE
−1 ⊕ ˜gV
−1. Hence the following result is a special
case of the general prolongation procedures [21, 17, 7], see Section 4.7
of [4] for this speciﬁc case.
Theorem 2.3. Let ˜Z be a smooth manifold of dimension 2m + 1
endowed with transversal subbundles E and V in T ˜Z of rank 1 and
m, respectively, and put Ξ := E ⊕ V ⊂ T ˜Z. Suppose that the Lie
bracket of two sections of V is a section of Ξ and that the tensorial
map E ⊗ V → T ˜Z/Ξ induced by the Lie bracket of vector ﬁelds is an
isomorphism.
Then there exists a principal bundle ˜p : ˜G → ˜Z with structure group
˜P endowed with a Cartan connection ˜ω ∈ Ω1( ˜G, ˜g) such that E =
T ˜p(˜ω−1(˜gE
−1⊕˜p)) and V = T ˜p(˜ω−1(˜gV
−1⊕˜p)). The pair ( ˜G, ˜ω) is uniquely
determined up to isomorphism provided that ˜ω is required to satisfy a
normalization condition discussed in 3.6.
In particular, a family of paths on Z as before gives rise to a Cartan
geometry on PTZ. This immediately generalizes to the case of an open
subset of PTZ, i.e. the case where paths are only given through each
point in an open set of directions.
ON THE GEOMETRY OF CHAINS 9
It turns out that for m = 2, the assumptions of the theorem already
imply that the subbundle V ⊂ T ˜Z is involutive. Then ˜Z is automatically
locally diﬀeomorphic to a projectivized tangent bundle in such a
way that V is mapped to the vertical subbundle and Ξ to the tautological
subbundle. Hence for m = 2, the geometries discussed in the
theorem are locally isomorphic to path geometries.
2.4. The path geometry of chains. From 2.2 we see that for a manifold
M endowed with a parabolic contact structure the chains give rise
to a path geometry on the open subset ˜M := P0TM of the projectivized
tangent bundle of M. We can easily describe the corresponding conﬁguration
of bundles explicitly. Denoting by (p : G → M, ω) the Cartan
geometry induced by the parabolic contact structure, we know from 2.2
that ˜M = G/Q, where Q ⊂ P denotes the stabilizer of the line in g/p
corresponding to g−2 ⊂ g−2⊕g−1. In particular, G is a Q–principal bundle
over ˜M and ω is a Cartan connection on G → ˜M. This implies that
T ˜M = G ×Q g/q, and the tangent map to the projection π : ˜M → M
corresponds to the obvious projection g/q → g/p. In particular, the
vertical bundle V = ker(Tπ) corresponds to p/q ⊂ g/q. From the construction
of the isomorphism G/Q → ˜M in 2.2, it is evident that the
tautological bundle Ξ corresponds to (g−2 ⊕p)/q. By part (1) of Lemma
2.2, the subspace (g−2 ⊕q)/q ⊂ g/q is Q–invariant, thus it gives rise to a
line subbundle E in Ξ, which is complementary to V . By construction,
this exactly describes the path geometry determined by the chains.
If dim(M) = 2n + 1, then the dimension of ˜M is 4n + 1. Put
˜G := PGL(2n + 2, R) and let ˜P ⊂ ˜G be the subgroup described in
2.3. Then by Theorem 2.3 the path geometry on ˜M gives rise to a
canonical principal bundle ˜G → ˜M with structure group ˜P endowed
with a canonical normal Cartan connection ˜ω ∈ Ω1( ˜G, ˜g). The main
question now is whether there is a direct relation between the Cartan
geometries (G → ˜M, ω) and ( ˜G → ˜M, ˜ω).
The only reasonable way to relate these two Cartan geometries is
to consider a morphism j : G → ˜G of principal bundles and compare
the pull–back j∗ ˜ω to ω. This means that j is equivariant, so we ﬁrst
have to choose a group homomorphism i : Q → ˜P and require that
j(u · g) = j(u) · i(g) for all g ∈ Q. Having chosen i and j, we have
j∗ ˜ω ∈ Ω1(G, ˜g) and the only way to directly relate this to ω ∈ Ω1(G, g)
is to have j∗ ˜ω = α ◦ ω for some linear map α : g → ˜g. If we have such
a relation, then we can immediately recover ˜G from G. Consider the
map Φ : G × ˜P → ˜G deﬁned by Φ(u, ˜g) := j(u) · ˜g. Equivariancy of j
immediately implies that Φ(u · g, ˜g) = Φ(u, i(g)˜g), so Φ descends to a
bundle map G ×Q
˜P → ˜G, where the left action of Q on ˜P is deﬁned via
i. This is immediately seen to be an isomorphism of principal bundles,
so ˜G is obtained from G by an extension of structure group. Under this
10 A. ˇCAP & V. ˇZ´ADN´IK
isomorphism, the given morphism j : G → ˜G corresponds to the natural
inclusion G → G ×Q
˜P induced by u → (u, e).
3. Induced Cartan connections
In this section, we study the problem of extending Cartan connections.
We derive the basic results in the setting of general Cartan geometries,
and then specialize to the case of parabolic contact structures
and, in particular, Lagrangean contact structures. Some of the developments
in 3.1 and 3.3 below are closely related to [15, 22].
3.1. Extension functors for Cartan geometries. Motivated by the
last observations in 2.4, let us consider the following problem. Suppose
we have given Lie groups G and ˜G with Lie algebras g and ˜g, closed
subgroups Q ⊂ G and ˜P ⊂ ˜G, and a homomorphism i : Q → ˜P which
is inﬁnitesimally injective, i.e. such that i′ : q → ˜p is injective. Given a
Cartan geometry (p : G → N, ω) of type (G, Q), we put ˜G := G ×Q
˜P
and denote by j : G → ˜G the canonical map. Since i is inﬁnitesimally
injective, this is an immersion, i.e. Tuj is injective for all u ∈ G. We
want to understand under which conditions on a linear map α : g → ˜g
there is a Cartan connection ˜ω ∈ Ω1( ˜G, ˜g) such that j∗ ˜ω = α ◦ ω, and if
so, whether ˜ω is uniquely determined.
Proposition 3.1. There is a Cartan connection ˜ω on ˜G such that
j∗ ˜ω = α◦ω if and only if the pair (i, α) satisﬁes the following conditions:
(1) α ◦ Ad(g) = Ad(i(g)) ◦ α for all g ∈ Q.
(2) On the subspace q ⊂ g, the map α restricts to the derivative i′ of
i : Q → ˜P.
(3) The map α : g/q → ˜g/˜p induced by α is a linear isomorphism.
If these conditions are satisﬁed, then ˜ω is uniquely determined.
Proof. Let us ﬁrst assume that there is a Cartan connection ˜ω on ˜G
such that j∗ ˜ω = α ◦ ω. For u ∈ G, the tangent space Tj(u)
˜G is spanned
by Tuj(TuG) and the vertical subspace Vj(u)
˜G. The behavior of ˜ω on
the ﬁrst subspace is determined by the fact that j∗ ˜ω = α ◦ ω, while on
the second subspace ˜ω has to reproduce the generators of fundamental
vector ﬁelds. Hence the restriction of ˜ω to j(G) is determined by the
fact that j∗ ˜ω = α◦ω. By deﬁnition of ˜G, any point ˜u ∈ ˜G can be written
as j(u) · ˜g for some u ∈ G and some ˜g ∈ ˜P, so uniqueness of ˜ω follows
from equivariancy.
Still assuming that ˜ω exists, condition (1) follows from equivariancy
of j, ω, and ˜ω. Equivariancy of j also implies that for A ∈ q and
the corresponding fundamental vector ﬁeld ζA we get Tj ◦ ζA = ζi′(A).
Thus condition (2) follows from the fact that both ω and ˜ω reproduce
the generators of fundamental vector ﬁelds. Let p : G → N and ˜p :
˜G → N be the bundle projections, so ˜p ◦ j = p. For ξ ∈ TuG we
ON THE GEOMETRY OF CHAINS 11
have α(ω(ξ)) = ˜ω(Tuj · ξ), so if this lies in ˜p then Tuj · ξ is vertical.
But then ξ is vertical and hence ω(ξ) ∈ q. Therefore, the map α is
injective, and since both G and ˜G admit a Cartan connection, we must
have dim(g/q) = dim(N) = dim(˜g/˜p), so (3) follows.
Conversely, suppose that (1)–(3) are satisﬁed for (i, α) and ω is given.
For ˜u ∈ ˜G and ˜ξ ∈ T˜u
˜G we can ﬁnd elements u ∈ G, ξ ∈ TuG, A ∈ ˜p,
and ˜g ∈ ˜P such that ˜u = j(u) · ˜g and ˜ξ = Tr˜g · (Tj · ξ + ζA). Then
we deﬁne ˜ω(˜ξ) := Ad(˜g)−1(α(ω(ξ)) + A). Using properties (1) and (2)
one veriﬁes that this is independent of all choices. By (3), it deﬁnes a
linear isomorphism T˜u
˜G → ˜g, and the remaining properties of a Cartan
connection are easily veriﬁed directly. q.e.d.
Any pair (i, α) which satisﬁes the properties (1)–(3) of the proposition
gives rise to an extension functor from Cartan geometries of type (G, Q)
to Cartan geometries of type ( ˜G, ˜P): Starting from a geometry (p : G →
N, ω) of type (G, Q), one puts ˜G := G ×Q
˜P (with Q acting on ˜P via
i) and deﬁnes ˜ω ∈ Ω1( ˜G, ˜g) to be the unique Cartan connection on ˜G
such that j∗ ˜ω = α ◦ ω, where j : G → ˜G is the canonical map. For
a morphism ϕ : G1 → G2 between geometries of type (G, Q), we can
consider the principal bundle map Φ : ˜G1 → ˜G2 induced by ϕ × id ˜P . By
construction, this satisﬁes Φ ◦ j1 = j2 ◦ ϕ and we obtain
j∗
1Φ∗
˜ω2 = ϕ∗
j∗
2 ˜ω2 = ϕ∗
(α ◦ ω2) = α ◦ ϕ∗
ω2 = α ◦ ω1.
But ˜ω1 is the unique Cartan connection whose pull–back along j1 coincides
with α ◦ ω1, which implies that Φ∗ ˜ω2 = ˜ω1, and hence Φ is a
morphism of Cartan geometries of type ( ˜G, ˜P).
There is a simple notion of equivalence for pairs (i, α). We call (i, α)
and (ˆi, ˆα) equivalent and write (i, α) ∼ (ˆi, ˆα) if and only if there is an
element ˜g ∈ ˜P such that ˆi(g) = ˜g−1i(g)˜g and ˆα = Ad(˜g−1) ◦ α. Notice
that if (i, α) satisﬁes conditions (1)–(3) of the proposition, then so does
any equivalent pair. In order to distinguish between diﬀerent extension
functors, for a geometry (p : G → M, ω) of type (G, Q) we will often
denote the geometry of type ( ˜G, ˜P) obtained using (i, α) by (G×i
˜P, ˜ωα).
Lemma 3.1. Let (i, α) and (ˆi, ˆα) be equivalent pairs satisfying conditions
(1)–(3) of the proposition. Then the resulting extension functors
for Cartan geometries are naturally isomorphic.
Proof. By assumption, there is an element ˜g ∈ ˜P such that ˆi(g) =
˜g−1i(g)˜g and ˆα = Ad(˜g−1) ◦ α. Let j : G → G ×i
˜P and ˆj : G → G ×ˆi
˜P
be the natural inclusions, and consider the map r˜g ◦ j : G → G ×i
˜P.
Evidently, we have j(u · g) · ˜g = j(u) · ˜g · ˆi(g). Hence, by the last
observation in 2.4, we obtain an isomorphism Ψ : G ×ˆi
˜P → G ×i
˜P such
that Ψ ◦ ˆj = r˜g ◦ j. Now we compute
ˆj∗
Ψ∗
˜ωα = j∗
(r˜g
)∗
˜ωα = Ad(˜g−1
) ◦ j∗
˜ωα = ˆα ◦ ω.
12 A. ˇCAP & V. ˇZ´ADN´IK
By uniqueness, Ψ∗ ˜ωα = ˜ωˆα, so Ψ is a morphism of Cartan geometries. It
is clear from the construction that this deﬁnes a natural transformation
between the two extension functors and an inverse can be constructed
in the same way using ˜g−1 rather than ˜g. q.e.d.
3.2. The relation to the Feﬀerman construction. There is a simple
source of pairs (i, α) which satisfy conditions (1)–(3) of Proposition
3.1. Suppose that ϕ : G → ˜G is an inﬁnitesimally injective homomorphism
of Lie groups such that ϕ(Q) ⊂ ˜P. Then i := ϕ|Q : Q → ˜P is an
inﬁnitesimally injective homomorphism and α := ϕ′ : g → ˜g is a Lie algebra
homomorphism. Then condition (2) of Proposition 3.1 is satisﬁed
by construction, while condition (1) easily follows from diﬀerentiating
the equation ϕ(ghg−1) = ϕ(g)ϕ(h)ϕ(g)−1. Hence the only nontrivial
condition is (3). Note that if (i, α) is obtained from ϕ in this way, than
any pair equivalent to (i, α) is obtained in the same way from the map
g → ˜gϕ(g)˜g−1 for some ˜g ∈ ˜G. The main feature of such pairs is that α
is a homomorphism of Lie algebras.
In this setting, one may actually go one step further. Suppose we have
ﬁxed an inﬁnitesimally injective ϕ : G → ˜G and a closed subgroup ˜P ⊂
˜G. Then we put Q := ϕ−1( ˜P) ⊂ G to obtain a pair (i := ϕ|Q, α := ϕ′)
and hence, if (3) holds, an extension functor from Cartan geometries
of type (G, Q) to geometries of type ( ˜G, ˜P). For a closed subgroup
P ⊂ G with Q ⊂ P, one gets a functor from geometries of type (G, P)
to geometries of type (G, Q) as described in 2.2. Given a geometry
(p : G → M, ω) of type (G, P), one deﬁnes ˜M := G/Q = G ×P (P/Q)
and (G → ˜M, ω) is a geometry of type (G, Q). Combining with the
above, one gets a functor from geometries of type (G, P) to geometries
of type ( ˜G, ˜P).
The most important example of this is the Cartan geometry interpretation
of Feﬀerman’s construction of a Lorentzian conformal structure
on the total space of a certain circle bundle over a CR manifold, see
[13]. In this case G = SU(n + 1, 1), ˜G = SO(2n + 2, 2), and ϕ is the
evident inclusion. Putting ˜P the stabilizer of a real null line ℓ ⊂ R2n+4
in ˜G, the group Q = G ∩ ˜P is the stabilizer of ℓ in G. Evidently, this is
contained in the stabilizer P ⊂ G of the complex null line spanned by
ℓ, and P/Q ∼= RP1 ∼= S1. Hence the above procedure deﬁnes a functor,
which to a parabolic geometry of type (G, P) on M associates a parabolic
geometry of type ( ˜G, ˜P) on the total space ˜M of a circle bundle
over M. More details about this can be found in [3, 5, 6]. Recently,
analogs of the Feﬀerman construction have been developed for certain
generic distributions in low dimensions, see [18, 2].
3.3. The eﬀect on curvature. We next discuss the eﬀect of extension
functors of the type discussed in 3.1 on the curvature of Cartan
ON THE GEOMETRY OF CHAINS 13
geometries. This will show speciﬁc features of the special case discussed
in 3.2.
For a Cartan connection ω on a principal P–bundle G → M with values
in g, one initially deﬁnes the curvature K ∈ Ω2(G, g) by K(ξ, η) :=
dω(ξ, η)+[ω(ξ), ω(η)]. This measures the amount to which the Maurer–
Cartan equation fails to hold. The deﬁning properties of a Cartan connection
immediately imply that K is horizontal and P–equivariant. In
particular, K(ξ, η) = 0 for all η provided that ξ is vertical or, equivalently,
that ω(ξ) ∈ p.
Using the trivialization of TG provided by ω, one can pass to the
curvature function κ : G → L(Λ2(g/p), g), which is characterized by
κ(u)(X + p, Y + p) := K(u)(ω−1
(X), ω−1
(Y )).
This is well deﬁned by horizontality of K, and equivariancy of K easily
implies that κ is equivariant for the natural P–action on the space
L(Λ2(g/p), g), which is induced from the adjoint action on all copies of
g.
Using the setting of 3.1, suppose that (i : Q → ˜P, α : g → ˜g) is
a pair satisfying the conditions (1)–(3) of Proposition 3.1. Consider
the map g × g → ˜g deﬁned by (X, Y ) → [α(X), α(Y )]˜g − α([X, Y ]g),
which measures the deviation from α being a homomorphism of Lie
algebras. This map is evidently skew symmetric. By condition (1),
α◦Ad(g) = Ad(i(g))◦α for all g ∈ Q, which inﬁnitesimally implies that
α ◦ ad(X) = ad(i′(X)) ◦ α for all X ∈ q, and by condition (2) we have
i′(X) = α(X) in this case. Hence this map vanishes if one of the entries
is from q ⊂ g, and we obtain a well deﬁned linear map Λ2(g/q) → ˜g.
By condition (3), α induces a linear isomorphism α : g/q → ˜g/˜p, and
we conclude that we obtain a well deﬁned map Ψα : Λ2(˜g/˜p) → ˜g by
putting
Ψα( ˜X + ˜p, ˜Y + ˜p) = [α(X), α(Y )] − α([X, Y ]),
where α(X) + ˜p = ˜X + ˜p and α(Y ) + ˜p = ˜Y + ˜p.
Proposition 3.3. Let (i, α) be a pair satisfying conditions (1)–(3)
of Proposition 3.1. Let (p : G → N, ω) be a Cartan geometry of type
(G, Q), let (G ×i
˜P, ˜ωα) be the geometry of type ( ˜G, ˜P) obtained using
the extension functor associated to (i, α), and let j : G → G ×i
˜P be the
natural map.
Then the curvature functions κ and ˜κ of the two geometries satisfy
˜κ(j(u))( ˜X, ˜Y ) = α(κ(u)(α−1
( ˜X), α−1
( ˜Y ))) + Ψα( ˜X, ˜Y ),
for any ˜X, ˜Y ∈ ˜g/˜p, and this completely determines ˜κ.
In particular, if ω is ﬂat, then ˜ω is ﬂat if and only if α is a homomorphism
of Lie algebras.
14 A. ˇCAP & V. ˇZ´ADN´IK
Proof. By deﬁnition, j∗ ˜ωα = α ◦ ω, and hence j∗d˜ωα = α ◦ dω. This
immediately implies that for the curvatures K and ˜K and ξ, η ∈ X(G)
we get
˜K(j(u))(Tj · ξ, Tj · η) = α(dω(u)(ξ, η)) + [α(ω(u)(ξ)), α(ω(u)(η))].
On the other hand, we get
α(K(u)(ξ, η)) = α(dω(u)(ξ, η)) + α([ω(u)(ξ), ω(u)(η)]).
Now the formula for ˜κ(j(u)) follows immediately from the deﬁnition
of the curvature functions. Since ˜κ is ˜P–equivariant, it is completely
determined by its restriction to j(G). The ﬁnal claim follows directly,
since Ψα vanishes if and only if α is a homomorphism of Lie algebras.
q.e.d.
3.4. Uniqueness. A crucial fact for the further development is that,
passing from parabolic contact structures to the associated path geometries
of chains, there is actually no freedom in the choice of the pair (i, α)
up to equivalence as introduced in 3.1 above. This result certainly is
valid in a more general setting but it seems to be diﬃcult to give a nice
formulation for conditions one has to assume.
Therefore we return to the setting of Section 2, i.e. G is semisimple,
P ⊂ G is obtained from a contact grading, Q is the subgroup described
in 2.2, and ˜G and ˜P correspond to path geometries in the appropriate
dimension as in 2.3. In this setting we can now prove:
Theorem 3.4. Let (i, α) and (ˆi, ˆα) be pairs satisfying conditions (1)–
(3) of Proposition 3.1. Suppose that there is a Cartan geometry (p : G →
M, ω) of type (G, Q) such that there is an isomorphism between the
geometries of type ( ˜G, ˜P) obtained using (i, α) and (ˆi, ˆα), which covers
the identity on M. Then (i, α) and (ˆi, ˆα) are equivalent.
Proof. Using the notation of the proof of Lemma 3.1, suppose that
we have an isomorphism Ψ : G ×i
˜P → G ×ˆi
˜P of principal bundles which
covers the identity on M and has the property that Ψ∗ ˜ωˆα = ˜ωα. Let
us denote by j and ˆj the natural inclusions of G into the two extended
bundles. Since Ψ covers the identity on M, there must be a smooth
function ϕ : G → ˜P such that Ψ(j(u)) = ˆj(u) · ϕ(u).
By construction we have j(u·g) = j(u)·i(g) and ˆj(u·g) = ˆj(u)·ˆi(g),
and using the fact that Ψ is ˜P–equivariant we obtainˆi(g) = ϕ(u)i(g)ϕ(u·
g)−1. On the other hand, diﬀerentiating the equation Ψ(j(u)) = ˆj(u) ·
ϕ(u), we obtain
(TΨ ◦ Tj) · ξ = (Trϕ(u)
◦ Tˆj) · ξ + ζδϕ(u)(ξ)(Ψ(j(u)))
where δϕ ∈ Ω1(G, ˜p) denotes the left logarithmic derivative of ϕ : G →
˜P. Applying ˜ωˆα to the left hand side of this equation, we simply get
(j∗
Ψ∗
˜ωˆα)(ξ) = (j∗
˜ωα)(ξ) = α(ω(ξ)).
ON THE GEOMETRY OF CHAINS 15
Applying ˜ωˆα to the right hand side, we obtain
(ˆj∗
(rϕ(u)
)∗
˜ωˆα)(ξ) + δϕ(u)(ξ) =
Ad(ϕ(u)−1
)((ˆj∗
˜ωˆα)(ξ)) + δϕ(u)(ξ) =
Ad(ϕ(u)−1
)(ˆα(ω(ξ))) + δϕ(u)(ξ),
and we end up with the equation
(∗) α(ω(ξ)) = Ad(ϕ(u)−1
)(ˆα(ω(ξ))) + δϕ(u)(ξ)
for all ξ ∈ TG. Together with the relation between i and ˆi derived
above, this shows that it suﬃces to show that ϕ(u) is constant to prove
that (i, α) ∼ (ˆi, ˆα).
By construction, δϕ(u) has values in ˜p, so projecting equation (∗) to
˜g/˜p implies that
α(ω(ξ)) + ˜p = Ad(ϕ(u)−1
)(ˆα(ω(ξ)) + ˜p),
for all ξ ∈ TuG, where Ad is the action of ˜P on ˜g/˜p induced by the
adjoint action. By property (3) from Proposition 3.1 this implies that
α = Ad(ϕ(u)−1) ◦ ˆα, so we see that Ad(ϕ(u)−1) must be independent
of u. Hence we must have ϕ(u) = ˜g1ϕ1(u) for some element ˜g1 ∈ ˜P
and a smooth function ϕ1 : G → ˜P which has values in the kernel of
Ad. As in 2.2, any element of ˜P can be uniquely written in the form
˜g0 exp( ˜Z1) exp( ˜Z2) with ˜g0 ∈ ˜G0 and ˜Zi ∈ ˜gi, and such an element lies
in the kernel of Ad if and only if Ad(˜g0) restricts to the identity on ˜g−
and ˜Z1 = 0. Since ˜p+ is dual to ˜g− and ˜g0 injects into L(˜g−, ˜g−) the
ﬁrst condition implies that Ad(˜g0) = id˜g. Since ˜G = PGL(k, R) for
some k, this implies that ˜g0 is the identity.
Hence ϕ1 has values in exp(˜g2) and therefore δϕ(u) has values in ˜g2.
Projecting equation (∗) to ˜g/˜g2, we obtain
α(ω(ξ)) + ˜g2 = Ad(ϕ(u)−1
)(ˆα(ω(ξ)) + ˜g2),
where this time Ad denotes the natural action on ˜g/˜g2. But by [23,
Lemma 3.2] an element of ˜g2 vanishes provided that all brackets with
elements of ˜g−1 vanish, and this easily implies that ϕ1(u) is the identity
and so ϕ is constant. q.e.d.
This result has immediate consequences on the problem of describing
the path geometry of chains associated to a parabolic contact structure.
If we start with the homogeneous model G/P for a parabolic contact
geometry, the induced path geometry of chains is deﬁned on the homogeneous
space G/Q. To obtain this by an extension functor as described
in 3.1, we need a homomorphism i : Q → ˜P and a linear map α : g → ˜g,
where ( ˜G, ˜P) gives rise to path geometries in the appropriate dimension.
The pair (i, α) has to satisfy conditions (1)–(3) of Proposition
3.1 in order to give rise to an extension functor. The only additional
16 A. ˇCAP & V. ˇZ´ADN´IK
condition is that the extended geometry (G ×i
˜P, ˜ωα) obtained from
(G → G/Q, ωMC ) is regular and normal. (Here ωMC denotes the left
Maurer–Cartan form.) By Theorem 2.3, a regular normal parabolic geometry
of type ( ˜G, ˜P) is uniquely determined by the underlying path
geometry, which is encoded into (G → G/Q, ωMC), see 2.4.
The theorem above then implies that (i, α) is uniquely determined up
to equivalence. In view of Lemma 3.1, the extension functor obtained
from (i, α) is (up to natural isomorphism) the only extension functor
of the type discussed in 3.1 which produces the right result for the
homogeneous model (and hence for locally ﬂat geometries).
The ﬁnal step is then to study under which conditions on a geometry
of type (G, P), the extension functor associated to (i, α) produces a
regular normal geometry of type ( ˜G, ˜P).
3.5. Let us return to the case of Lagrangean contact structures as
discussed in 2.1. By deﬁnition, we have G = PGL(n + 2, R) and P ⊂ G
is the subgroup of all matrices which are block upper triangular with
blocks of sizes 1, n, and 1. From part (1) of Lemma 2.2 one immediately
concludes that Q ⊂ P is the subgroup formed by all matrices of the block
form 

p 0 s
0 R 0
0 0 q

 ,
such that |pq det(R)| = 1. Since the corresponding manifolds have dimension
2n + 1, the right group for the path geometry deﬁned by the
chains is ˜G = PGL(2n + 2, R). The subgroup ˜P ⊂ ˜G is given by the
classes of those matrices which are block upper triangular with blocks
of sizes 1, 1, 2n. In the sequel, we will always further split the last block
into two blocks of size n.
Consider the (well deﬁned) smooth map i : Q → ˜P and the linear
map α : g → ˜g deﬁned by
i



p 0 s
0 R 0
0 0 q


 :=






sgn(q
p )β sgn(q
p )s
p β 0 0
0 β−1 0 0
0 0 q−1β−1R 0
0 0 0 pβ−1(R−1)t






,
α



a u d
x B v
z y c


 :=






a−c
2 d 1
2u 1
2vt
z c−a
2
1
2y −1
2xt
x v B − a+c
2 id 0
yt −ut 0 −Bt + a+c
2 id






,
where β := |p/q|, and id denotes the n × n identity matrix.
ON THE GEOMETRY OF CHAINS 17
Proposition 3.5. The map i : Q → ˜P is an injective group homomorphism
and the pair (i, α) satisﬁes conditions (1)–(3) of Proposition
3.1. Hence it gives rise to an extension functor from Cartan geometries
of type (G, Q) to Cartan geometries of type ( ˜G, ˜P).
Proof. All these facts are veriﬁed by straightforward computations,
some of which are a little tedious. q.e.d.
3.6. Regularity and normality. We next have to discuss the conditions
on the curvature of a Cartan connection which were used in
Theorems 2.1 and 2.3. If G is a semisimple group and P ⊂ G is parabolic,
then one can identify (g/p)∗ with p+, the sum of all positive
grading components, via the Killing form, see [23, Lemma 3.1]. Hence
we can view the curvature function deﬁned in 3.3 as having values in
Λ2p+ ⊗ g. Via the gradings of p+ and g, this space is naturally graded,
and the Cartan connection ω is called regular if its curvature function
has values in the part of positive homogeneity. Otherwise put, if X ∈ gi
and Y ∈ gj, then κ(u)(X + p, Y + p) ∈ gi+j+1 ⊕ · · · ⊕ gk.
Recall that a Cartan geometry is torsion free, if and only if κ has
values in Λ2p+ ⊗ p. Since elements of p+ have strictly positive homogeneity,
this subspace is contained in the part of positive homogeneity,
and any torsion free Cartan geometry is automatically regular. Hence
regularity should be viewed as a condition which avoids particularly bad
types of torsion.
On the other hand, there is a natural map ∂∗ : Λ2p+ ⊗ g → p+ ⊗ g
deﬁned by
∂∗
(Z ∧ W ⊗ A) := −W ⊗ [Z, A] + Z ⊗ [W, A] − [Z, W] ⊗ A
for decomposable elements. This is the diﬀerential in the standard complex
computing the Lie algebra homology of p+ with coeﬃcients in the
module g. This map is evidently equivariant for the natural P–action,
so in particular, ker(∂∗) ⊂ Λ2p+ ⊗ g is a P–submodule. The Cartan
connection ω is called normal if and only if its curvature has values in
this submodule.
To proceed with the program set out in the end of 3.4 we next have to
analyze the map Ψα : Λ2(˜g/˜p) → ˜g introduced in 3.3 in the special case
of the pair (i, α) from 3.5. As a linear space, we may identify ˜g/˜p with
˜g− = ˜gE
−1 ⊕ ˜gV
−1 ⊕ ˜g−2. Note that using brackets in ˜g, we may identify
˜gV
−1 with ˜gE
1 ⊗ ˜g−2 if necessary. We will view ˜g−2 as R2n = Rn ⊕ Rn
and correspondingly write X ∈ ˜g−2 as (X1, X2). By , we denote the
standard inner product on Rn.
Lemma 3.6. Viewing Ψα as an element of Λ2(˜g−)∗ ⊗˜g, it lies in the
subspace (˜gV
−1)∗ ∧ (˜g−2)∗ ⊗ ˜g0. Denoting by W0 ∈ ˜gE
1 the element whose
unique nonzero entry is equal to 1, the trilinear map ˜g−2 × ˜g−2 × ˜g−2 →
18 A. ˇCAP & V. ˇZ´ADN´IK
˜g−2 deﬁned by (X, Y, Z) → [Ψα(X, [Y, W0]), Z] is (up to a nonzero multiple)
the complete symmetrization of the map
(X, Y, Z) → X1, Y2
Z1
−Z2
.
Proof. Let x ∈ ˜gE
−1 be the element whose unique nonzero entry is
equal to 1. Then an arbitrary element of ˜g− can be written uniquely as
X + [Y, W0] + ax for X, Y ∈ ˜g−2 and a ∈ R. From the deﬁnition of α in
3.5 we obtain
α




0 −Y t
2 0
X1 0 Y1
a Xt
2 0



 =







0 0 −1
2Y t
2
1
2 Y t
1
a 0 1
2Xt
2 −1
2Xt
1
X1 Y1 0 0
X2 Y2 0 0







,
so this is congruent to X + [Y, W0] + ax modulo ˜p. Using this, one
can now insert into the deﬁning formula for Ψα from 3.3 and compute
directly that the result always has values in ˜g0, and indeed only in the
lower right 2n × 2n block. Moreover, all the entries in that block are
made up from bilinear expressions involving one entry from ˜g−2 and one
entry from ˜gV
−1, so we see that Ψα ∈ (˜g−2)∗ ∧ (˜gV
−1)∗ ⊗ ˜g0.
For X, Y ∈ ˜g−2, one next computes that the only nonzero block in
Ψα(X, [Y, W0]) (which is a 2n × 2n–matrix) is explicitly given by
1
2
X1Y t
2 + Y1Xt
2 + (Y t
2 X1 + Xt
2Y1) id X1Y t
1 + Y1Xt
1
−X2Y t
2 − Y2Xt
2 −Y2Xt
1 − X2Y t
1 − (Y t
2 X1 + Xt
2Y1) id
!
.
To obtain [Ψα(X, [Y, W0]), Z] ∈ ˜g−2 for another element Z ∈ ˜g−2, we
now simply have to apply this matrix to Z1
Z2
. Taking into account that
v, w = vtw = wtv for v, w ∈ Rn we obtain half the sum of all cyclic
permutations of
( X1, Y2 + Y1, X2 )
Z1
−Z2
,
which is three times the symmetrization of (X, Y, Z) → X1, Y2
Z1
−Z2
.
q.e.d.
Using this we can now complete the ﬁrst part of the program outlined
in the end of 3.4:
Theorem 3.6. The extension functor associated to the pair (i, α)
from 3.5 maps locally ﬂat Cartan geometries of type (G, Q) to torsion
free (and hence regular), normal parabolic geometries of type ( ˜G, ˜P).
Proof. Let (p : G → N, ω) be a locally ﬂat Cartan geometry of type
(G, Q). This means that ω has trivial curvature, so by Proposition 3.3,
ON THE GEOMETRY OF CHAINS 19
the curvature function ˜κ of the parabolic geometry (G ×i
˜P, ˜ωα) has the
property that
˜κ(j(u)) = Ψα : Λ2
(˜g/˜p) → ˜g,
where j : G → G ×i
˜P is the natural map. By the lemma above, ˜κ(j(u))
has values in ˜g0 ⊂ ˜p, and since having values in ˜p is a ˜P–invariant
property, torsion freeness follows.
Similarly, since ker(∂∗) is a ˜P–submodule in Λ2(˜g/˜p)∗⊗˜g, it suﬃces to
show that ∂∗(Ψα) = 0 to complete the proof of the theorem. This may
be checked by a direct computation, but there is a more conceptual
argument. Tracefree matrices in the lower right 2n × 2n block of ˜g0
form a Lie subalgebra isomorphic to sl(2n, R) which acts on each of the
spaces Λk(˜g/˜p)∗⊗˜g. Hence we may decompose each of them into a direct
sum of irreducible representations. Since ∂∗ is a ˜P–homomorphism, it
is equivariant for this action of sl(2n, R), and hence it can be nonzero
only between isomorphic irreducible components.
In the proof of the lemma we have noted that ˜g−2 is the standard
representation of sl(2n, R), so the explicit formula for Ψα shows that
it sits in a component isomorphic to S3R2n∗ ⊗ R2n. There is a unique
trace from this representation to S2R2n∗, and the kernel of this is well
known to be irreducible. One immediately checks that (˜g/˜p)∗ ⊗˜g cannot
contain an irreducible component isomorphic to the kernel of this trace.
Hence we can ﬁnish the proof by showing that Ψα lies in the kernel of
that trace, which is a simple direct computation. q.e.d.
This has a nice immediate application.
Corollary 3.6. Consider the homogeneous model G → G/P of Lagrangean
contact structures. Then the resulting path geometry of chains
is non–ﬂat and hence not locally isomorphic to ˜G/ ˜P, but its automorphism
group contains G. In particular, for each n ≥ 1, we obtain an
example of a non–ﬂat torsion free path geometry on a manifold of dimension
2n+1 whose automorphism group has dimension at least n2+4n+3.
Remark. (1) In [14], the author directly constructed a torsion free
path geometry from the homogeneous model of three–dimensional Lagrangean
contact structures. This construction was one of the motivations
for this paper and one of the guidelines for the right choice of the
pair (i, α). The other main guideline for this choice are the computations
needed to show that Ψα has values in ˜g0.
(2) We shall see later that in the situation of the corollary, the dimension
of the automorphism group actually equals the dimension of G. In
particular, for n = 1, one obtains a non–ﬂat path geometry on a three
manifold with automorphism group of dimension 8. To our knowledge,
this is the maximal possible dimension for the automorphism group of
a non–ﬂat path geometry in this dimension.
20 A. ˇCAP & V. ˇZ´ADN´IK
Via the interpretation of path geometries in terms of systems of second
order ODEs, we obtain examples of nontrivial systems of such ODEs
with large automorphism groups.
3.7. More on curvatures of regular normal geometries. We have
completed half of the program outlined in the end of 3.4 at this point.
Theorem 3.6 shows that the extension functor associated to the pair
(i, α) deﬁned in 3.5 produces the regular normal parabolic geometry
determined by the path geometry of chains for locally ﬂat Lagrangean
contact structures. In view of Theorem 3.4 and Lemma 3.1 this pins
down the pair (i, α) up to equivalence and hence the associated extension
functor up to isomorphism.
Hence it only remains to clarify under which conditions on a Lagrangean
contact structure this extension procedure produces a regular
normal parabolic geometry. This then tells us the most general situation
in which a direct relation (as discussed in 2.4 and 3.1) between the
two parabolic geometries can exist. As can already be expected from
the case of the Feﬀerman construction (see [3, 6]) this is a rather subtle
question. Moreover, the result cannot be obtained by algebraically
comparing the two normalization conditions, but one needs more information
on the curvature of regular normal and torsion free normal
geometries. In particular, the proof of part (2) of the Lemma below
needs quite a lot of deep machinery for parabolic geometries.
As discussed in 3.6, the curvature function of a parabolic geometry
of type (G, P) has values in Λ2p+ ⊗ g. Since both p+ and g are graded,
there is a natural notion of homogeneity on this space. While being of
some ﬁxed homogeneity is not a P–invariant property, the fact that all
nonzero homogeneous components have at least some given homogeneity
is P–invariant. This is used in the deﬁnition of regularity in 3.6, which
simply says that all nonzero homogeneous components are in positive
homogeneity.
The map ∂∗ used in the deﬁnition of normality in 3.6 actually extends
to a family of maps ∂∗ : Λℓp+ ⊗ g → Λℓ−1p+ ⊗ g. These are the differentials
in the standard complex computing the Lie algebra homology
H∗(p+, g). By deﬁnition, the curvature function κ of a normal parabolic
geometry of type (G, P) has values in ker(∂∗) ⊂ Λ2p+ ⊗g. Hence we can
naturally project to the quotient to obtain a function κH with values in
ker(∂∗)/ im(∂∗) = H2(p+, g). Equivariancy of κ implies that κH can be
viewed as a smooth section of the bundle G ×P H2(p+, g). This section
is called the harmonic curvature of the normal parabolic geometry. It
turns out (see [8]) that P+ acts trivially on H∗(p+, g), so this bundle
admits a direct interpretation in terms of the underlying structure. As
we shall see below, this bundle is algorithmically computable.
Now from 3.6 we know that p+
∼= (g/p)∗ as a P–module, and since
g− ⊂ g is a complementary subspace (and G0–module) to p ⊂ g we
ON THE GEOMETRY OF CHAINS 21
can identify p+ with (g−)∗ as a G0–module. Hence we can also view
the spaces Λℓp+ ⊗ g as L(Λℓg−, g), which are the chain spaces in the
standard complex computing the Lie algebra cohomology of g− with
coeﬃcients in g. The diﬀerentials ∂ : L(Λℓg−, g) → L(Λℓ+1g−, g) in
that complex turn out to be adjoint to the maps ∂∗ with respect to a
certain inner product.
Hence we obtain an algebraic Hodge theory on each of the spaces
Λℓp+ ⊗ g, with algebraic Laplacian = ∂∗ ◦ ∂ + ∂ ◦ ∂∗. This construction
is originally due to Kostant (see [16]), whence is usually called
the Kostant Laplacian. The kernel of is a G0–submodule called the
harmonic subspace of Λℓp+ ⊗ g. Kostant’s version of the Bott–Borel–
Weil theorem in [16] gives a complete algorithmic description of the
G0–module ker( ). By the Hodge decomposition, ker( ) is isomorphic
to the homology group of the appropriate dimension.
We will need two general facts about the curvature of regular normal
respectively torsion free normal parabolic geometries in the sequel.
Lemma 3.7. Let (p : G → M, ω) be a regular normal parabolic geometry
of type (G, P) with curvature functions κ : G → Λ2p+ ⊗ g and
κH : G → H2(p+, g). Then we have:
(1) The lowest nonzero homogeneous component of κ has values in the
subset ker( ) ⊂ Λ2p+ ⊗ g.
(2) Suppose that (p : G → M, ω) is torsion free and that E0 ⊂ ker( ) ⊂
Λ2p+ ⊗ g is a G0–submodule such that κH has values in the image of
E0 under the natural isomorphism ker( ) → H2(p+, g) (induced by projecting
ker( ) ⊂ ker(∂∗) to the quotient). Then κ has values in the
P–submodule of Λ2p+ ⊗ g generated by E0.
Proof. (1) is an application of the Bianchi identity, which goes back
to [21], see also [7, Corollary 4.10]. (2) is proved in [4, Corollary 3.2].
q.e.d.
The ﬁnal bit of information we need is the explicit form of ker( ) for
the pairs (g, p) and (˜g, ˜p) corresponding to Lagrangean contact structures
on manifolds of dimension 2n + 1 respectively path geometries in
dimension 4n + 1. Obtaining the explicit description of the irreducible
components of these submodules is an exercise in the application of
Kostant’s results from [16] and the algorithms from the book [1], see
also [4]. The results are listed in the tables below. The ﬁrst column contains
the homogeneity of the component and the second column contains
the subspace that it is contained in. The actual component is always
the highest weight part in that subspace, so in particular, it lies in the
kernel of all traces one can form.
22 A. ˇCAP & V. ˇZ´ADN´IK
(g, p), n = 1
homog. contained in
4 gR
1 ∧ g2 ⊗ gR
1
4 gL
1 ∧ g2 ⊗ gL
1
(g, p), n > 1
homog. contained in
2 gL
1 ∧ gR
1 ⊗ g0
1 Λ2gL
1 ⊗ gR
−1
1 Λ2gR
1 ⊗ gL
−1
(˜g, ˜p), n = 1
homog. contained in
3 ˜gV
1 ∧ ˜g2 ⊗ ˜g0
2 ˜gE
1 ∧ ˜g2 ⊗ ˜gV
−1
1 Λ2˜gV
1 ⊗ ˜gE
−1
(˜g, ˜p), n > 1
homog. contained in
3 ˜gV
1 ∧ ˜g2 ⊗ ˜g0
2 ˜gE
1 ∧ ˜g2 ⊗ ˜gV
−1
0 Λ2˜gV
1 ⊗ ˜g−2
3.8. We are now ready to prove the main result of this article:
Theorem 3.8. Let (p : G → M, ω) be a regular normal parabolic
geometry of type (G, P) and let ( ˜G := G ×Q
˜P → P0(TM), ˜ωα) be the
parabolic geometry obtained using the extension functor associated to
the pair (i, α) deﬁned in 3.5. Then this geometry is regular and normal
if and only if (p : G → M, ω) is torsion free.
Proof. We ﬁrst prove necessity of torsion freeness. From the tables in
3.7 we see that for n = 1 a regular normal parabolic geometry of type
(G, P) is automatically torsion free, so we only have to consider the
case n > 1. If ˜ωα is regular and normal, then all nonzero homogeneous
components of ˜κ are homogeneous of positive degrees. The table in
3.7 shows that then the homogeneity is at least two, and by part (1)
of Lemma 3.7 the homogeneous component of degree two sits in the
subspace ˜gE
1 ∧ ˜g2 ⊗ ˜gV
−1. In particular, for any ˜u ∈ ˜G, the restriction of
˜κ(˜u) to Λ2˜g−2 is homogeneous of degree at least three, which implies
that ˜κ(˜u) has values in ˜g−1 ⊕ ˜p, i.e. for the natural projection π : ˜g →
˜g/(˜g−1 ⊕ ˜p) we get π ◦ ˜κ(˜u) = 0.
Using the notation of the proof of Lemma 3.6, consider two elements
X, Y ∈ ˜g−2. From that proof we see that
(π ◦ ˜κ(j(u)))(X, Y ) = (π ◦ α ◦ κ(u))




0 0 0
X1 0 0
0 Xt
2 0

 ,


0 0 0
Y1 0 0
0 Y t
2 0



 .
By regularity, κ(u)(Λ2g−1) ⊂ g−1 ⊕ p. From the deﬁnition in 3.5 it is
evident that α induces a linear isomorphism g/(g−2 ⊕p) → ˜g/(˜g−1 ⊕˜p).
Hence we conclude that if ˜ωα is regular and normal, then κ(u)(Λ2g−1) ⊂
p. From the table in 3.7 we see that this implies that the homogeneous
component of degree one of κ has to vanish identically, and then further
ON THE GEOMETRY OF CHAINS 23
that the homogeneous component of degree two has values in p. Since
Λ2g−2 = 0, components of homogeneity at least three automatically
have values in p, so we see that ω is torsion free.
To prove suﬃciency, we ﬁrst need two facts on the curvature function
κ of a torsion free normal parabolic geometry of type (G, P). On the
one hand, the map ∂∗ as deﬁned in 3.6 can be written as the sum
∂∗
1+∂∗
2 of two P–equivariant maps, with ∂∗
1 corresponding to the ﬁrst two
summands and ∂∗
2 corresponding to the last summand in the deﬁnition.
We claim that κ has values in the kernels of both operators ∂∗
i . On the
other hand, one easily veriﬁes that the subspace p ⊂ p formed by all
matrices of the form


0 u d
0 B v
0 0 0


is a P–submodule. (Indeed, this is the preimage in p of the semisimple
part of the reductive algebra g0 = p/p+.) Our second claim is that
κ(u)(X, Y ) ∈ p for all u ∈ G and all X, Y .
To prove both claims, it suﬃces to show that κ has values in the
P–submodule Λ2
0p+ ⊗ p ⊂ Λ2p+ ⊗ p. Here Λ2
0p+ is the kernel of the
P–homomorphism Λ2p+ → p+ deﬁned by the Lie bracket on p+, so
Λ2
0p+ ⊗ g = ker(∂∗
2).
In the case n = 1, this is evident, since from the table in 3.7 we see
that the lowest nonzero homogeneous component of κ(u) is of degree 4,
vanishes on Λ2g−1 and has values in p+. For homogeneous components
of higher degree, these two properties are automatically satisﬁed, and
we conclude that κ(u) ∈ g1 ∧ g2 ⊗ p+ ⊂ Λ2
0p+ ⊗ p.
In the case n > 1, we see from the table in 3.7 that by torsion freeness
the lowest homogeneous component of κ(u) must be of homogeneity 2.
By part (1) of Lemma 3.7 it has values in ker( ) ⊂ Λ2g1 ⊗g0. Since this
component of ker( ) is a highest weight part, it lies in the kernel of all
possible traces, and hence it must be contained in the tensor product of
Λ2g1∩Λ2
0p+ with the semisimple part of g0. Hence ker( ) is contained in
the P–submodule Λ2
0p+ ⊗ p so, by part (2) of Lemma 3.7, the curvature
function κ has values in that submodule.
In view of Proposition 3.3 and the proof of Theorem 3.6, to prove
that ˜ωα is normal, it suﬃces to verify that the map F(u) : Λ2˜g− → ˜g
deﬁned by F(u)(X, Y ) := α(κ(u)(α−1(X), α−1(Y ))) lies in the kernel
of ∂∗ for all u ∈ G. To compute ∂∗F(u), it is better to view F(u) as
an element of Λ2˜p+ ⊗ ˜g, and we want to relate this to κ(u), viewed as
an element of Λ2p+ ⊗ g. We have to compute the map ϕ : p+ → ˜p+,
which is dual to the composition of the canonical projection g/q → g/p
with α−1 : ˜g/˜p → g/q, since by construction F(u) = (Λ2ϕ ⊗ α)(κ(u)).
Recall that the duality between g/p and p+ (and likewise for the other
algebra) is induced by the Killing form. Since the Killing form of a
24 A. ˇCAP & V. ˇZ´ADN´IK
simple Lie algebra is uniquely determined up to a nonzero multiple by
invariance, we may as well use the trace form on both sides, which leads
to a nonzero multiple of ϕ. But then the computation is very easy,
showing that
ϕ


0 Z ψ
0 0 W
0 0 0

 =




0 ψ Z Wt
0 0 0 0
0 0 0 0
0 0 0 0



 .
In particular, ϕ(p+) ⊂ ˜gE
1 ⊕ ˜g2, which implies that ∂∗
2(F(u)) = 0 for all
u.
On the other hand, the formula for α from 3.5 shows that α(p) ⊂
˜gV
−1⊕˜g0⊕˜gE
1 ⊕˜g2, and the ˜g0–component is contained in the bottom right
2n × 2n block. This shows that ˜ωα is regular, and that for Z ∈ p+ and
A ∈ p we have [ϕ(Z), α(A)] ∈ ˜gE
1 ⊕˜g2. One immediately veriﬁes directly
that the ˜gE
1 –component of [ϕ(Z), α(A)] equals the ˜gE
1 –component of
α([Z, A]), while the ˜g2–component of [ϕ(Z), α(A)] equals twice the ˜g2–
component of α([Z, A]). From the deﬁnition of ∂∗
1 we now conclude that
Λ2ϕ ⊗ α maps ker(∂∗
1) to ker(∂∗
1), so we also get ∂∗
1(F(u)) = 0 for all u.
q.e.d.
4. Applications
For torsion free Lagrangean contact structures, Theorem 3.8 provides
us with an explicit description of the parabolic geometry determined by
the path geometry of chains. In particular, we obtain an explicit formula
for the Cartan curvature which is the basis for the applications discussed
in this section. The main result is that one can essentially reconstruct
the torsion free Lagrangean contact structure from the harmonic curvature
of this parabolic geometry. In particular, this implies that a contact
diﬀeomorphism which maps chains to chains has to either preserve or
swap the subbundles deﬁning the Lagrangean contact structure. On the
way, we can prove that chains can never be described by linear connections
and that only locally ﬂat Lagrangean contact structures give rise
to torsion free path geometries of chains.
4.1. Decomposing the Cartan curvature. For a torsion free Lagrangean
contact structure with curvature κ, the curvature ˜κ of the
normal Cartan connection associated to the path geometry of chains
is determined by the formula from Proposition 3.3, which holds on
j(G) ⊂ G ×i
˜P. In this formula, there are two terms, one of which
depends on κ while the other one only comes from the map α. Our
main task is to extract parts of ˜κ which only depend on one of the two
terms. The diﬃculty is that this has to be done in a geometric way
without knowing the subset j(G) in advance.
ON THE GEOMETRY OF CHAINS 25
The curvature function ˜κ has values in the P–module Λ2˜p+ ⊗ ˜g, and
using the map ϕ from the proof of Theorem 3.8, the formula from Proposition
3.3 reads as ˜κ(j(u)) = (Λ2ϕ ⊗ α)(κ(u)) + Ψα. Now ˜p+ contains
the P–invariant subspace ˜g2. Correspondingly, we obtain P–invariant
subspaces Λ2˜g2 ⊂ ˜p+ ∧˜g2 ⊂ Λ2˜p+. In the proof of Theorem 3.8, we have
seen that ϕ has values in ˜gE
1 ⊕ ˜g2, whence Λ2ϕ has values in ˜p+ ∧ ˜g2.
From Lemma 3.6 we know that Ψα ∈ ˜p+ ∧ ˜g2 ⊗ ˜g, so we conclude that
˜κ(j(u)) lies in this ˜P–submodule. By equivariancy, all values of the
curvature function lie in ˜p+ ∧ ˜g2 ⊗ ˜g ⊂ Λ2˜p+ ⊗ ˜g.
On the quotient ˜p+/˜g2, the subgroup ˜P+ ⊂ ˜P acts trivially, so we can
identify this quotient with the ˜G0–module ˜g1 = ˜gE
1 ⊕ ˜gV
1 . Correspondingly,
we get ˜P–equivariant projections
πE
: ˜p+ ∧ ˜g2 ⊗ ˜g → ˜gE
1 ∧ ˜g2 ⊗ ˜g,
πV
: ˜p+ ∧ ˜g2 ⊗ ˜g → ˜gV
1 ∧ ˜g2 ⊗ ˜g.
From the description of the image of ϕ in the proof of Theorem 3.8 we
conclude that (Λ2ϕ ⊗ α)(κ(u)) ∈ ker(πV ). On the other hand, Lemma
3.6 in particular shows that πV (Ψα) = 0 and Ψα ∈ ker(πE).
Theorem 4.1. Let (M, L, R) be a torsion free Lagrangean contact
structure.
(1) There is no linear connection on the tangent bundle TM which has
the chains among its geodesics.
(2) The parabolic geometry associated to the path geometry of chains
on ˜M = P0(TM) is torsion free if and only if (M, L, R) is locally ﬂat,
i.e. locally isomorphic to the homogeneous model G/P.
Proof. (1) Suppose that ∇ is a linear connection on TM whose geodesics
in directions transverse to L⊕R are parametrizations of the chains.
Since symmetrizing a connection does not change the geodesics, we may
without loss of generality assume that ∇ is torsion free. Then we can
look at the associated projective structure [∇] on M and use the machinery
of correspondence spaces from [4]. The fact that the geodesics
of ∇ are the chains exactly means that the path geometry of chains on
˜M is isomorphic to an open subgeometry of the correspondence space
C(M, [∇]), see 4.7 of [4]. In particular, the Cartan curvature ˜κ is the
restriction of the curvature of this correspondence space. By [4, Proposition
2.4] this curvature has the property that it vanishes upon insertion
of any tangent vector contained in the vertical bundle of ˜M → M. But
this contradicts the fact that πV ◦ ˜κ = 0 we have observed above.
(2) By Theorem 3.6, the path geometry of chains associated to a locally
ﬂat Lagrangean contact structure is torsion free. Conversely, if the
Cartan connection ˜ω is torsion free, then according to part (1) of Lemma
3.7 and the tables in 3.7, the lowest nonzero homogeneous component
of ˜κ must be of degree at least three, and the harmonic curvature must
26 A. ˇCAP & V. ˇZ´ADN´IK
have values in ˜gV
1 ∧ ˜g2 ⊗ ˜g0 ⊂ ker(πE). By part (2) of Lemma 3.7 the
whole curvature ˜κ has values in ker(πE). Above, we have observed that
Ψα ∈ ker(πE) so we conclude that for each u ∈ G we get πE ◦ (Λ2ϕ ⊗
α)(κ(u)) = 0.
In the proof of Theorem 3.8 we have seen that ϕ is a linear isomorphism
p+ → ˜gE
1 ⊕ ˜g2, and hence ˜gE
1 ∧ ˜g2 is contained in the image of
Λ2ϕ. Hence we conclude that α ◦ κ(u) = 0 and since α is injective, the
result follows. q.e.d.
4.2. Harmonic curvature. We have discussed the deﬁnition of harmonic
curvature already in 3.7. Let πH be the natural projection from
ker(∂∗) ⊂ Λ2˜p+ ⊗ ˜g to the quotient ker(∂∗)/ im(∂∗). Since this is a ˜P–
equivariant map, the composition ˜κH = πH ◦ ˜κ : ˜G → ker(∂∗)/ im(∂∗)
deﬁnes a smooth section of the associated bundle ˜G × ˜P ker(∂∗)/ im(∂∗),
which is the main geometric invariant of the parabolic geometry associated
to the path geometry of chains.
From 3.7 we also know that on the quotient ker(∂∗)/ im(∂∗) the group
˜P+ acts trivially, and we may identify it with the ˜G0–module ker( ) ⊂
Λ2˜p+ ⊗ ˜g. From the table in 3.7, we see that this module contains two
irreducible components in positive homogeneity, which are the highest
weight components of the subrepresentations ˜gE
1 ∧ ˜g2 ⊗ ˜gV
−1 respectively
˜gV
1 ∧˜g2 ⊗˜g0. Correspondingly, we obtain decompositions πH = πE
H +πV
H
and ˜κH = ˜κE
H + ˜κV
H .
Lemma 4.2. Let πE and πV be the projections on ˜p+ ∧˜g2 ⊗˜g deﬁned
in 4.1. Then the restriction of πE
H (respectively πV
H) to
ker(∂∗
) ∩ (˜p+ ∧ ˜g2 ⊗ ˜g)
factorizes through πE (respectively πV ).
Proof. By Kostant’s version of the Bott–Borel–Weil theorem, see [16],
the ˜G0–irreducible components contained in ker( ) occur with multiplicity
one, even within Λ∗˜p+ ⊗ ˜g. To obtain πE and πV , we used the
projection ˜p+ ∧ ˜g2 ⊗ ˜g → ˜g1 ∧ ˜g2 ⊗ ˜g with kernel Λ2˜g2 ⊗ ˜g. By the
multiplicity one result and the fact that both components of ker( )
are contained in ˜g1 ∧ ˜g2 ⊗ ˜g, there is no nonzero ˜G0–equivariant map
Λ2˜g2 ⊗ ˜g → ker(∂∗)/ im(∂∗). Hence each of the projections πH, πE
H and
πV
H factorizes through ˜g1 ∧ ˜g2 ⊗ ˜g. Looking at the resulting map for πE
H,
we see that again by multiplicity one, the subspace ˜gV
1 ∧ ˜g2 ⊗ ˜g must be
contained in the kernel, so we conclude that πE
H factorizes through πE.
In the same way one shows that πV
H factorizes through πV . q.e.d.
Proposition 4.2. Let (M, L, R) be a torsion free Lagrangean contact
structure, and let ˜κH = ˜κE
H +˜κV
H be the harmonic curvature of the regular
normal parabolic geometry determined by the path geometry of chains.
ON THE GEOMETRY OF CHAINS 27
Then the function ˜G → ˜gV
1 ∧˜g2 ⊗˜g0 corresponding to ˜κV
H is a nonzero
multiple of the unique equivariant extension of the constant function Ψα
(compare with Lemma 3.6) on j(G).
Proof. We have to compute the function πV
H ◦ ˜κ. By the lemma, πV
H
factorizes through the projection πV introduced in 4.1, and from there
we know that πV (˜κ(j(u))) = πV (Ψα). Hence we see that (πV
H ◦ ˜κ)|j(G) =
πV
H(Ψα). Now Ψα ∈ ˜gV
1 ∧ ˜g2 ⊗ ˜g0 by Lemma 3.6, and the values even
lie in the semisimple part of ˜g0, which may be identiﬁed with sl(˜g−2).
Evidently, ˜gV
1
∼= ˜gE
−1 ⊗ ˜g2 as a ˜G0–module, so we may interpret Ψα
as an element of ˜gE
−1 ⊗ (⊗3˜g2) ⊗ ˜g−2. In Lemma 3.6 and the proof of
Theorem 3.6 we have seen that in this picture Ψα lies in the irreducible
component ˜gE
−1 ⊗ (S3˜g2 ⊗ ˜g−2)0, where the subscript denotes the trace
free part. Passing back to ˜gV
1 ∧˜g2 ⊗˜g0 this exactly means that Ψα lies in
the highest weight subspace, which is the intersection with ker( ). Now
πV
H restricts to ˜G0–equivariant linear isomorphism on this intersection,
which implies the result. q.e.d.
Remark. Similarly to the proof above, one shows that the harmonic
curvature component ˜κE
H is the extension of a component of j(u) →
(Λ2ϕ ⊗ α)(κ(u)). Since we explicitly know Λ2ϕ ⊗ α, this can be used
to obtain a more explicit description of the second harmonic curvature
component. From part (2) of Theorem 4.1 and [4, 4.7] we see that
vanishing of ˜κE
H is equivalent to local ﬂatness of the original Lagrangean
contact structure, so κ is completely encoded in ˜κE
H.
4.3. Passing to the underlying manifold. The harmonic curvature
component determined by the function ˜κV
H is a section of the bundle
associated to ˜gV
1 ∧ ˜g2 ⊗ ˜g0. In the proof of Proposition 4.2 we have seen
that we can replace that space by ˜gE
−1⊗(⊗3˜g2)⊗˜g−2. The corresponding
bundle is E⊗⊗3F∗⊗F → ˜M, where F := T ˜M/(E⊕V ). Since E ⊂ T ˜M
is a line bundle, we can view ˜κV
H as a section of ⊗3F∗ ⊗ F which is
determined up to a nonzero multiple.
To relate this to the underlying manifold M, recall that ˜M is an open
subset in the projectivized tangent bundle of M. A point in ˜M is a line
in some tangent space TxM that is transversal to Lx ⊕ Rx. We have
noted in 2.4 that TM ∼= G ×P g/p and T ˜M ∼= G ×Q g/q, and the tangent
map of the projection π : ˜M → M corresponds to the natural projection
g/q → g/p. Fix a point ℓ ∈ π−1(x). Then for each ξ ∈ TxM there is a
lift ˜ξ ∈ Tℓ
˜M and we can consider the class of ˜ξ in Fℓ = Tℓ
˜M/(Eℓ ⊕ Vℓ).
Since Vℓ is the vertical subbundle, this class is independent of the choice
of the lift and from the explicit description of Tπ we see that restricting
to Lx ⊕ Rx, we obtain a linear isomorphism Lx ⊕ Rx
∼= Fℓ.
Fixing x and ℓ we therefore see that the harmonic curvature component
corresponding to ˜κV
H gives rise to an element of ⊗3(Lx ⊕ Rx)∗ ⊗
28 A. ˇCAP & V. ˇZ´ADN´IK
(Lx ⊕Rx), which is determined up to a nonzero multiple. To write down
this map explicitly, we ﬁrst need the Levi bracket
L : (Lx ⊕ Rx) × (Lx ⊕ Rx) → TxM/(Lx ⊕ Rx).
Since this has values in a one–dimensional space, we may view it as a
real valued bilinear map determined up to a nonzero multiple. Further,
we denote by J the almost product structure corresponding to the decomposition
L ⊕ R. This means that J is the endomorphism of L ⊕ R
which is the identity on L and minus the identity on R. Using this we
can now formulate:
Lemma 4.3. The element of ⊗3(Lx ⊕Rx)∗ ⊗(Lx ⊕Rx) obtained from
˜κV
H above is (a nonzero multiple of) the complete symmetrization of the
map
(ξ, η, ζ) → L(ξ, J(η))J(ζ).
Proof. This is a reinterpretation of the proof of Lemma 3.6. Observe
that J corresponds to the map
X1
X2
→
X1
−X2
in the notation there. Since L corresponds to [ , ] : g−1 × g−1 → g−2,
computing the bracket




0 0 0
X1 0 0
0 Xt
2 0

 ,


0 0 0
Y1 0 0
0 −Y t
2 0



 ,
we see that the expression X1, Y2 + Y1, X2 in the proof of Lemma
3.6 corresponds to L(ξ, J(η)). q.e.d.
4.4. Reconstructing the Lagrangean contact structure. Now we
can ﬁnally show that the Cartan curvature of the path geometry of
chains can be used to (almost) reconstruct the Lagrangean contact
structure on M that we have started from:
Theorem 4.4. Let (M, L, R) be a torsion free Lagrangean contact
structure. Then for each x ∈ M, the subset Lx ∪ Rx ⊂ TxM can be
reconstructed from the harmonic curvature of the normal parabolic geometry
associated to the path geometry of chains.
Proof. In view of the results in 4.2 and 4.3 it suﬃces to show that
Lx ∪ Rx can be recovered from the complete symmetrization S of the
map
(ξ, η, ζ) → L(ξ, J(η))J(ζ).
First we see that S(ξ, ξ, ξ) = 0 if and only if L(ξ, J(ξ)) = 0. Note that
this is always satisﬁed for ξ ∈ Lx ∪ Rx. Fixing an element ξ with this
property, we see that
S(ξ, ξ, η) = 2L(ξ, J(η))J(ξ).
ON THE GEOMETRY OF CHAINS 29
By non–degeneracy of L, given a nonzero element ξ we can always ﬁnd
η such that L(ξ, J(η)) = 0. Hence we see that ξ is an eigenvector for
J (which by deﬁnition is equivalent to ξ ∈ Lx ∪ Rx) if and only if
S(ξ, ξ, ξ) = 0 and there is an element η such that S(ξ, ξ, η) is a nonzero
multiple of ξ. q.e.d.
Corollary 4.4. Let (M, L, R) be a torsion free Lagrangean contact
structure and let f : M → M be a contact diﬀeomorphism which
maps chains to chains. Then either f is an automorphism or an anti–
automorphism of the Lagrangean contact structure. Here anti–automorphism
means that Txf(Lx) = Rf(x) and Txf(Rx) = Lf(x) for all
x ∈ M.
Proof. By assumption, f induces an automorphism ˜f of the path
geometry of chains associated to (M, L, R). This automorphism has to
pull back the Cartan curvature ˜κ and also the harmonic curvature ˜κH to
itself. From the theorem we conclude that this implies Txf(Lx ∪ Rx) =
Lf(x) ∪ Rf(x), and this is only possible if f is an automorphism or an
anti–automorphism. q.e.d.
5. Partially integrable almost CR structures
What we have done for Lagrangean contact structures so far can be
easily adapted to deal with partially integrable almost CR structure.
We will only brieﬂy sketch the necessary changes in this section.
5.1. A non–degenerate partially integrable almost CR structure on a
smooth manifold M is given by a contact structure H ⊂ TM together
with an almost complex structure J on H such that the Levi bracket
L has the property that L(Jξ, Jη) = L(ξ, η) for all ξ, η. Then L is the
imaginary part of a non–degenerate Hermitian form and we denote the
signature of this form by (p, q). Such a structure of signature (p, q) is
equivalent to a regular normal parabolic geometry of type (G, P), where
G = PSU(p + 1, q + 1) and P ⊂ G is the stabilizer of a point in CPn+1,
n = p+q, corresponding to a null line, see [7, 4.15]. The group G is the
quotient of SU(p + 1, q + 1) by its center (which is isomorphic to Zn+2)
and we will work with representative matrices as before.
We will use the Hermitian form of signature (p, q) on Cn+1 corresponding
to
(z0, . . . , zn+1) → z0 ¯zn+1 + zn+1¯z0 + p
j=1 |zj|2 − n
j=p+1 |zj|2.
Then the decomposition on sl(n + 2, C) with block sizes 1, n, and 1
restricts to a contact grading on the Lie algebra g of G. The explicit
form for signature (n, 0) can be found in [7, 4.15]. In general, g consists
30 A. ˇCAP & V. ˇZ´ADN´IK
of all matrices of the form


w Z iz
X A −IZ∗
ix −X∗I − ¯w


with blocks of sizes 1, n, and 1, w ∈ C, x, z ∈ R, X ∈ Cn, Z ∈ Cn∗, and
A ∈ u(p, q) such that w − ¯w + tr(A) = 0. Here I is the diagonal matrix
with the ﬁrst p entries equal to 1 and the remaining q entries equal to
−1.
It is easy to show that the subgroup Q ⊂ G corresponds to matrices
of the form 

ϕ 0 iaϕ
0 Φ 0
0 0 ¯ϕ−1

 ,
with ϕ ∈ C\{0}, a ∈ R and Φ ∈ U(p, q) such that (ϕ2/|ϕ|2) det(Φ) = 1.
5.2. Next we need an analog of the pair (i, α) introduced in 3.5. As
before we start with a manifold M of dimension 2n + 1, so again ˜G =
PGL(2n + 2, R). We will use a block decomposition into blocks of sizes
1, 1, n, and n as before. The right choice turns out to be
i




ϕ 0 iaϕ
0 Φ 0
0 0 ¯ϕ−1



 :=








|ϕ| −a|ϕ| 0 0
0 |ϕ|−1 0 0
0 0 ℜ |ϕ|
ϕ Φ −ℑ |ϕ|
ϕ Φ
0 0 ℑ |ϕ|
ϕ Φ ℜ |ϕ|
ϕ Φ








,
α




w Z iz
X A −IZ∗
ix −X∗I − ¯w



 :=






ℜ(w) −z ℜ(Z) −ℑ(Z)
x −ℜ(w) −ℑ(X∗I) −ℜ(X∗I)
ℜ(X) ℑ(IZ∗) ℜ(A) −ℑ(A) + ℑ(w)
ℑ(X) −ℜ(IZ∗) ℑ(A) − ℑ(w) ℜ(A)






,
where ℜ and ℑ denote real and imaginary part, respectively, and we
write ℑ(w) for the appropriate multiple of the identity matrix.
There is an analog of Lemma 3.6 (with similar proof), the only change
one has to make is that the map whose alternation has to be used is
given by
(X, Y, Z) → ( X1, IY1 + X2, IY2 )
−Z2
Z1
.
ON THE GEOMETRY OF CHAINS 31
This map has similar properties as the one from 3.6 so the analogs of
Theorem 3.6 and Corollary 3.6 hold.
Concerning the structure of ker( ) the situation is also similar to
the case of Lagrangean contact structures, since the decomposition of
ker( ) can be determined from the complexiﬁcations of g and p which
are the same in both cases. The only diﬀerence is that the two irreducible
components for n = 1 respectively the two irreducible components
contained in homogeneity 1 in the case n > 1 in the Lagrangean
case correspond to only one component here. This component however
has a complex structure and it consists of maps g−1 ∧ g−2 → g1 which
are complex linear in the ﬁrst variable respectively maps Λ2g−1 → g−1,
which are conjugate linear in both variables. For n > 1 this component
is a torsion which is up to a nonzero multiple given by the Nijenhuis
tensor. Vanishing of this component is equivalent to torsion freeness
and to integrability of the almost CR structure, see [7, 4.16].
Theorem 5.2. Let (M, H, J) be a partially integrable almost CR
structure and let (p : G → M, ω) be the corresponding regular normal
parabolic geometry of type (G, P). Then the parabolic geometry (G ×Q
˜P → P0(TM), ˜ωα) constructed using the extension functor associated to
the pair (i, α) from 5.1 is regular and normal if and only if ω is torsion
free, i.e. the almost CR structure is integrable.
Proof. Apart from some numerical factors which cause no problems,
this is completely parallel to the proof of Theorem 3.8. q.e.d.
Hence the direct relation between the regular normal parabolic geometries
associated to a partially integrable almost CR structure respectively
to the associated path geometry of chains works exactly on
the subclass of CR structures.
5.3. Applications. The developments of Section 4 can be applied to
the CR case with only minimal changes. In analog of Lemma 4.3, one
obtains S ∈ ⊗3H∗
x ⊗ Hx, which is the complete symmetrization of
(ξ, η, ζ) → L(ξ, J(η))J(ζ),
where J is the almost complex structure on H.
Theorem 5.3. Let (M, H, J) be a CR structure.
(1) There is no linear connection on TM which has the chains among
its geodesics.
(2) The path geometry of chains is torsion free if and only if the CR
structure is locally ﬂat.
(3) The almost complex structure J can be reconstructed up to sign from
the harmonic curvature of the associated path geometry of chains.
Proof. The only change compared to Section 4 is that one has to
extend S to the complexiﬁed bundle H ⊗C. As in the proof of Theorem
32 A. ˇCAP & V. ˇZ´ADN´IK
4.4 one then reconstructs the subset H1,0
x ∪ H0,1
x ⊂ Hx ⊗ C for each
x ∈ M, i.e. the union of the holomorphic and the anti–holomorphic
part. This union determines J up to sign. q.e.d.
This theorem now also implies that the signature of the CR structure,
which is encoded in L(−, J(−)), can be reconstructed from the path
geometry of chains. As a corollary, we obtain a completely independent
proof of the analog of Corollary 4.4, which is due to [10] for CR
structures:
Corollary 5.3. A contact diﬀeomorphism between two CR manifolds
which maps chains to chains is either a CR isomorphism or a CR anti–
isomorphism.
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Fakult¨at f¨ur Mathematik,
Universit¨at Wien,
Nordbergstraße 15,
A–1090 Wien, Austria
and
International Erwin Schr¨odinger Institute
for Mathematical Physics,
Boltzmanngasse 9,
A–1090 Wien, Austria
E-mail address: Andreas.Cap@esi.ac.at
International Erwin Schr¨odinger Institute
for Mathematical Physics,
Boltzmanngasse 9,
A–1090 Wien, Austria
and
Faculty of Education,
Masaryk University,
Poˇr´ıˇc´ı 31,
60300 Brno, Czech Republic
E-mail address: zadnik@math.muni.cz
98 Attachments
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 081, 33 pages
A Projective-to-Conformal Fefferman-Type
Construction
Matthias HAMMERL †1
, Katja SAGERSCHNIG †2
, Josef ˇSILHAN †3
,
Arman TAGHAVI-CHABERT †4
and Vojtˇech ˇZ ´ADN´IK †5
†1
University of Vienna, Faculty of Mathematics,
Oskar-Morgenstern-Platz 1, 1010 Vienna, Austria
E-mail: matthias.hammerl@univie.ac.at
†2
INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail: katja.sagerschnig@univie.ac.at
†3
Masaryk University, Faculty of Science, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic
E-mail: silhan@math.muni.cz
†4
Universit`a di Torino, Dipartimento di Matematica “G. Peano”,
Via Carlo Alberto 10, 10123 Torino, Italy
E-mail: ataghavi@unito.it
†5
Masaryk University, Faculty of Education, Poˇr´ıˇc´ı 31, 60300 Brno, Czech Republic
E-mail: zadnik@mail.muni.cz
Received February 09, 2017, in final form October 09, 2017; Published online October 21, 2017
https://doi.org/10.3842/SIGMA.2017.081
Abstract. We study a Fefferman-type construction based on the inclusion of Lie groups
SL(n + 1) into Spin(n + 1, n + 1). The construction associates a split-signature (n, n)conformal
spin structure to a projective structure of dimension n. We prove the existence
of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed
conformal space. We obtain a complete characterisation of the constructed conformal spaces
in terms of these solutions to overdetermined equations and an integrability condition on the
Weyl curvature. The Fefferman-type construction presented here can be understood as an
alternative approach to study a conformal version of classical Patterson–Walker metrics as
discussed in recent works by Dunajski–Tod and by the authors. The present work therefore
gives a complete exposition of conformal Patterson–Walker metrics from the viewpoint of
parabolic geometry.
Key words: parabolic geometry; projective structure; conformal structure; Cartan connection;
Fefferman spaces; twistor spinors
2010 Mathematics Subject Classiﬁcation: 53A20; 53A30; 53B30; 53C07
1 Introduction
In conformal geometry the geometric structure is given by an equivalence class of pseudoRiemannian
metrics: two metrics g and ˆg are considered to be equivalent if they differ by
a positive smooth rescaling, ˆg = e2f g. In projective geometry the geometric structure is given by
an equivalence class of torsion-free affine connections: two connections D and ˆD are considered
as equivalent if they share the same geodesics (as unparametrised curves). While conformal and
projective structures both determine a corresponding class of affine connections, neither of them
induces a single distinguished connection on the tangent bundle. Instead, both structures have
canonically associated Cartan connections that govern the respective geometries and encode
2 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
prolonged geometric data of the respective structures. It is therefore often useful when studying
projective and conformal structures to work in the framework of Cartan geometries.
The present paper investigates a geometric construction that produces a conformal class of
split-signature metrics on a 2n-dimensional manifold arising naturally from a projective class
of connections on an n-dimensional manifold. Split-signature conformal structures of this type
have appeared in several places in the literature before. The projective-to-conformal construction
studied in this paper should be understood as a generalisation of the classical Riemann
extensions of affine spaces by E.M. Patterson and A.G. Walker [26]. One of the main authors
motivations for the present study was the article [15] by M. Dunajski and P. Tod, where the
Patterson–Walker construction was generalised to a projectively invariant setting in dimension
n = 2. On the other hand, in [25] conformal structures of signature (2, 2) were constructed
using Cartan connections that contain the conformal structures arising from 2-dimensional projective
structures as a special case. A generalisation of this Cartan-geometric approach to higher
dimensions can be found in [24].
In this paper the construction is studied as an instance of a Feﬀerman-type construction, as
formalised in [6, 11], based on an inclusion of the respective Cartan structure groups SL(n+1) →
Spin(n + 1, n + 1). We show that in the general situation n ≥ 3 the induced conformal Cartan
geometry is non-normal. To obtain information on the conformal structure it is thus important
to understand how the normal conformal Cartan connection differs from the induced one, and
the main part of the paper concerns the study of this modification. We may summarise the
main contributions of the paper as follows:
• A comprehensive treatment of the projective-to-conformal Fefferman-type construction
including a discussion of the intermediate Lagrangean contact structure (Section 3) and
a comparison with Patterson–Walker metrics (Section 6.1).
• A thorough study of the normalisation process (Section 4) and an explicit formula for the
modification needed to obtain the normal conformal Cartan connection (Section 5.2).
• The characterisation of the conformal structures obtained via our Fefferman-type construction
(culminating in Theorem 4.14).
Let us comment upon the characterisation in more detail. This is formulated in terms of
a conformal Killing field k and a twistor spinor χ on the conformal space together with a (conformally
invariant) integrability curvature condition. In Theorem 4.14 the properties of k and χ
are specified in terms of corresponding conformal tractors, which nicely reflects the algebraic
setup of the Fefferman-type construction in geometric terms.
An alternative equivalent characterisation theorem was obtained by the authors in [20, Theorem
1] by different means, namely, by direct computations based on spin calculus in the spirit
of [28, 29]. The conformal properties are given purely in underlying terms and do not refer to
tractors. In Section 6.2 (Theorem 6.3) we indicate how this alternative characterisation can be
obtained in the current framework.
We remark that, to our knowledge, the present work is the first comprehensive treatment of
a non-normal Fefferman-type construction and we expect that the techniques developed should
have considerable scope for applications to other similar constructions. A particularly interesting
case of this sort is the Fefferman construction for (non-integrable) almost CR-structures.
Possible further applications concern relations between solutions of so-called BGG-equations and
special properties of the induced conformal structures. Several such relationships were already
obtained by the authors in [20]. For instance, we can give a full description of Einstein metrics
contained in the resulting conformal class in terms of the initial projective structure. Moreover,
in [21] we were able to show that the obstruction tensor of the induced conformal structure
vanishes.
A Projective-to-Conformal Fefferman-Type Construction 3
2 Projective and conformal parabolic geometries
The standard reference for the background material on Cartan and parabolic geometries presented
here is [11].
2.1 Cartan and parabolic geometries
Let G be a Lie group with Lie algebra g and P ⊆ G a closed subgroup with Lie algebra p.
A Cartan geometry (G, ω) of type (G, P) over a smooth manifold M consists of a P-principal
bundle G → M together with a Cartan connection ω ∈ Ω1(G, g). The canonical principal bundle
G → G/P endowed with the Maurer–Cartan form constitutes the homogeneous model for Cartan
geometries of type (G, P).
The curvature of a Cartan connection ω is the 2-form
K ∈ Ω2
(G, g), K(ξ, η) := dω(ξ, η) + [ω(ξ), ω(η)], for all ξ, η ∈ X(G),
which is equivalently encoded in the P-equivariant curvature function
κ: G → Λ2
(g/p)∗
⊗ g, κ(u)(X + p, Y + p) := K ω−1
(u)(X), ω−1
(u)(Y ) . (2.1)
The curvature is a complete obstruction to a local equivalence with the homogeneous model. If
the image of κ is contained in Λ2(g/p)∗ ⊗ p the Cartan geometry is called torsion-free.
A parabolic geometry is a Cartan geometry of type (G, P), where G is a semi-simple Lie group
and P ⊆ G is a parabolic subgroup. A subalgebra p ⊆ g is parabolic if and only if its maximal
nilpotent ideal, called nilradical p+, coincides with the orthogonal complement p⊥ of p ⊆ g with
respect to the Killing form. In particular, this yields an isomorphism (g/p)∗ ∼= p+ of P-modules.
The quotient g0 = p/p+ is called the Levi factor; it is reductive and decomposes into a semisimple
part gss
0 = [g0, g0] and the center z(g0). The respective Lie groups are Gss
0 ⊆ G0 ⊆ P and
P+ ⊆ P so that P = G0 P+ and P+ = exp(p+). An identification of g0 with a subalgebra in
p yields a grading g = g−k ⊕ · · · ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ · · · ⊕ gk, where p+ = g1 ⊕ · · · ⊕ gk. We set
g− = g−k ⊕· · ·⊕g−1. If k is the depth of the grading the parabolic geometry is called |k|-graded.
The grading of g induces a grading on Λ2p+⊗g ∼= Λ2(g/p)∗⊗g. A parabolic geometry is called
regular if the curvature function κ takes values only in the components of positive homogeneity.
In particular, any torsion-free or |1|-graded parabolic geometry is regular.
Given a g-module V , there is a natural p-equivariant map, the Kostant co-diﬀerential,
∂∗
: Λk
(g/p)∗
⊗ V → Λk−1
(g/p)∗
⊗ V, (2.2)
defining the Lie algebra homology of p+ with values in V ; see, e.g., [11, Section 3.3.1] for
the explicit form. For V = g, this gives rise to a natural normalisation condition: parabolic
geometries satisfying ∂∗(κ) = 0 are called normal. The harmonic curvature κH of a normal
parabolic geometry is the image of κ under the projection ker ∂∗ → ker ∂∗/ im ∂∗. For regular
and normal parabolic geometries, the entire curvature κ is completely determined just by κH.
A Weyl structure j : G0 →G of a parabolic geometry (G, ω) over M is a reduction of the Pprincipal
bundle G → M to the Levi subgroup G0 ⊆ P. The class of all Weyl structures, which
are parametrised by one-forms on M, includes a particularly important subclass of exact Weyl
structures, which are parametrised by functions on M: For |1|-graded parabolic geometries,
these correspond to further reductions of G0 → M just to the semi-simple part Gss
0 of G0 or,
equivalently, to sections of the principal R+-bundle G0/Gss
0 → M. The latter bundle is called
the bundle of scales and its sections are the scales.
For a Weyl structure j : G0 → G, the pullback j∗ω = j∗ω− + j∗ω0 + j∗ω+ of the Cartan
connection may be decomposed according to g = g− ⊕ g0 ⊕ p+. The g0-part j∗ω0 is a principal
connection on the G0-bundle G0 → M; it induces connections on all associated bundles, which
are called (exact) Weyl connections. The p+-part j∗ω+ is the so-called Schouten tensor.
4 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
2.2 Tractor bundles and BGG operators
Every Cartan connection ω on G → M naturally extends to a principal connection ˆω on the
G-principal bundle ˆG := G ×P G → M, which further induces a linear connection V on any
associated vector bundle V := G ×P V = ˆG ×G V for a G-representation V . Bundles and
connections arising in this way are called tractor bundles and tractor connections. The tractor
connections induced by normal Cartan connections are called normal tractor connections.
In particular, for the adjoint representation we obtain the adjoint tractor bundle AM :=
G ×P g. The canonical projection g → g/p and the identification TM ∼= G ×P (g/p) yield
a bundle projection Π: AM → TM; the inclusion p+ ⊆ g and the identification p+
∼= (g/p)∗
yield a bundle inclusion T∗M → AM. This allows us to interpret the Cartan curvature κ
from (2.1) as a 2-form Ω on M with values in AM.
The holonomy group of the principal connection ˆω is by definition the holonomy of the Cartan
connection ω, i.e., Hol(ω) := Hol(ˆω) ⊆ G. By the holonomy of a geometric structure we mean
the holonomy of the corresponding normal Cartan connection.
In [12], and later in a simplified manner in [4], it was shown that for a tractor bundle
V = G ×P V one can associate a sequence of differential operators, which are intrinsic to the
given parabolic geometry (G, ω),
Γ(H0)
ΘV
0
→ Γ(H1)
ΘV
1
→ · · ·
ΘV
n−1
→ Γ(Hn).
The operators ΘV
k are the BGG-operators and they operate between the sections of subquotients
Hk = ker ∂∗/im ∂∗ of the bundles of V-valued k-forms, where ∂∗ : ΛkT∗M ⊗ V → Λk−1T∗M ⊗ V
denotes the bundle map induced by the Kostant co-differential (2.2).
The first BGG-operator ΘV
0 : Γ(H0) → Γ(H1) is constructed as follows. The bundle H0
is simply the quotient V/V , where V ⊆ V is the subbundle corresponding to the largest Pinvariant
filtration component in the G-representation V . It turns out, there is a distinguished
differential operator that splits the projection Π0 : V → H0, namely, the splitting operator, which
is the unique map LV
0 : Γ(H0) → Γ(V) satisfying
Π0(LV
0 (σ)) = σ, ∂∗
(d
V
LV
0 (σ)) = 0, for all σ ∈ Γ(H0).
The latter condition allows to define the first BGG-operator by ΘV
0 := Π1 ◦ d
V
◦ LV
0 , where
Π1 : ker ∂∗ → Γ(H1). The first BGG-operator defines an overdetermined system of differential
equations on σ ∈ Γ(H0), ΘV
0 (σ) = 0, which is termed the ﬁrst BGG-equation.
2.3 Further notations and conventions
In order to distinguish various objects related to projective and conformal structures, the symbols
referring to conformal data will always be endowed with tildes. To write down explicit formulae,
we employ abstract index notation, cf., e.g., [27]. Furthermore, we will use different types
of indices for projective and conformal manifolds. E.g., on a projective manifold M we write
EA := T∗M, EA := TM, and multiple indices denote tensor products, as in E B
A := T∗M ⊗TM.
Indices between squared brackets are skew, as in E[AB] := Λ2T∗M, and indices between round
brackets are symmetric, as in E(AB) := S2TM. Analogously, on a conformal manifold M we write
Ea := T∗M, Ea := TM etc. By E(w) and E[w] we denote the density bundle over M and M,
respectively. Tensor products with other natural bundles are denoted as EA(w) := EA ⊗ E(w),
E[ab][w] := E[ab] ⊗ E[w], and the like.
2.4 Projective structures
Let M be a smooth manifold of dimension n ≥ 2. A projective structure on M is given by
a class, p, of torsion-free projectively equivalent affine connections: two connections D and ˆD
A Projective-to-Conformal Fefferman-Type Construction 5
are projectively equivalent if they have the same geodesics as unparametrised curves. This is
the case if and only if there is a one-form ΥA ∈ Γ(EA) such that, for all ξA ∈ Γ EA ,
ˆDAξB
= DAξB
+ ΥAξB
+ ΥP ξP
δ B
A .
An oriented projective structure (M, p), which is a projective structure p on an oriented
manifold M, is equivalently encoded as a normal parabolic geometry of type (G, P), where
G = SL(n + 1) and P = GL+(n) Rn∗ is the stabiliser of a ray in the standard representation
Rn+1.
Affine connections from the projective class p are precisely the Weyl connections of the
corresponding parabolic geometry. Exact Weyl connections are those D ∈ p which preserve
a volume form — these are also known as special affine connections. In particular, a choice
of D ∈ p reduces the structure group to G0 = GL+(n), if D is special, the structure group is
further reduced to Gss
0 = SL(n).
For later purposes we now give explicit expressions of the main curvature quantities, cf., e.g.,
[2, 17]. For D ∈ p, the Schouten tensor is determined by the Ricci curvature of D; if D is special,
then the Schouten tensor is PAB = 1
n−1R P
PA B, in particular, it is symmetric. The projective
Weyl curvature and the Cotton tensor are
W C
AB D = R C
AB D + PADδC
B − PBDδC
A, YCAB = 2D[APB]C.
Henceforth, we use a suitable normalisation of densities so that the line bundle associated to
the canonical one-dimensional representation of P has projective weight −1. Hence, comparing
with the usual notation, the density bundle of projective weight w, denoted by E(w), is just the
bundle of ordinary −w
n+1 -densities. As an associated bundle to G → M, E(w) corresponds to
the 1-dimensional representation of P given by
GL+(n) Rn∗
→ R+, (A, X) → det(A)w
. (2.3)
The projective standard tractor bundle is the tractor bundle associated to the standard representation
of G = SL(n + 1). The projective dual standard tractor bundle is denoted by T ∗,
i.e., T ∗ := G ×P Rn+1∗
. With respect to a choice of D ∈ p, we write
T ∗
=
EA(1)
E(1)
, T ∗
C
ϕA
σ
=
DCϕA + PCAσ
DCσ − ϕC
.
2.5 Conformal spin structures and tractor formulas
Let M be a smooth manifold of dimension 2n ≥ 4. A conformal structure of signature (n, n)
on M is given by a class, c, of conformally equivalent pseudo-Riemannian metrics of signature
(n, n): two metrics g and ˆg are conformally equivalent if ˆg = f2g for a nowhere-vanishing
smooth function f on M. It may be equivalently described as a reduction of the frame bundle
of M to the structure group CO(n, n) = R+ × SO(n, n). An oriented conformal structure of
signature (n, n) is a conformal structure of signature (n, n) together with fixed orientations both
in time-like and space-like directions, equivalently, a reduction of the frame bundle to the group
COo(n, n) = R+ × SOo(n, n), the connected component of the identity. An equivariant lift of
such a reduction with respect to the 2-fold covering CSpin(n, n) = R+ ×Spin(n, n) → COo(n, n)
is referred to as a conformal spin structure M, c of signature (n, n).
A conformal spin structure of signature (n, n) is equivalently encoded as a normal parabolic
geometry of type G, P , where G = Spin(n + 1, n + 1) and P = CSpin(n, n) Rn,n∗ is the
stabiliser of an isotropic ray in the standard representation Rn+1,n+1.
6 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
A general Weyl connection is a torsion-free affine connection D such that Dg ∈ c for any
g ∈ c. If Dg = 0, i.e., D is the Levi-Civita connection of a metric g ∈ c, it is an exact Weyl
connection. A choice of Weyl connection reduces the structure group to G0 = CSpin(n, n). If
the Weyl connection is exact the structure group is further reduced to Gss
0 = Spin(n, n).
Now we briefly introduce the main curvature quantities of conformal structures, cf., e.g., [16].
For g ∈ c, the Schouten tensor,
P = P(g) =
1
2n − 2
Ric(g) −
Sc(g)
2(2n − 1)
g ,
is a trace modification of the Ricci curvature Ric(g) by a multiple of the scalar curvature Sc(g);
its trace is denoted J = gpqPpq. The conformal Weyl curvature and the Cotton tensors are
W c
ab d = R c
ab d − 2δc
[aPb]d + 2gd[aP
c
b], Ycab = 2D[aPb]c.
As for projective structures, we will employ a suitable parametrisation of densities so that
the canonical 1-dimensional representation of P has conformal weight −1. Hence, the density
bundle of conformal weight w, denoted as E[w], is just the bundle of ordinary −w
2n -densities.
As an associated bundle to the Cartan bundle G → M, it corresponds to the 1-dimensional
representation of P given by
(R+ × Spin(n, n)) R2n∗
→ R+, (a, A, Z) → a−w
. (2.4)
In particular, the conformal structure may be seen as a section of E(ab)[2], which is called the
conformal metric and denoted by gab.
The spin bundles corresponding to the irreducible spin representations of Spin(n, n) are denoted
by Σ+ and Σ−, and Σ = Σ+ ⊕ Σ−. We employ the weighted conformal gamma matrix
γ ∈ Γ Ea ⊗ End Σ [1] such that γpγq + γqγp = −2gpq. For ξ ∈ X M and χ ∈ Γ Σ , the
Clifford multiplication of ξ on χ is then written as ξ · χ = ξpγpχ.
The conformal standard tractor bundle is the associated bundle T := G ×P
Rn+1,n+1 with
respect to the standard representation. It carries the canonical tractor metric h and the conformal
standard tractor connection T , which preserves h. With respect to a metric g ∈ c, we
have
T =



E[−1]
Ea[1]
E[1]


 , h =


0 0 1
0 g 0
1 0 0

 , T
c


ρ
ϕa
σ

 =



Dcρ − P
b
c ϕb
Dcϕa + σPca + ρgca
Dcσ − ϕc


 . (2.5)
The BGG-splitting operator is given by
LT
0 : Γ E[1] → Γ T , σ →


1
2n −DpDp − J σ
Daσ
σ

 . (2.6)
The spin tractor bundle is the associated bundle S := G ×P
∆n+1,n+1, where ∆n+1,n+1 is the
spin representation of G = Spin(n + 1, n + 1). Since we work in even signature, it decomposes
into irreducibles ∆n+1,n+1 = ∆n+1,n+1
+ ⊕ ∆n+1,n+1
− ; the corresponding bundles are denoted by
S± = G ×P
∆n+1,n+1
± . Under a choice of g ∈ c, these decompose as S± =
Σ [−1
2
]
Σ±[ 1
2
]
, where Σ±
A Projective-to-Conformal Fefferman-Type Construction 7
are the natural spin bundles as before. For later use we record the formulas for the Clifford
action of T on S and for the spin tractor connections on S = S+ ⊕ S−,


ρ
ϕa
σ

 ·
τ
χ
=
−ϕaγaτ +
√
2ρχ
ϕaγaχ −
√
2στ
, S
c
τ
χ
=
Dcτ + 1√
2
Pcpγpχ
Dcχ + 1√
2
γcτ
, (2.7)
cf. [19]. The BGG-splitting operator of S± is
L
S±
0 : Γ Σ±
1
2 → Γ S± , χ →
1√
2n
D/ χ
χ
, (2.8)
where D/ : Γ Σ± → Γ Σ , D/ := γpDp, is the Dirac operator. The first BGG-operator associated
to S± is the twistor operator
ΘS
0 : Γ Σ±
1
2 → Γ Ea ⊗ Σ±
1
2 , χ → Daχ + 1
2n γaD/ χ,
cf., e.g., [3]. Elements in the kernel of ΘS
0 are called twistor spinors. It is well known that ΠS
0
induces an isomorphism between S-parallel sections of S with ker ΘS
0 .
The adjoint tractor bundle is the associated bundle AM := G ×P
˜g with respect to the adjoint
representation of G on ˜g = so(n+1, n+1) ∼= Λ2Rn+1,n+1. The standard pairing on AM induced
by the Killing form on ˜g is denoted as ·, · : AM × AM → R. Henceforth we identify AM
with Λ2T . With respect to a metric g ∈ c,
AM =



Ea[0]
E[a0a1][2] E[1]
Ea[2]


 .
The standard representation of g on Rn+1,n+1 gives rise to the map
•: AM ⊗ T → T ,


ρa
µa0a1 ϕ
βa

 •


ν
ωb
σ

 =


ρrωr − ϕν
µb
rωr − σρb − νβb
βrωr + ϕσ

 . (2.9)
The normal tractor connection is given by
AM
c


ρa
µa0a1 ϕ
ka

 =





Dcρa − P
p
c µpa − Pcaϕ
Dcµa0a1 + 2gc[a0
ρa1]
+2Pc[a0
ka1]
Dcϕ − P
p
c kp + ρc
Dcka − µca + gcaϕ





. (2.10)
Written as a two-form Ω with values in Λ2T , the curvature of T is
Ωc0c1 =


−Yac0c1
Wc0c1a0a1 0
0

 ∈ Γ E[c0c1] ⊗ AM . (2.11)
The BGG-splitting operator
LAM
0 : Γ Ea
= Γ Ea[2] → Γ AM , ka →


ρa
µa0a1 ϕ
ka

 ,
8 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
is determined by
µa0a1 = D[a0
ka1], ϕ = −
1
2n
gpq
Dpkq, (2.12)
ρa = −
1
4n
Dp
Dpka +
1
4n
Dp
Dakp +
1
4n2
DaDp
kp +
1
n
P
p
akp −
1
2n
Jka,
and the corresponding first BGG-operator of AM is computed as
ΘAM
0 : Γ Ea[2] → Γ E(ab)0
[2] , ξa → D(cξa)0
,
where the subscript 0 denotes the trace-free part. Thus ΘAM
0 is the conformal Killing operator
and solutions to the first BGG-equation are conformal Killing fields. In a prolonged form, the
conformal Killing equation is equivalent to
AM
b s = ξa
Ωab, (2.13)
where s = LAM
0 (ξ), see [7, 18].
3 The Fefferman-type construction
The construction of split-signature conformal structures from projective structures discussed in
this section fits into a general scheme relating parabolic geometries of different types. Namely, it
is an instance of the so-called Fefferman-type construction, whose name and general procedure
is motivated by Fefferman’s construction of a canonical conformal structure induced by a CR
structure, see [6] and [11] for a detailed discussion.
3.1 General procedure
Suppose we have two pairs of semi-simple Lie groups and parabolic subgroups, (G, P) and G, P ,
and a Lie group homomorphism i: G → G such that the derivative i : g → ˜g is injective. Assume
further that the G-orbit of the origin in G/P is open and that the parabolic P ⊆ G contains
Q := i−1 P , the preimage of P ⊆ G.
Given a parabolic geometry (G → M, ω) of type (G, P), one first forms the Feﬀerman space
M := G/Q = G ×P P/Q. (3.1)
Then G → M, ω is automatically a Cartan geometry of type (G, Q). As a next step, one
considers the extended bundle G := G ×Q P with respect to the homomorphism Q → P. This is
a principal bundle over M with structure group P and j : G → G denotes the natural inclusion.
The equivariant extension of ω ∈ Ω1(G, g) yields a unique Cartan connection ωind ∈ Ω1 G, ˜g
of type G, P such that j∗ωind = i ◦ ω. Altogether, one obtains a functor from parabolic
geometries (G → M, ω) of type (G, P) to parabolic geometries G → M, ωind of type G, P .
The relation between the corresponding curvatures is as follows: The previous assumptions
yield a linear isomorphism ˜g/˜p ∼= g/q and an obvious projection g/q → g/p, where q ⊆ p is the
Lie algebra of Q ⊆ P. Composing these two maps one obtains a linear projection ˜g/˜p → g/p,
whose dual map is denoted as ϕ: (g/p)∗ → (˜g/˜p)∗. Since i : g → ˜g is a homomorphism of Lie
algebras, the curvature function κind : G → Λ2(˜g/˜p)∗ ⊗ ˜g is related to κ: G → Λ2(g/p)∗ ⊗ g by
κind ◦ j = (Λ2ϕ ⊗ i ) ◦ κ. We note that κind is fully determined by this formula.
Since i is an embedding, the notation is in most cases simplified such that we write g ⊆ ˜g,
q = g ∩ ˜p, etc.
A Projective-to-Conformal Fefferman-Type Construction 9
3.2 Algebraic setup and the homogeneous model
Here we specify the general setup for Fefferman-type constructions from Section 3.1 according
to the description of oriented projective and conformal spin structures given in Sections 2.4
and 2.5, respectively. Let Rn+1,n+1 be the real vector space R2n+2 with an inner product, h, of
split-signature. Let ∆n+1,n+1
+ and ∆n+1,n+1
− be the irreducible spin representations of
G := Spin(n + 1, n + 1)
as in Section 2.5. We fix two pure spinors sF ∈ ∆n+1,n+1
− and sE ∈ ∆n+1,n+1
± with non-trivial
pairing, which is assigned for later use to be sE, sF = −1
2. Note that sE lies in ∆n+1,n+1
+ if n
is even or in ∆n+1,n+1
− if n is odd.
Let us denote by E, F ⊆ Rn+1,n+1 the kernels of sE, sF with respect to the Clifford multiplication,
i.e.,
E := X ∈ Rn+1,n+1
: X · sE = 0 , F := X ∈ Rn+1,n+1
: X · sF = 0 .
The purity of sE and sF means that E and F are maximally isotropic subspaces in Rn+1,n+1.
The other assumptions guarantee that E and F are complementary and dual each other via the
inner product h. Hence we use the decomposition
Rn+1,n+1
= E ⊕ F ∼= Rn+1
⊕ Rn+1∗
(3.2)
to identify the spinor representation ∆n+1,n+1 = ∆n+1,n+1
+ ⊕ ∆n+1,n+1
− with the exterior power
algebra Λ•E ∼= Λ•Rn+1, whose irreducible subrepresentations are ∆n+1,n+1
−
∼= ΛevenRn+1 and
∆n+1,n+1
+
∼= ΛoddRn+1. When n is even, respectively, odd, we can identify ∆n+1,n+1
−
∗ ∼=
∆n+1,n+1
+ , respectively ∆n+1,n+1
−
∗ ∼= ∆n+1,n+1
− .
Now, let us consider the subgroup in G defined by
G := {g ∈ Spin(n + 1, n + 1): g · sE = sE, g · sF = sF }.
This subgroup preserves the decomposition (3.2) so that the restriction of the action to F is dual
to the restriction to E. It further preserves the volume form on E, respectively F ∼= E∗, which
is determined by sE and sF according to the previous identifications. Hence G ∼= SL(n + 1) and
this defines an embedding i: SL(n + 1) → Spin(n + 1, n + 1).1
The G-invariant decomposition (3.2) determines a G-invariant skew-symmetric involution
K ∈ so(n + 1, n + 1) acting by the identity on E and minus the identity on F. The relationship
among K, sE and sF may be expressed as
h(X, K(Y )) = −h(K(X), Y ) = 2 sE, (X ∧ Y ) · sF , (3.3)
where
(X ∧ Y ) · sF =
1
2
(X · Y · sF − Y · X · sF ) = X · Y · sF + h(X, Y )sF .
The spin action of ˜g is denoted by •, and thus A • s = −1
4A · s, for any A ∈ ˜g and s ∈ ∆. In
particular, K •sF = −1
2(n+1)sF and K •sE = 1
2(n+1)sE. Here we identify ˜g = so(n+1, n+1)
with Λ2Rn+1,n+1. It is convenient to split ˜g in terms of irreducible g-modules as
˜g = Λ2
(E ⊕ F) = (E ⊗ F)0
g=sl(n+1)
⊕ (E ⊗ F)Tr ⊕ Λ2
E ⊕ Λ2
F
g⊥
, (3.4)
1
Instead of the embedding SL(n + 1) → Spin(n + 1, n + 1) we could also consider the embedding SL(n + 1) →
SO(n + 1, n + 1). The advantage of employing the embedding into the spin group is two-fold: on the one hand, it
is then seen directly that the induced conformal structure has a canonical spin structure, and, on the other hand,
we can then use convenient spinorial objects for its characterisation.
10 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
where (E ⊗ F)Tr = RK, and K acts as [K, φ] = 2φ, [K, ψ] = −2ψ, [K, λ] = 0, for any φ ∈ Λ2E,
ψ ∈ Λ2F and λ ∈ E ⊗ F. Further, the annihilators of sE and sF in ˜g are the subalgebras
ker sE = sl(n + 1) ⊕ Λ2E and ker sF = sl(n + 1) ⊕ Λ2F.
The homogeneous model for conformal spin structures of signature (n, n) is the space of
isotropic rays in Rn+1,n+1, G/P ∼= Sn × Sn. The subgroup G ⊆ G does not act transitively on
that space. According to the decomposition (3.2), there are three orbits: the set of rays contained
in E, the set of rays contained in F, and the set of isotropic rays that are neither contained in E
nor in F. Note that only the last orbit is open in G/P, which is one of the requirements from
Section 3.1. Therefore, we define P ⊆ G to be the stabiliser of a ray through a light-like vector
˜v ∈ Rn+1,n+1 \(E ∪F). Denoting by Q = i−1(P) the stabiliser of the ray R+˜v in G, we have the
identification of G/Q with the open orbit of the origin in G/P. The subgroup Q, which is not
parabolic, is contained in the parabolic subgroup P ⊆ G defined as the stabiliser in G of the
ray through the projection of ˜v to E. In particular, G/P is the standard projective sphere Sn,
the homogeneous model of oriented projective structures of dimension n, and G/Q → G/P is
the canonical fibration with the standard fibre P/Q, whose total space is the model Fefferman
space.
Let us denote by L = R˜v the line spanned by the light-like vector ˜v and let L⊥ be the
orthogonal complement in Rn+1,n+1 with respect to h. The tangent space of G/Q at the origin
can be seen in three different ways, namely,
L⊥
/L [1] ∼= g/q ∼= ˜g/˜p.
The latter isomorphism is induced by the embedding g ⊆ ˜g, the former one by the standard
action of g ⊆ ˜g on the vector ˜v ∈ Rn+1,n+1. Both these identifications are Q-equivariant.
There are several natural Q-invariant objects that in turn yield distinguished geometric objects
on the general Fefferman space. The n-dimensional Q-invariant subspace
f := ¯F + L /L [1] ⊆ L⊥
/L [1], where ¯F := F ∩ L⊥
,
which is isomorphic to p/q ⊆ g/q, the kernel of the projection g/q → g/p. Another n-dimensional
Q-invariant subspace is
e := ¯E + L /L [1] ⊆ L⊥
/L [1], where ¯E := E ∩ L⊥
.
The intersection e ∩ f is 1-dimensional with a distinguished Q-invariant generator that corresponds
to the G-invariant involution K ∈ ˜g,
k := K + ˜p ∈ ˜g/˜p.
Note that all these objects are isotropic with respect to the natural conformal class induced
by the restriction of h to L⊥ ⊆ Rn+1.n+1. In particular, both e and f are maximally isotropic
subspaces such that
k ∈ e ∩ f ⊆ k⊥
= e + f. (3.5)
In Section 3.1 we introduced a map ϕ: (g/p)∗ → (˜g/˜p)∗, the dual map to the projection
˜g/˜p ∼= g/q → g/p. The kernel of this projection is just f and the image of ϕ is identified with its
annihilator, which will be denoted by f◦. Since f is a maximally isotropic subspace in ˜g/˜p ∼= g/q,
f◦ ∼= f[−2].
Since (˜g/˜p)∗ ∼= ˜p+, we may conclude with the help of explicit matrix realisations from Appendix
A that f◦ = ˜p+ ∩ ker sF . Moreover, we note that
˜p+ ∩ ker sF E⊗F
= p+, ˜p ∩ ker sF E⊗F
= p, (3.6)
Λ2
F ∩ ˜p = Λ2 ¯F ⊆ ˜g0, ˜p+, Λ2 ¯F = f◦
, f◦
, Λ2 ¯F = 0. (3.7)
A Projective-to-Conformal Fefferman-Type Construction 11
3.3 The Fefferman space and induced structure
The pairs of Lie groups (G, P) and G, P from the previous subsection satisfy all the properties
to launch the Fefferman-type construction.
Proposition 3.1. The Feﬀerman-type construction for the pairs of Lie groups (G, P) and
G, P yields a natural construction of conformal spin structures M, c of signature (n, n)
from n-dimensional oriented projective structures (M, p). The Feﬀerman space M is identiﬁed
with the total space of the weighted cotangent bundle without the zero section T∗M(2)\{0}.
Proof. The first part of the statement is obvious from the general setting for Fefferman-type
constructions and the Cartan-geometric description of oriented projective and conformal spin
structures.
The second part is shown due to two natural identifications: On the one hand, the Fefferman
space is by (3.1) equal to the total space of the associated bundle M ∼= G ×P P/Q over M. On
the other hand, the weighted cotangent bundle to M is identified with the associated bundle
T∗M(2) ∼= G ×P (g/p)∗(2) with respect to action of P induced by the adjoint action and the
representation (2.3) for w = 2. Hence it remains to verify that the action of P on (g/p)∗(2)\{0}
is transitive and Q is a stabiliser of a non-zero element. But this is a purely algebraic task,
which may be easily checked in a concrete matrix realisation.
From the algebraic setup in Section 3.2 we easily conclude number of specific features of the
induced conformal structure on M:
Proposition 3.2. The conformal spin structure M, c induced from an oriented projective
structure (M, p) by the Feﬀerman-type construction admits the following tractorial objects that
are all parallel with respect to the induced tractor connection:
(a) pure tractor spinors sE ∈ Γ S± and sF ∈ Γ S− with non-trivial pairing,
(b) a tractor endomorphism K ∈ Γ AM which is an involution, i.e., K2 = idT
, and which
acts by the identity, respectively minus the identity on the maximally isotropic complementary
subbundles E := ker sE, respectively F := ker sF of T .
The corresponding underlying objects η = ΠS
0 (sE), χ = ΠS
0 (sF ) and k = ΠAM
0 (K) satisfy:
(c) η ∈ Γ Σ±
1
2 and χ ∈ Γ Σ−
1
2 are pure spinors, whose kernels e := ker η and f := ker χ
have 1-dimensional intersection and f coincides with the vertical subbundle of M → M,
(d) k ∈ Γ TM is a nowhere-vanishing light-like vector ﬁeld generating the intersection e ∩ f.
Proof. The G-invariant spinor sE ∈ ∆± gives rise to the tractor spinor sE ∈ Γ S± = G ×Q
∆± such that it corresponds to the constant (Q-equivariant) map G → ∆±. Hence sE is
automatically parallel with respect to the induced tractor connection on S±. Similar reasoning
for other G-invariant objects and their compatibility described above yield the first part of
the statement. In particular, E = G ×Q E, F = G ×Q F and the decomposition T = E ⊕ F
corresponds to the decomposition (3.2).
The filtration L ⊆ L⊥ ⊆ Rn+1,n+1 gives rise to the filtration of the standard tractor bundle,
which can be written as


E[−1]
0
0

 ⊆


E[−1]
Ea[1]
0

 ⊆



E[−1]
Ea[1]
E[1]


 = T .
12 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
In particular, the subbundles associated to ¯E, ¯F ⊆ L⊥ are distinguished by the middle slot. The
corresponding Q-invariant maximally isotropic subspaces e, f ⊆ g/q determine the distributions
G ×Q e and G ×Q f in TM = G ×Q g/q. According to the tractor Clifford action (2.7) it
follows that these are precisely the kernels of the spinors η and χ. Since these subspaces are
maximally isotropic, the corresponding spinors are pure. Since f ∼= p/q is the kernel of the
projection g/q → g/p, the corresponding subbundle f is identified with the vertical subbundle
of the projection M → M. The intersection e ∩ f is 1-dimensional and it is generated by the
projection of K ∈ ˜g to ˜g/˜p. Indeed, K cannot be contained in ˜p, since K acts by the identity
on E and minus the identity on F and ˜p is the stabiliser of a line that is neither contained in E
nor in F. Altogether, the corresponding vector field k on M is a nowhere-vanishing generator
of e ∩ f, in particular, it is light-like.
3.4 Relating tractors, Weyl structures and scales
As a technical preliminary for further study we now relate natural objects associated to the
original projective Cartan geometry (G, ω) on M and the induced conformal geometry (G, ωind)
on the Fefferman space M.
Since G ⊆ G, any G-representation V is also a G-representation, which yields compatible
tractor bundles over M and M with compatible tractor connections: V = G ×P V → M
with the tractor connection induced by ω and V = G ×P
V = G ×Q V → M with the
tractor connection ind induced by ωind. Sections of V bijectively correspond to P-equivariant
functions ϕ: G → V , while sections of V correspond to Q-equivariant functions ϕ: G → V . Since
Q ⊆ P, every section of V gives rise to a section of V, and we can view Γ(V) ⊆ Γ V . Now,
Proposition 3.2 in [8] admits a straightforward generalisation to Fefferman-type constructions
for which P/Q is connected and thus, in particular, to the one studied in this article:
Proposition 3.3.
(a) A section s ∈ Γ V is contained in Γ(V) (i.e., the corresponding Q-equivariant function ϕ
is indeed P-equivariant) if and only if inds is strictly horizontal (i.e., va ind
a s = 0 for all
va ∈ Γ f ).
(b) The restriction of ind to Γ(V) ⊆ Γ V coincides with the tractor connection .
Remark 3.4. Another instance of compatible bundles over M and M is provided by the density
bundles E(w) and E[w], which are defined via the representation of P and P as in (2.3) and (2.4),
respectively. Restricting these representations to Q, it easily follows that the notation is indeed
compatible so that we can view Γ(E(w)) ⊆ Γ E[w] .
Both projective and conformal density bundles can be described as associated bundles to
the respective bundles of scales. Hence everywhere positive sections of density bundles are
considered as scales. In particular, the inclusion Γ(E+(1)) ⊆ Γ E+[1] may be interpreted so
that any projective scale induces a conformal one. Such conformal scales will be called reduced
scales. An intrinsic characterisation of reduced scales among all conformal ones is formulated in
Proposition 5.2.
The previous remark yields that any projective exact Weyl structure on M induces a conformal
exact Weyl structure on M. This fact can be generalised as follows:
Proposition 3.5. Any projective (exact) Weyl structure on M induces a conformal (exact)
Weyl structure on the Feﬀerman space M.
Proof. A version of this result in a more general context was proved in [1, Proposition 6.1]: any
Weyl structure for ω induces a Weyl structure for ωind if P+ ⊆ P and G0 ∩ P ⊆ G0. But both
A Projective-to-Conformal Fefferman-Type Construction 13
these conditions are satisfied as follows from the setup in Section 3.2 and explicit realisations in
Appendix A.
Conformal Weyl structures induced by projective ones as above will be called reduced Weyl
structures.
3.5 Normality
Here we show that our Fefferman-type construction does not preserve the normality in general,
see Proposition 3.8. This can be shown directly as we did in a previous version of the article,
see arXiv:1510.03337v2. Alternatively, we can treat the construction as the composition of two
other constructions via a natural intermediate Lagrangean contact structure.
A Lagrangean contact structure on M consists of a contact distribution H ⊆ TM together
with a decomposition H = e ⊕f into two subbundles that are maximally isotropic with respect
to the Levi form H × H → TM /H. Such structure on a manifold M of dimension 2n − 1 is
equivalently encoded as a normal parabolic geometry of type (G, P ), where G = SL(n + 1) and
P ⊆ G is the stabiliser of a flag of type line-hyperplane in the standard representation Rn+1.
For n > 2 there are three harmonic curvatures, two of which are torsions whose vanishing is
equivalent to the integrability of the respective subbundles e , f ⊆ H. For n = 2 there are two
harmonic curvatures of homogeneity 4, hence the Cartan connection is torsion-free. In that case
both e and f are 1-dimensional and thus automatically integrable.
On the one hand, P is contained in P, where P ⊆ G is the stabiliser of a ray in Rn+1. For
suitable choices as in Appendix A, the Lie algebra to P consists of matrices of the form
p =


a Ut w
0 B V
0 0 c

 .
Given a projective Cartan geometry (G → M, ω) of type (G, P), it turns out that the correspondence
space M := G/P can be identified with the projectivised cotangent bundle P(T∗M).
The Cartan geometry (G → M , ω) of type (G, P ) is regular and thus it covers a natural Lagrangean
contact structure on M . In particular, the canonical contact distribution on P(T∗M)
coincides with H and the vertical subbundle of the projection M → M coincides with one of the
two distinguished subbundles, say f ⊆ H. As in general, this construction preserves normality.
In accord with [5], respectively [11, Section 4.4.2] we may state:
Proposition 3.6. Let (G → M, ω) be a normal projective parabolic geometry and let (G →
M , ω) be the corresponding normal Lagrangean contact parabolic geometry. The latter geometry
is torsion-free if and only if n = 2 or it is ﬂat, i.e., the initial projective structure is ﬂat.
On the other hand, P contains Q, where Q = G ∩ P as before. This allows us to consider
the Fefferman-type construction for the pairs (G, P ) and (G, P). Given a Lagrangean contact
structure on M , it induces a conformal spin structure on M = G/Q. This construction is indeed
very similar to the original Fefferman construction; one deals with different real forms of the
same complex Lie groups in the two cases. That is why the following statement and its proof
is analogous to the one for the CR case. Following [8], respectively [11, Section 4.5.2] we may
state:
Proposition 3.7. Let (G → M , ω) be the normal Lagrangean contact parabolic geometry and let
G → M, ωind be the conformal parabolic geometry obtained by the Feﬀerman-type construction.
Then ωind is normal if and only if ω is torsion-free.
14 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
Altogether, composing the two previous steps we obtain our projective-to-conformal Fefferman-type
construction with the desired control of the normality. Note that from (3.5) and the
respective matrix realisations it follows that the induced objects on M = T∗M(2) \ {0} from
Proposition 3.2 correspond to the induced objects on M = P(T∗M). In particular, the vertical
subbundle of the projection M → M is spanned by k and the decomposition k⊥ = e ⊕ f ⊆ TM
descends to the decomposition H = e ⊕ f ⊆ TM
G
P

G
B
UU
Q
BBP
((
P

M
yy
M
xx
M.
Proposition 3.8. Let (G → M, ω) be a normal projective parabolic geometry and let G →
M, ωind be the conformal parabolic geometry obtained by the Feﬀerman-type construction.
(a) If dim M = 2 then ωind is normal.
(b) If dim M > 2 then ωind is normal if and only if ω is ﬂat.
Moreover, independently of the dimension of M, ωind is ﬂat if and only if ω is ﬂat.
3.6 Remarks on torsion-free Lagrangean contact structures
At this stage it is easy to formulate a local characterisation of split-signature conformal structures
arising from torsion-free Lagrangean contact structures, see Proposition 3.10. As before, the
results and their proofs are very analogous to those in the CR case, therefore we just quickly
indicate the reasoning and point to differences.
As in Proposition 3.2, the G-invariant algebraic objects induce the tractor fields sE, sF and K
on the conformal Fefferman space that are parallel with respect to the induced tractor connection
and have the required compatibility properties. But, starting with a torsion-free Lagrangean
contact structure, the induced connection is already normal. In particular, the corresponding
underlying objects χ, η and k are pure twistor spinors and a light-like conformal Killing field,
respectively.
The existence of parallel tractors sE, sF and K with the algebraic properties as in Proposition
3.2 are by no means independent conditions:
Proposition 3.9. Let M, c be a conformal spin structure of split-signature (n, n). Then the
following conditions are locally equivalent:
(a) The spin tractor bundle admits two pure parallel tractor spinors sE ∈ Γ S± and sF ∈
Γ S− with non-trivial pairing.
(b) The conformal holonomy Hol(c) reduces to SL(n + 1) ⊆ Spin(n + 1, n + 1) preserving
a decomposition into maximally isotropic subspaces E ⊕ F = Rn+1,n+1.
(c) The adjoint tractor bundle admits a parallel involution K ∈ Γ AM , i.e., K2 = idT
.
A Projective-to-Conformal Fefferman-Type Construction 15
The only subtle point within the proof concerns the consequences of property (c). The
existence of a parallel skew-symmetric involution K on the standard tractor bundle immediately
implies that the conformal holonomy Hol(c) is reduced to GL(n + 1). But, analogously to the
corresponding discussion for the CR case in [9] or [23], one can show that Hol(c) is actually
contained in SL(n + 1). The rest follows easily.
It turns out that conformal spin structures induced by torsion-free Lagrangean contact structures
are locally characterised by any of the three equivalent conditions above. Indeed, according
to results from [10], the holonomy reduction of the conformal structure to G = SL(n + 1) ⊆
Spin(n + 1, n + 1) = G yields the so-called curved orbit decomposition of M, which corresponds
to the decomposition of the homogeneous model G/P with respect to the action of G. Each
subset from the decomposition of M, provided it is non-empty, further carries a geometry of
the same type as its counterpart in the homogeneous model. From Section 3.2 we know there
is one open and two closed n-dimensional orbits. The closed n-dimensional orbits carry Cartan
geometries of type (G, P), and thus inherit projective structures, the open orbit carries a Cartan
geometry of type (G, Q). Note that the two closed orbits coincide with the zero sets of χ and η,
the open subset is the one where both spinors, and thus k, are non-vanishing. Since k is the
conformal Killing field corresponding to the parallel adjoint tractor K, it inserts trivially into
the curvature of the normal Cartan connection, cf. (2.13). Hence, according to [5], the Cartan
geometry of type (G, Q) on the open orbit of M descends to a Cartan geometry of type (G, P )
on the local leaf space M determined by k. It follows that this Cartan geometry is torsion-free
and thus determines a torsion-free Lagrangean contact structure. Altogether, following [9] we
may state the following characterisation:
Proposition 3.10. A split-signature conformal spin structure is locally induced by a torsionfree
Lagrangean contact structure via the Feﬀerman-type construction if and only if any of the
equivalent conditions from Proposition 3.9 holds and the underlying twistor spinors χ and η and
the conformal Killing ﬁeld k are nowhere-vanishing.
3.7 The exceptional case: dimension n = 2
From Section 3.5 we know that the intermediate 3-dimensional Lagrangean contact structure
on M induced by a 2-dimensional projective structure on M is torsion-free. Hence the induced
conformal Cartan geometry on M is normal and thus all the equivalent conditions from Proposition
3.9 are satisfied. Moreover, the fact that it comes from a projective structure implies that
any vertical vector of the projection M → M inserts trivially into the Cartan curvature, i.e.,
iXκ(u) = 0, for all X ∈ f, u ∈ G. (3.8)
Analogously to the discussion before Proposition 3.10 we may conclude:
Proposition 3.11. A conformal spin structure of signature (2, 2) is locally induced by a 2dimensional
projective structure via the Feﬀerman-type construction if and only if any of the
equivalent conditions from Proposition 3.9 holds, the underlying twistor spinors χ and η and
the conformal Killing ﬁeld k are nowhere-vanishing and the curvature of the normal conformal
Cartan connection satisﬁes (3.8).
Remark 3.12. Conformal structures induced from 2-dimensional projective structures are wellstudied,
see, e.g., [14, 15, 25]. Notably, the intermediate 3-dimensional Lagrangean contact
structure can be equivalently viewed as a path geometry (or the geometry associated to second
order ODEs modulo point transformations). Such structure is induced by a projective structure
(i.e., the paths are the unparametrised geodesics of the projective class of connections) if and
only if one of the two harmonic curvatures vanishes. It follows from [25] that this is equivalent to
16 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
vanishing of the self-dual, respectively anti-self-dual part of the Weyl curvature of the induced
conformal structure. In particular, the condition (3.8) in the previous proposition can be replaced
by the condition that the conformal structure is half-flat.
4 Normalisation and characterisation
By Proposition 3.8, for n ≥ 3, the induced conformal Cartan connection associated to a non-flat
n-dimensional projective structure differs from the normal conformal Cartan connection for the
induced conformal structure. In this section we will analyse the form of the difference and
thus derive properties of the induced conformal structures. Furthermore, we will show that any
split-signature conformal manifold having these properties is locally equivalent to the conformal
structure on the Fefferman space over a projective manifold.
4.1 The normalisation process
We are going to normalise the conformal Cartan connection ωind ∈ Ω1 G, ˜g that is induced by
a normal projective Cartan connection ω ∈ Ω1(G, g). Any other conformal Cartan connection ω
differs from ωind by some Ψ ∈ Ω1 G, ˜g so that ω = ωind + Ψ. This Ψ must vanish on vertical
fields and be P-equivariant. The condition on ω to induce the same conformal structure on M
as ωind is that Ψ has values in ˜p ⊆ ˜g. One can therefore regard Ψ as a P-equivariant function
Ψ: G → (˜g/˜p)∗ ⊗ ˜p. According to the general theory as outlined in [11, Section 3.1.13] there is
a unique such Ψ such that the curvature function κ of ω satisfies ∂∗κ = 0, and then ω is the
normal conformal Cartan connection ωnor.
The failure of ωind to be normal is given by ∂∗κind : G → (˜g/˜p)∗ ⊗˜p. The normalisation of ωind
proceeds by homogeneity of (˜g/˜p)∗ ⊗ ˜p, which decomposes into two homogeneous components
according to the decomposition ˜p = ˜g0 ⊕ ˜p+. In the first step of normalisation one looks for
a Ψ1 such that ω1 = ω + Ψ1 has ∂∗κ1 taking values in the highest homogeneity, i.e., ∂∗κ1 : G →
(˜g/˜p)∗ ⊗ ˜p+.
To write down this first normalisation we employ Weyl structures G0 →G. By Proposition 3.5
we can take a reduced Weyl structure, i.e., one that is induced by a reduction G0 →G → G with
respect to the structure group Q0 := Q∩G0. This allows us to project ∂∗κind to ∂∗κind
0
: G0 →
(˜g/˜p)∗ ⊗ ˜g0 and to employ the G0-equivariant Kostant Laplacian : (˜g/˜p)∗ ⊗ ˜g0 → (˜g/˜p)∗ ⊗ ˜g0,
:= ∂ ◦ ∂∗ + ∂∗ ◦ ∂. For the first normalisation step we need to form a map Ψ1 : G → (˜g/˜p)∗ ⊗ ˜p
that agrees with − −1 ∂∗κind
0
in the ˜g0-component. If we have formed any such Ψ1 along
G0 →G we can just equivariantly extend this to all of G.
To proceed with the analysis of the normalisation we need to establish a couple of technical
lemmas. As before, we denote by f◦ ⊂ ˜p+
∼= (˜g/˜p)∗ the annihilator of f = p/q ⊂ g/q ∼= ˜g/˜p.
Recall that f◦ = ϕ(p+) ∼= f[−2].
Lemma 4.1. Let V be a g-representation contained in a ˜g-representation V and denote by
φ → φ the inclusion Λkp+ ⊗ V → Λk˜p+ ⊗ V induced by ϕ: p+ → ˜p+ and V → V . Then, for
any φ ∈ Λkp+ ⊗ V ,
∂∗φ − ∂∗
φ ∈ Λk−1
f◦
⊗ Λ2 ¯F • V ⊆ Λk−1˜p+ ⊗ V .
In particular, for the adjoint representations, ∂∗φ = 0 if and only if ∂∗φ ∈ Λk−1f◦ ⊗ Λ2 ¯F.
Proof. For the sake of presentation, assume that φ is decomposable, i.e., of the form φ =
Z1 ∧ · · · ∧ Zk ⊗ v, where Zi ∈ p+ and v ∈ V . Let us denote by the same symbols also the images
of these elements under the inclusion g → ˜g and V → V , i.e., Zi ∈ ˜p and v ∈ V , respectively.
Let Zi ∈ f◦ be the images of Zi under the inclusion ϕ: p+ → ˜p+. Now, by definition of the
A Projective-to-Conformal Fefferman-Type Construction 17
Kostant co-differential, the difference ∂∗φ − ∂∗φ evaluated on any k − 1 elements from ˜g/˜p is
a linear combination of terms of the form
Zi − Zi • v. (4.1)
However, the differences Zi −Zi ∈ ˜p are represented by the matrices as in (A.1) in the Appendix
where only the Z-entries are non-vanishing and hence contained in Λ2F ∩ ˜p = Λ2 ¯F. Thus (4.1)
belong to the image of •: Λ2 ¯F × V → V and the first claim follows.
For the second claim we use that Λ2 ¯F • g = Λ2 ¯F, g ⊆ Λ2F and Λ2F ∩ g = 0: since ∂∗φ
(evaluated on any k − 1 elements from ˜g/˜p) has values in g ⊂ ˜g, vanishing of ∂∗φ is equivalent
to ∂∗φ having values in Λ2F. But ∂∗φ has generally values in ˜p and Λ2F ∩ ˜p = Λ2 ¯F, hence the
claim follows.
Lemma 4.2. If ψ ∈ ˜p+ ∧ f◦ ⊗ Λ2 ¯F ⊆ Λ2˜p+ ⊗ ˜p then ∂∗ψ ∈ f◦ ⊗ f◦ ⊆ ˜p+ ⊗ ˜p+.
Proof. ψ is a sum of terms of the form Z1 ∧ Z2 ⊗ A, where Z1 ∈ ˜p+, Z2 ∈ f◦ and A ∈ Λ2 ¯F.
Applying the Kostant co-differential gives
∂∗
(Z1 ∧ Z2 ⊗ A) = Z1 ⊗ [Z2, A] − Z2 ⊗ [Z1, A].
Now [Z2, A] belongs to f◦, Λ2 ¯F = 0 and [Z1, A] belongs to ˜p+, Λ2 ¯F = f◦, hence the claim
follows.
The following lemma contains the crucial information which is necessary to perform our
normalisation. We are going to specify the curvature function κind (later also κnor) by describing
its values along the natural Q-reduction G → G over M. Recall from Section 3.2 that Λ2 ¯F is
a Q-invariant subspace in ˜g0, which can be identified with (Λ2f)[−2].
Lemma 4.3. For any u ∈ G, we have
∂∗
κind
(u) ∈ f◦
⊗ Λ2 ¯F ⊆ ˜p+ ⊗ ˜g0.
Identifying Λ2 ¯F ∼= Λ2f [−2] and f◦ ∼= f[−2], we have in fact ∂∗κind(u) ∈ f Λ2f [−4], i.e.,
∂∗κind(u) is contained in the kernel of the alternation map
alt: f ⊗ Λ2
f [−4] → Λ3
f [−4].
Proof. It is a general assumption that ωind is induced by a normal projective Cartan connection
on G, i.e., ∂∗κ(u) = 0, for any u ∈ G. Hence it follows from Lemma 4.1 that ∂∗κind(u) belongs
to f◦ ⊗ Λ2 ¯F ∼= f ⊗ Λ2f [−4].
Further we need a finer discussion involving the properties of κ: G → Λ2p+ ⊗ g to show
that κ(u) belongs to the kernel of the Q-equivariant map Λ2p+ ⊗ g → Λ3f [−4] given by
φ → alt ∂∗
φ . (4.2)
Note that any element φ ∈ Λ2p+ ⊗ p+ for which ∂∗φ = 0 is mapped to zero: since φ ∈ Λ2˜p+ ⊗ ˜p
and [˜p+, ˜p] = ˜p+, the co-differential ∂∗φ has values in f◦ ⊗ ˜p+. But, by Lemma 4.1, it also has
values in f◦ ⊗ Λ2 ¯F and ˜p+ ∩ Λ2 ¯F = 0.
Thus it suffices to consider the harmonic elements from Λ2p+⊗g0, i.e., the ones corresponding
to the projective Weyl tensor. For that purpose we consider the simple part of Q0 = Q∩G0 which
is isomorphic to SL(n−1), cf. the matrix realisation (A.2) in the Appendix where it corresponds
to the A-block. Considering both Λ2p+ ⊗ g0
∼= Λ2Rn∗ ⊗ Rn∗ ⊗ Rn and Λ3f [−4] ∼= Λ3Rn∗ as
18 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
representations of SL(n−1), the map (4.2) is either trivial or an isomorphism on each SL(n−1)irreducible
component.
One can check that there is only one SL(n − 1)-irreducible component that occurs in both
spaces, and it is isomorphic to Λ2Rn−1∗
. Hence it suffices to compute (4.2) on one element
contained in such component. Let Xn ∈ g− and Zn ∈ p+ be the two dual basis vectors stabilised
by SL(n − 1) and consider an element
φ = Z1 ∧ Z2 ⊗ Xn ⊗ Zn − Z1 ∧ Z2 ⊗ X1 ⊗ Z1 + Zn ∧ Z2 ⊗ Xn ⊗ Z1.
Indeed φ is completely trace-free, satisfies the algebraic Bianchi identity and the SL(n−1)-orbit
of φ is isomorphic to Λ2Rn−1∗
. Now,
∂∗
φ = −Z1 ⊗ Zn ∧ Z2 − Zn ⊗ Z1 ∧ Z2,
which indeed lies in the kernel of the alternation map. Hence the statement follows.
We can now determine the form of the normal conformal Cartan connection:
Proposition 4.4. The normal conformal Cartan connection is of the form
ωnor
= ωind
+ Ψ1
+ Ψ2
,
where Ψ1 = −1
2∂∗κind ∈ Ω1
hor G, ˜p and Ψ2 ∈ Ω1
hor G, ˜p+ . Furthermore, along the reduction
G → G we have Ψ1 ∈ Ω1
hor G, Λ2 ¯F , Ψ2 ∈ Ω1
hor(G, f◦).
Remark 4.5. Since Ψ1 and Ψ2 are horizontal, they may equivalently be regarded as bundlevalued
1-forms on M. Denoting by Λ2F the associated bundle G ×Q Λ2 ¯F over M and by
f◦ ⊆ T∗M the annihilator of f = ker χ ⊆ TM, Proposition 4.4 says
Ψ1
∈ Ω1
M, Λ2
F , Ψ2
∈ Ω1
M, f◦
, Ψ1
(v) = Ψ2
(v) = 0, for all v ∈ Γ(ker χ).
Below we also use the corresponding frame forms, i.e., the P-equivariant functions φ1 : G →
(˜g/˜p)∗ ⊗ ˜p and φ2 : G → (˜g/˜p)∗ ⊗ ˜p+ such that, for any u ∈ G, Ψ1 = φ1(u) ◦ ωind and Ψ2 =
φ2(u) ◦ ωind. In these terms, the proposition means that along the reduction G → G these maps
restrict to Q-equivariant functions
φ1
: G → f◦
⊗ Λ2 ¯F, φ2
: G → f◦
⊗ f◦
.
Further we put Ψ = Ψ1 + Ψ2 and φ = φ1 + φ2.
Proof. The Kostant Laplacian restricts to an invertible endomorphism of ((˜g/˜p)∗⊗˜g0)∩im ∂∗
that acts by scalar multiplication on each of the G0-irreducible components. Now, restricting to
G → G and suppressing all arguments u ∈ G, it was shown in Lemma 4.3 that ∂∗κind is contained
in one of the irreducible components, namely in f Λ2f [−4]. On this component acts by
multiplication by 2. Thus, the modification map accomplishing the first normalisation step is
φ1
: G → f◦
⊗ Λ2 ¯F, φ1
:= −
1
2
∂∗
κind
= − −1
∂∗
κind
.
Now, let ω1 := ωind + φ1 ◦ ωind be the modified Cartan connection. The corresponding
curvature function κ1 can be expressed in terms of κind, φ1 and its differential dφ1 so that
κ1
(X, Y ) = κind
(X, Y ) + X, φ1
(Y ) − Y, φ1
(X)
+ dφ1
(ξ)(Y ) − dφ1
(η)(X) − φ1
([X, Y ]) + φ1
(X), φ1
(Y ) , (4.3)
A Projective-to-Conformal Fefferman-Type Construction 19
where X, Y ∈ g and ξ = ωind −1
(X), η = ωind −1
(Y ), cf. [11, formula (3.1)]. For the last term
we have φ1(X), φ1(Y ) = 0 since φ1(X) has values in Λ2 ¯F. The first three terms are
κind
(X, Y ) + X, φ1
(Y ) − Y, φ1
(X) = κind
(X, Y ) + ∂φ1
(X, Y ),
which by construction vanishes upon application of the Kostant co-differential, i.e., ∂∗ κind +
∂φ1 = 0. The remaining terms in (4.3) can be combined into a map Λ2g → Λ2 ¯F,
(X, Y ) → dφ1
(ξ)(Y ) − dφ1
(η)(X) − φ1
([X, Y ]),
which vanishes upon insertion of two elements X, Y ∈ p. Therefore, applying Lemma 4.2, we
conclude that ∂∗κ1 has values in f◦ ⊗ f◦. Thus the second modification map is
φ2
: G → f◦
⊗ f◦
, φ2
:= − −1
∂∗
κ1
.
4.2 Properties
The information provided in the previous proposition allows us to determine the properties
satisfied by the normal conformal Cartan curvature:
Proposition 4.6. The normal conformal Cartan curvature κnor restricts to a map
κnor
: G → Λ2
(˜g/˜p)∗
⊗ sl(n + 1) ⊕ Λ2
F . (4.4)
Moreover, the following integrability condition holds:
iXκnor
(u) ∈ f◦
⊗ Λ2 ¯F ⊕ f◦
, for all X ∈ f, u ∈ G. (4.5)
Proof. Let κnor be the curvature function of the normal Cartan connection ωnor = ωind + φ ◦
ωind, where φ = φ1 + φ2. With the same conventions as in the proof of Proposition 4.4, [11,
formula (3.1)] yields
κnor
(X, Y ) = κind
(X, Y ) + [X, φ(Y )] − [Y, φ(X)]
+ dφ(ξ)(Y ) − dφ(η)(X) − φ([X, Y ]) + [φ(X), φ(Y )].
Clearly, κind(X, Y ) has values in sl(n + 1) and vanishes upon insertion of X ∈ p. A term of
the form [X, φ(Y )] vanishes if Y ∈ p and has values in p, Λ2 ¯F ⊕ f◦ ⊆ Λ2 ¯F ⊕ f◦ for X ∈ p.
A term of the form dφ(ξ)(Y ) has values in Λ2 ¯F ⊕f◦ and vanishes for Y ∈ p. The term φ([X, Y ])
has values in Λ2 ¯F ⊕ f◦ and vanishes for X, Y ∈ p. The last term [φ(X), φ(Y )] vanishes for all
X, Y ∈ g since φ(X) has values in Λ2 ¯F ⊕ f◦. Altogether, we obtain (4.4) and (4.5).
We observe here that it follows directly from (4.4) that the pairing of κnor with the involution
K vanishes, κnor, K = 0.
To derive properties of induced tractorial and underlying objects on the conformal structure
we will need the following preparatory lemma.
Lemma 4.7. Let V be a G-representation and v ∈ V an element which is stabilised under
G ⊆ G. Let v ∈ Γ V be the section of the associated tractor bundle G ×P
V corresponding to
the constant function G → V , u → v, along G. Then the covariant derivative norv corresponds
to the Q-equivariant function
G → f◦
⊗ V, u → φ1
(u) • v + φ2
(u) • v.
20 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
Proof. The covariant derivative norv corresponds to the map
X ∈ g → ωnor −1
(X) · v + X • v. (4.6)
The first term in (4.6) vanishes since it is the directional derivative of the constant function v.
Now ωnor = ωind + φ1 + φ2, and since X • v = 0 the claim follows.
We now show that the distinguished tractors sE, sF and K on the Fefferman space are all
given as BGG-splittings from their underlying objects. Moreover, several stronger properties
hold:
Proposition 4.8. Let sE ∈ Γ S± , sF ∈ Γ S− and K ∈ Γ AM be the tractor spinors and
the adjoint tractor, respectively, and let η = ΠS
0 (sE), χ = ΠS
0 (sF ) and k = ΠAM
0 (K) be the
corresponding underlying objects as in Proposition 3.2.
(a) The tractor spinor sF is parallel, i.e., norsF = 0. In particular, χ is a pure twistor
spinor, sF = L
S−
0 (χ) and Hol(c) ⊆ SL(n + 1) Λ2 Rn+1 ∗
⊆ Spin(n + 1, n + 1).
(b) The tractor spinor sE is a BGG-splitting, i.e., ∂∗ norsE = 0 and sE = L
S±
0 (η).
(c) The adjoint tractor K is a BGG-splitting and k is a conformal Killing ﬁeld, i.e., ∂∗ norK
= 0, K = LAM
0 (k) and norK = ikΩnor. Moreover, we have
ikκnor
= 2φΛ2F . (4.7)
Proof. (a) Since φ1, φ2 have values in ker sF we have φ1 • sF + φ2 • sF = 0. Thus, according
to Lemma 4.7, we have norsF = 0 and the rest is obvious.
(b) The spinor sE is of the form sE = ( ∗
η ). According to Lemmas 4.3 and 4.7, φ1 has values
in f Λ2f [−4] and ∂∗ norsE corresponds to
∂∗
φ1
• sE =
φ1 • η Σ [−1
2
]
0
.
The projection φ1 • η Σ [−1
2
]
can be realised as the full (triple) Clifford action on φ1(u) ∈
3
f [−4], where u ∈ G. Now it is easy to see that this action must vanish for a φ1(u) ∈
f Λ2f [−4]: We realise φ1(u) equivalently in S2f ⊗ f [−4] by symmetrisation in the first
two slots, then the complete Clifford action on η vanishes because the action of the first two
slots is just a (trivial) trace multiplication.
(c) According to Lemma 4.7, norK corresponds to φ1 • K + φ2 • K. Since K/˜p = k ∈ f,
the previous element lies in ˜p. In particular, norK has trivial projecting slot, and thus k =
Π0(K) is a conformal Killing field. Since φ2 • K ∈ ˜p+, we have that ∂∗ norK corresponds to
∂∗ φ1 • K . Now φ1 • K = −K • φ1 = 2φ1, since K acts by multiplication with −2 on Λ2 ¯F.
But φ1 ∈ im ∂∗ ⊆ ker ∂∗, and the expression ∂∗ φ1 • K therefore vanishes. The equality
norK = ikΩnor is just (2.13) for the conformal Killing field k with its BGG-splitting K. In
terms of the Q-equivariant functions φ = φ1 + φ2 and κnor along G → G, this can be expressed
as φ • K = ikκnor, which yields (4.7).
We now collect the essential information about the induced conformal structure (M, c) which
we derived:
Proposition 4.9. Let M, c be the conformal spin structure induced from an oriented projective
structure (M, p) via the Feﬀerman-type construction. Then the following properties are satisﬁed:
A Projective-to-Conformal Fefferman-Type Construction 21
(a) M, c admits a nowhere-vanishing light-like conformal Killing ﬁeld k such that the corresponding
tractor endomorphism K = LAM
0 (k) is an involution, i.e., K2 = idT
.
(b) M, c admits a pure twistor spinor χ ∈ Γ Σ−
1
2 with k ∈ Γ(ker χ) such that the corresponding
parallel tractor spinor sF = L
S−
0 (χ) is pure.
(c) K acts by minus the identity on ker sF .
(d) The following integrability condition holds:
va
wc
Wabcd = 0, for all v, w ∈ Γ(ker χ). (W)
The only thing left to show for Proposition 4.9 is that the integrability condition (4.5) is
equivalent to the condition (W) on the Weyl tensor:
Lemma 4.10. Let M, c be a split-signature conformal spin structure endowed with tractors
sE, sF and K satisfying conditions (a) and (b) from Proposition 4.9. Then condition (4.5)
is equivalent to (W).
Proof. The implication (4.5) =⇒ (W) is obvious. It remains to prove the converse implication
(W) =⇒ (4.5).
By (W), one has that iXκnor
˜g0
(u) ∈ f ⊗ Λ2f [−4] ⊆ f◦ ⊗ Λ2 ¯F for X ∈ f, u ∈ G. Since sF
is parallel with respect to nor, we have κnor(u) ∈ Λ2(˜g/˜p)∗ ⊗ ˜p ∩ ker sF . The projection of
˜p ∩ ker sF to ˜p+ is precisely f◦, hence it follows that iXκnor
˜p+
(u) ∈ (˜g/˜p)∗ ⊗ f◦, and we obtain
iXκnor(u) ∈ (˜g/˜p)∗ ⊗ Λ2 ¯F ⊕ f◦ .
We now prove that iX1 iX2 κnor = 0 for all X1, X2 ∈ f. For this purpose it will be useful
to work with the curvature form Ωnor, which we can represent as in (2.11). By (W) and the
algebraic Bianchi identity, Wabcd vanishes upon insertion of v, w ∈ Γ(ker χ) into any two slots,
and in particular vawbWabcd = 0. Thus, it remains to check that vawbYdab = 0. As in the
proof of Proposition 3.2, a vector field w ∈ Γ(ker χ) corresponds to a section
∗
wd
0
∈ Γ F .
According to (2.9),
va
Ωnor
ab •


∗
wd
0

 =


−vaYrabwr
vaW c
ab dwd
0

 ∈ Γ Eb ⊗ T .
Since ivΩnor annihilates F, it follows that vawrYrab = 0. Using Yrab = −Ybra − Yabr, we obtain
also vawbYrab = 0.
4.3 Characterisation
We are now going to characterise the induced conformal structures. For this purpose we will
introduce the following (’intermediate’) Cartan connection form:
ω := ωnor
−
1
2
ikκnor
. (4.8)
The following observation then follows immediately from Proposition 4.4 and formula (4.7):
Lemma 4.11. The pullbacks of the Cartan connection forms ω ∈ Ω1
hor G, ˜g , ωnor ∈ Ω1
hor G, ˜g
and ωind ∈ Ω1
hor G, ˜g to G → G agree modulo forms with values in p+ ⊆ sl(n + 1) ⊆ g.
22 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
For the rest of this section, we will start with a given split-signature conformal spin structure
M, c satisfying all the properties of Proposition 4.9. In particular, M is endowed with
a conformal Killing field k ∈ Γ TM , and we can still use formula (4.8) to define a Cartan
connection ω . The corresponding tractor connection will be denoted by and the curvature
by Ω or κ . The following proposition now shows that the so constructed Cartan connection ω
is in fact an SL(n + 1)-connection.
Proposition 4.12. Let M, c be a split-signature conformal (spin) structure satisfying all the
properties of Proposition 4.9. Then the sections sF and K are parallel with respect to the tractor
connection , i.e., sF = 0 and K = 0.
In particular, Hol(ω ) ⊆ SL(n + 1) ⊆ Spin(n + 1, n + 1) and ω pulls back to a Cartan
connection of type (SL(n + 1), Q) with respect to the Q-reduction G → G. Along that reduction,
the curvature functions κ and κnor are related according to κ = κnor
sl(n+1)
and κ satisﬁes
the following integrability condition:
iXκ (u) ∈ f◦
⊗ p+, for all X ∈ f, u ∈ G. (4.9)
Proof. A tractor connection induced by ω can be written as = nor +Ψ with Ψ = −1
2ikΩ1.
That sF = 0 follows immediately from the fact that − nor = −1
2ikΩnor has values in Λ2F.
Since k is a conformal Killing field we have norK = ikΩnor. By definition
K = nor
K −
1
2
ikΩnor
• K,
which vanishes, since ikΩnor has values in Λ2F and therefore 1
2ikΩnor • K = ikΩnor. As in
Proposition 4.9, we write the decomposition of T into maximally isotropic eigenspaces of K
with eigenvalues ±1 as E ⊕ F. Since K is -parallel, it follows that this decomposition is
preserved by . Moreover, since F is the kernel of the pure tractor spinor sF it follows that
Hol(ω ) ⊆ SL(n + 1). In particular, ω reduces to a Cartan connection of type (SL(n + 1), Q) on
a Q-principal bundle G ⊆ G.
We further compute that
Ω = Ωnor
−
1
2
d
nor
ikΩnor
= Ωnor
−
1
2
d
nor nor
K
= Ωnor
−
1
2
Ωnor
• K = Ωnor
+
1
2
K • Ωnor
= Ωnor
(E⊗F)0
,
where we are again using norK = ikΩnor for the conformal Killing field k and that Ωnor has
values in E ⊗ F ⊕ Λ2F. Stated for the corresponding curvature functions, this yields κ =
κnor
sl(n+1)
. Moreover, since κnor has values in ker sF ∩ p, it follows from (3.6) that (˜p ∩
ker sF )sl(n+1) = p, thus κ has values in p.
We know from (4.5) that iXκnor has values in Λ2 ¯F ⊕ f◦ for X ∈ f. But since Λ2 ¯F sl(n+1)
=
0 and (˜p+ ∩ ker sF )sl(n+1) = p+, we obtain that (iXκnor)sl(n+1) has values in p+. Finally,
iX1 iX2 κnor
sl(n+1)
= 0 for X1, X2 ∈ f follows immediately from (4.5), and altogether we obtain
(4.9).
Next, before proving the main characterisation Theorem 4.14, we will show the following
proposition on factorisations of particular Cartan geometries. This proposition can be understood
as an adapted variant of [5, Theorem 2.7].
Proposition 4.13. Let G → M, ω be a Cartan geometry of type (SL(n+1), Q) with curvature
κ: G → Λ2(g/q)∗ ⊗ g and let the following conditions be satisﬁed:
iX1 iX2 κ(u) ∈ p, for all X1, X2 ∈ g/q, u ∈ G,
A Projective-to-Conformal Fefferman-Type Construction 23
iX1 iX2 κ(u) ∈ p+, for all X1 ∈ p/q, X2 ∈ g/q, u ∈ G,
iX1 iX2 κ(u) = 0, for all X1, X2 ∈ p/q, u ∈ G.
Then G is locally a P-bundle over M = G/P and ω deﬁnes a canonical projective structure
on M.
Proof. The third of the above listed conditions implies that G is locally a P-bundle G → M
by [5]. We will restrict G to assume this globally. We define M = G/P and G0 = G/P+.
Let σ: G0 → G be a G0-equivariant splitting. It follows from the second of the above listed
conditions that
LζX
ω = −ad(X) ◦ ω mod p+, for all X ∈ p.
Now define θ ∈ Ω1(G0, g−), γ ∈ Ω1(G0, g0) and ρ ∈ Ω1(G0, p+) via the decomposition σ∗ω =
θ⊕γ ⊕ρ. Since σ is G0-equivariant and the Lie derivative is compatible with pullbacks it follows
that
L¯ζX
(θ ⊕ γ) = −ad(X) ◦ (θ ⊕ γ), for all X ∈ g0.
In particular, θ and γ are G0-equivariant and define a (reductive) Cartan geometry (G0 →
M, θ ⊕ γ) of type (Rn SL(n), SL(n)), i.e., an affine connection on M. Since by assumption Ω
has values in p, θ ⊕ γ is torsion-free and so is the affine connection.
Now take another splitting σ = σ · exp Υ, for some Υ: G → p+. Since Ad(exp Υ) acts by the
identity on g− = g/p, one has (rexp Υ)∗ω = ω mod p, and thus θ is independent of the choice of
splitting. Then σ ∗ω = θ⊕γ ⊕ρ and θ⊕γ = Ad(exp Υ)◦(θ⊕γ), projected to g− ⊕g0. But since
exp Υ ∈ P+, this shows that γ is projectively equivalent to γ. We thus obtain a well-defined
projective structure on M.
Since ω is P-torsion-free and P-equivariant modulo p+, it can be (uniquely) modified to
a normal Cartan connection ωnor ∈ Ω1(G, g) with ωnor − ω ∈ Ω1(G, p+). In particular, each
splitting σ: G0 → G is in fact a Weyl structure of the projective structure on M.
Theorem 4.14. A split-signature (n, n) conformal spin structure c on a manifold M is (locally)
induced by an n-dimensional projective structure via the Feﬀerman-type construction if and only
if the following properties are satisﬁed:
(a) M, c admits a nowhere-vanishing light-like conformal Killing ﬁeld k such that the corresponding
tractor endomorphism K = LAM
0 (k) is an involution, i.e., K2 = idT
.
(b) M, c admits a pure twistor spinor χ ∈ Γ Σ−
1
2 with k ∈ Γ(ker χ) such that the corresponding
parallel tractor spinor sF = L
S−
0 (χ) is pure.
(c) K acts by minus the identity on ker sF .
(d) The following integrability condition holds:
va
wc
Wabcd = 0, for all v, w ∈ Γ(ker χ).
Proof. Starting with a projective structure (M, p), it follows from Proposition 4.9 that the
induced conformal structure M, c has all the stated properties. On the other hand, let M, c
be a conformal structure with the stated properties. Then, by Proposition 4.12, ω restricts
to a Q-equivariant Cartan connection form with values in sl(n + 1) on the reduction G → G.
The corresponding curvature κ takes values in p and for X ∈ f we have that iXκ takes values
in p+. It follows from Proposition 4.13 that ω factorises to a projective structure p on the leaf
space M.
24 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
Let us now show that the two constructions are inverse to each other. Assume first that
a conformal structure (M, c) is induced by a projective structure (M, p). Then according to
Lemma 4.11 ω and ωind agree modulo p+. This implies that the projective structure defined
by ω is equal to the original projective structure. Conversely, assume now that (M, p) is
a projective structure with associated Cartan geometry (G, ω ) that is induced from a conformal
structure M, c with associated Cartan geometry G, ωnor . Since ω is not normal, but torsionfree,
there is ϕ ∈ Ω1
hor(G, p+) such that ω +ϕ is the normal projective Cartan connection. Since
p+ ⊆ p the induced conformal structure on M agrees with the original conformal structure.
Thus, the Fefferman-type construction (with normalisation) and the described factorisation are
(locally) inverse to each other.
For a reformulation of the characterisation theorem in terms of underlying objects, see Section
6.2.
5 Reduced scales and explicit normalisation
Although we obtained the desired characterisation in Theorem 4.14, we do not yet know the
explicit relationship between the induced Cartan connection form ωind and the normal conformal
Cartan connection form ωnor. One of the aims of the present section is to obtain a formula for
this difference, which is achieved in Theorem 5.7. As a consequence, we also obtain an explicit
formula for the curvature Ωind in terms of the normal conformal Cartan curvature Ωnor in
Corollary 5.8. In this more refined analysis, reduced scales will play an important role.
5.1 Characterisation of reduced scales
The notion of reduced Weyl structures and reduced scales is introduced in Section 3.4. Here
we shall find an intrinsic characterisation (i.e., using the conformal structure only) of reduced
scales and discuss their properties.
As the scale bundle on the projective manifold M we may consider the positive elements in
the density bundle E(1), which is the projecting part of the dual standard tractor bundle T ∗,
see Section 2.4. Similarly, on the Fefferman space M we take the positive elements in the
density bundle E(1), the projecting part of the conformal standard tractor bundle T . Hence for
a projective scale ρ ∈ Γ(E+(1)) we have the tractor LT ∗
0 (ρ) ∈ Γ(T ∗); similarly, for a conformal
scale σ ∈ Γ E+(1) we have the tractor LT
0 (σ) ∈ Γ T . These will be termed scale tractors.
On the one hand, reduced scales correspond to the sections of E+(1) → M seen as a subset
of all sections of E+(1) → M, see Remark 3.4. On the other hand, sections of T ∗ → M are
understood as specific sections of the bundle F → M, which is a subbundle in T → M, see the
generalities in Section 3.4 and the setup of our construction in Section 3.2. It follows that these
two natural inclusions commute with the BGG-splitting operators.
Lemma 5.1. The full arrows in the following diagram commute:
Γ(T ∗) 1  GG
Π0

Γ T
Π0

Γ(E+(1))
LT ∗
0
ii
1  GG Γ E+[1] .
LT
0

Proof. Consider a projective density ρ ∈ Γ(E+(1)) on M, the corresponding tractor LT ∗
0 (ρ) ∈
Γ(T ∗), and its extension to F ⊆ T , which is denoted by s . The extension of ρ ∈ Γ(E+(1))
to E+[1] obviously coincides with the projection Π0(s ), and it is denoted by σ. We need to show
A Projective-to-Conformal Fefferman-Type Construction 25
that s = LT
0 (σ), i.e., that ∂∗ nors = 0. According to Proposition 4.4, ωnor = ωind + Ψ1 + Ψ2
with Ψ1 ∈ Ω1 M, Λ2F , Ψ2 ∈ Ω1 M, f◦ , hence we have
nor
s = ind
s + Ψ1
• s + Ψ2
• s .
Since Λ2F acts trivially on F ⊆ T , we have Ψ1 • s = 0. Since f◦ ⊆ T∗M, it follows that
∂∗ Ψ2 • s = 0. It thus follows that ∂∗ nors = ∂∗ inds . Let φ be the frame form
of LT ∗
0 (ρ). Then, according to Lemma 4.1, we have that ∂∗φ = 0 since Λ2F • F = 0, and in
particular ∂∗ nors = 0.
We can now characterise reduced scales in terms of the corresponding scale tractors:
Proposition 5.2. Let M, c be a conformal spin structure associated to an oriented projective
structure (M, p) via the Feﬀerman-type construction. Let σ ∈ Γ E+[1] be a conformal scale
and let s := LT
0 (σ) ∈ Γ T be the corresponding scale tractor. Then the following statements
are equivalent:
(a) The scale σ is reduced.
(b) The tractor s is a section of F ⊆ T .
(c) The twistor spinor χ is parallel with respect to the Levi-Civita connection D of the metric
corresponding to the scale σ.
Furthermore, in a reduced scale, the Schouten tensor is strictly horizontal, i.e., it satisﬁes
va
Pab = 0, for all va
∈ Γ(ker χ), (5.1)
and the scalar curvature J vanishes.
Proof. (a) =⇒ (b): This follows from definitions and Lemma 5.1.
(b) =⇒ (c): The condition (b) means that s · sF = 0. According to (2.6), (2.8) and (2.7),
this condition expanded in slots yields
s · sF = LT
0 (σ) · LS
0 (χ) =


− 1
2n Jσ
0
σ

 ·
1
n
√
2
D/ χ
χ
=
−
√
2
2n Jχσ
− 1
nD/ χσ
=
0
0
,
where we use the Levi-Civita connection D corresponding to σ. In particular, D/ χ = 0 and,
since χ is a twistor spinor, the condition (c) follows.
(c) =⇒ (b): The condition (c) yields sF = 0
χ according to (2.8). The fact that asF = 0
yields Pacγcχ = 0 according to (2.7). Hence (5.1) holds, which in particular means J = 0.
Summarising, we have
s = LT
0 (σ) =


0
0
σ

 and sF = LS
0 (χ) =
0
χ
. (5.2)
Hence s · sF = 0 and the condition (b) follows.
(b) and (c) =⇒ (a): According to the previous reasoning and (2.5), we have
nor
a s =


0
σPab
0

 .
Hence nors is strictly horizontal, i.e., va nor
a s = 0 for every va ∈ Γ(ker χ). Since nor =
ind+Ψ and Ψ is horizontal, the horizontality of nors is equivalent to the horizontality of inds.
Altogether, the condition (a) follows from Proposition 3.3 and Lemma 5.1.
26 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
We will need some finer discussion on the slots of the distinguished tractor
K =


ρa
µab | ϕ
ka

 ∈ Γ AM (5.3)
in reduced scales. From Proposition 4.8 we know that K is the BGG-splitting LAM
0 (k), which
in particular means that µab = D[akb] and ϕ = − 1
2nDrkr according to (2.12). However, in the
following statement we only exploit the algebraic properties of K, namely, that it acts by minus
and plus the identity on F and E, respectively.
Lemma 5.3. Let us ﬁx a reduced scale. Then the expression of K as in (5.3) satisfy ρa = 0, ϕ =
−1, µa
rvr = −va for every va ∈ Γ(ker χ) and µa
rµrb = gab. Further we have µab = γ[aγb]χ, ¯η
for some ¯η ∈ Γ Σ −1
2 .
Proof. Firstly, we use K • s = −s for any s ∈ Γ F . The scale tractor s = LT
0 (σ) of a reduced
scale σ is a section of F and it has the form as in (5.2). Thus it follows from (2.9) that ρa = 0
and ϕ = −1. Next, for every va ∈ Γ(ker χ), the tractor s =
0
va
0
is clearly a section of F, since
s · sF = 0. Thus it follows from (2.9) that µa
rvr = −va.
Secondly, we use K•s = s for any s ∈ Γ E . Considering the tractor s =
0
ωa
0
with arbitrary
ωa ∈ Γ TM , the tractor ¯s := s + K • s is a section of E, whose middle slot is ωa + µa
rωr. It
follows again from (2.9) that µa
rµrb = gab.
Thirdly, we use (3.3) which shows how K is built from sE and sF . Since the top slot of sF
vanishes, middle slots of K are given by a suitable tensor product of χ and the top slot of sE.
We will also need more properties of conformal curvature quantities in reduced scales.
Lemma 5.4. In a reduced scale,
Wabcdµcd
= 0, where µab = D[akb], (5.4)
vc
Yabc = 0, for all va ∈ Γ(ker χ). (5.5)
Proof. In slots, the condition Ωnor
ab • sF = 0 implies that Wabcdγcγdχ = 0. Pairing both sides
of the latter equality with a spinor ¯η from Lemma 5.3 yields (5.4).
Consider two arbitrary sections va, wb of f = ker χ. Conditions (W) and (5.1) imply
Rabcdvawd = 0. Now, inserting va and we into the Bianchi identity D[aRbc]de = 0, we obtain
vaweDaRbcde = 0, where we used the fact that f is parallel. Since we can always regard gab
as a section of f ⊗ f∗, this implies 0 = gecvaDaRbcde = vaDaRicbc where Ricbc is the Ricci
tensor of Da. Since J = 0, we have that Pab is proportional to Ricab by a constant factor.
Thus vaDaPbc = 0. From (5.1) and since f is parallel we also have vbDaPbc = 0. Altogether,
(5.5) follows by the definition of the Cotton tensor.
5.2 Explicit normalisation formula
So far we discussed three Cartan connections on the Fefferman space M: the induced one ωind
(Section 3.3), the corresponding normal one ωnor (Section 4.1) and the modified auxiliary one ω
(Section 4.3). Various properties of these and derived objects are enumerated in Propositions 4.9
and 4.12. The following proposition refines the integrability conditions included there.
A Projective-to-Conformal Fefferman-Type Construction 27
Proposition 5.5. Let M, c be the conformal spin structure induced from an oriented projective
structure (M, p) via the Feﬀerman-type construction. Then, along the reduction G → G,
iXκnor
(u) ∈ f◦
⊗ Λ2 ¯F, for all X ∈ f, u ∈ G, (5.6)
iXκ (u) = 0, for all X ∈ f, u ∈ G. (5.7)
Proof. From (4.5) we already know that iXκnor has values in f◦ ⊗ Λ2 ¯F ⊕f◦ . We note that the
top slot of sections of Λ2F vanishes in reduced scales, cf. (3.7). Thus the part in f◦ corresponds
to vrYabr for a v ∈ Γ(f), which however has to vanish by (5.5). Hence (5.6) follows. The last
condition (5.7) follows from κ = κnor
sl(n+1)
, cf. Proposition 4.12.
Since ω is an SL(n + 1)-connection on G → M, it is the extension of a Cartan connection ω ,
on G → M. Now, due to (5.7), any section v ∈ Γ(ker χ) inserts trivially into its curvature.
But this is the standard condition on the connection ω to be a Cartan connection also on the
bundle G → M, i.e., to be a projective Cartan connection, cf. [5].
Furthermore, we will show that the descended Cartan connection is normal, i.e., ω = ω. To
do this, we first compute ∂∗κ and then use the relation between the co-differentials ∂∗ on M
and ∂∗ on M discussed in Lemma 4.1.
Proposition 5.6. The curvature κ satisﬁes
∂∗
κ (u) = ikκnor
(u) ∈ f◦
⊗ Λ2 ¯F, for all u ∈ G. (5.8)
Proof. We shall compute ∂∗Ω directly. First observe that using Proposition 4.12 we have
Ω = Ωnor
sl(n+1)
= Ωnor + 1
2K • Ωnor. Hence ∂∗Ω = 1
2∂∗ K • Ωnor , since ∂∗Ωnor = 0.
Using (5.3) and (2.11), we compute K • Ωnor
ab as


ρc
µc0c1 | ϕ
kc

 •


−Ydab
Wabd0d1 | 0
0

 =



ρrWabrc − µc
rYrab + ϕYcab
−2Wab
r
[c0
µc1]r + 2k[c0
Yc1]ab | krYrab
krWabrc


 .
In a reduced scale, from the previous display together with Lemmas 5.3 and 5.4 we compute
∂∗
K • Ωnor
ab =


0
2krWrac0c1 | 0
0

 = 2kr
Ωnor
ra ,
which yields (5.8).
Theorem 5.7. Let (G, ω) be a projective normal Cartan geometry over M and let G, ωind be
the conformal Cartan geometry over M induced via the Feﬀerman-type construction. Then
(a) ωind = ω = ωnor − 1
2ikκnor,
(b) ωnor = ωind + Ψ1, where Ψ1 = −1
2∂∗κind = 1
2ikκnor.
Proof. (a) We use that iXκ = 0 for all X ∈ f according to (5.7). Then Proposition 5.6
together with Lemma 4.1 imply that ∂∗κ = 0. Thus ω is projectively normal, and therefore we
obtain ω = ωind.
(b) The normalisation process of Proposition 4.4 provides Ψ = Ψ1 + Ψ2 such that ωnor =
ωind + Ψ, where Ψ1, Ψ2 are the first and second normalisation steps. However since ω = ωind,
it follows from Proposition 5.6 and (4.8) that ∂∗κ = ∂∗κind is, up to a constant multiple, the
difference between ωnor and ωind. Therefore already the first normalisation step completes the
normalisation, i.e., Ψ2 = 0.
28 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
Using the explicit relationship provided in Theorem 5.7 we can also obtain a detailed description
of the difference between the induced and the normal Cartan curvatures:
Corollary 5.8. In a reduced scale, we have the following relation between the curvatures of the
induced and the normal conformal Cartan connection:
Ωind
ab = Ωnor
ab +
1
2
K • Ωnor
ab =



−Ycab
Wabc0c1 − Wab
r
[c0
µc1]r + k[c0
Yc1]ab | 0
1
2krWabrc


 . (5.9)
In particular, 1
2ikW is the torsion of the induced Cartan connection ωind.
Proof. We obtained the concrete expression of K • Ωnor in the proof of Proposition 5.6. Now
Lemmas 5.3 and 5.4, and a short computation yields (5.9).
6 Comparison with Patterson–Walker metrics
and alternative characterisation
In this section we will show that the Fefferman-type construction studied in this article is closely
related to the construction of so-called Patterson–Walker metrics. These are the Riemann
extensions of affine connected spaces, firstly described in [26]. A conformal version of this
construction was obtained by [15] for dimension n = 2, and was treated by the authors of the
present article in general dimension in [20].
6.1 Comparison
Let M be a smooth manifold and p: T∗M → M its cotangent bundle. The vertical subbundle
V ⊆ T(T∗M) of this projection is canonically isomorphic to T∗M. An affine connection D
on M determines a complementary horizontal distribution H ⊆ T(T∗M) that is isomorphic
to TM via the tangent map of p.
Deﬁnition 6.1. The Riemann extension or the Patterson–Walker metric associated to a torsion-free
affine connection D on M is the pseudo-Riemannian metric g on T∗M fully determined
by the following conditions:
(a) both V and H are isotropic with respect to g,
(b) the value of g with one entry from V and another entry from H is given by the natural
pairing between V ∼= T∗M and H ∼= TM.
It follows that V is parallel with respect to the Levi-Civita connection of the just constructed
metric. Hence Patterson–Walker metrics are special cases of Walker metrics, i.e., metrics admitting
a parallel isotropic distribution. The explicit description of the metric g in terms of the
Christoffel symbols of D can be found in [20, 26].
The previous definition can be adapted to weighted cotangent bundles T∗M(w) = T∗M ⊗
E(w), provided that M is oriented and D is special, i.e., preserving a volume form on M,
which allows a trivialisation of E(w). It turns out that Patterson–Walker metrics induced by
projectively equivalent connections are conformally equivalent if and only if w = 2 (interpreted
as a projective weight according to the conventions from Section 2.4). Altogether, we have
a natural split-signature conformal structure on T∗M(2) induced by an oriented projective
structure (M, p).
From Section 3.3 we know that M = T∗M(2) \ {0} is the Fefferman space. Special affine
connections from p are just the exact Weyl connections of the corresponding parabolic geometry.
A Projective-to-Conformal Fefferman-Type Construction 29
The corresponding objects on M are the reduced Weyl connections, respectively reduced scales,
which correspond to distinguished metrics in the conformal class, see Section 3.4. We are going
to show that these metrics are just the Patterson–Walker metrics.
Proposition 6.2. Let M, c be the conformal structure of signature (n, n) associated to an ndimensional
projective structure (M, p) via the Feﬀerman-type construction. Then any metric
in c corresponding to a reduced scale is a Patterson–Walker metric.
Proof. Within the proof we refer to the notation and explicit matrix realisations from Appendix
A. By definition, the Fefferman space is M = G/Q, which yields TM ∼= G ×Q g/q.
Conformally invariant objects on M, respectively objects related to a choice of reduced scale,
correspond to data on g/q ∼= ˜g/˜p that are invariant under the action of Q, respectively Gss
0 ∩ Q.
Elements in g/q will be represented by matrices of the form


−z
2 ∗ ∗
X ∗ ∗
w Y t −z
2

 ,
where z, w ∈ R and X, Y ∈ Rn−1. Firstly, one verifies that
Y t
X − zw, (6.1)
is the only quadratic form that is invariant under Gss
0 ∩ Q. Hence any reduced-scale metric in c
corresponds to the quadratic form (6.1) in a suitable frame. Secondly, the vertical subbundle
V ⊆ TM corresponds to the Q-invariant subspace f = p/q ⊆ g/q given by X = 0 and w = 0.
The horizontal distribution H ⊆ TM induced by a linear connection from p corresponds to the
unique (Gss
0 ∩ Q)-invariant subspace h ⊆ g/q complementary to f, which is given by Y = 0 and
z = 0. Obviously, both f and h are isotropic with respect to (6.1). Hence any reduced-scale
metric in c satisfies the condition (a) from Definition 6.1. Thirdly, the canonical identification
V ∼= T∗M(2) corresponds to an isomorphism f ∼= (g/p)∗(2) of Q-modules. Identifying (g/p)∗(2)
with p+(2), it turns out to be given by


−z
2 ∗ ∗
0 ∗ ∗
0 Y t −z
2

 →


0 Y t −z
0 0 0
0 0 0

 .
Now, the inner product of any v ∈ f and u ∈ h coincides with the pairing of the corresponding
elements v ∈ p+(2) and u ∈ g/p. Hence any reduced-scale metric in c satisfies also the
condition (b) from Definition 6.1 and so it is a Patterson–Walker metric.
6.2 Alternative characterisation
We have characterised split-signature (n, n) conformal structures c on M induced by an ndimensional
projective structure via the Fefferman-type construction in Theorem 4.14. Now
we know these structures correspond to conformal Patterson–Walker metrics discussed in [20].
There we found the following characterisation in terms of underlying objects by direct computations
and spin calculus. Our aim here is to indicate how to reach the same result in the current
framework.
Theorem 6.3. A split-signature (n, n) conformal spin structure c on a manifold M is (locally)
induced by an n-dimensional projective structure via the Feﬀerman-type construction if and only
if the following properties are satisﬁed:
(a) M, c admits a nowhere-vanishing light-like conformal Killing ﬁeld k.
30 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
(b) M, c admits a pure twistor spinor χ such that f = ker χ is integrable and k ∈ Γ f .
(c) The Lie derivative of χ with respect to the conformal Killing ﬁeld k is Lkχ = −1
2(n + 1)χ.
(d) The following integrability condition holds:
vr
ws
Warbs = 0, for all vr
, ws
∈ Γ(ker χ). (W)
We now express the conditions from Proposition 4.9 in underlying terms:
(i) For a conformal Killing field k with the splitting K as in (5.3), a straightforward computation
shows that the condition K2 = idT
is equivalent to
ka
ka = 0, ρa
ρa = 0,
µa
bkb
= ϕka
, µa
bρb
= −ϕρa
,
ka
ρa = ϕ2
− 1, µ c
a µcb = gab + 2k(aρb). (6.2)
(ii) For a twistor spinor χ, the corresponding tractor spinor sF = ¯χ
χ ∈ Γ(S−) is parallel with
respect to nor. In particular, purity of sF can be checked at one point. If χ = 0, respectively
¯χ = 0, this tractor spinor is pure whenever ¯χ, respectively χ, is pure. If χ = 0 and ¯χ = 0, the
purity of ¯χ
χ is equivalent to χ and ¯χ being pure and their kernels having (n − 1)-dimensional
intersection, cf. [13, Proposition III-1.12] or [22, 30].
(iii) Let k be a conformal Killing field which splits to K and χ a twistor spinor which splits
to sF . Then the condition Lkχ = −1
2(n + 1)χ is equivalent to K • sF = −1
2(n + 1)sF . If the
tractor spinor sF is pure it has an (n + 1)-dimensional maximally isotropic kernel ker sF . Then
K•sF = −1
2(n+1)sF is equivalent to K acting by minus the identity on ker sF , which therefore
coincides with the eigenspace of K corresponding to −1.
The assumption on the pure twistor spinor χ in Theorem 6.3 guarantees the existence of a
suitable compatible metric for which χ is parallel. This result is proved in [20, Proposition 4.2].
Henceforth we shall assume Dχ = 0 where D is the corresponding Levi-Civita connection.
In particular, we have sF = L
S−
0 (χ) = 0
χ , which is pure and parallel by observation (ii).
Expanding the latter condition according to (2.7) yields
va
Pab = 0, for all va
∈ Γ(ker χ), (6.3)
For K = LAM
0 (k), we know from observation (iii) that K acts by − id on ker sF . By the very
same reasoning as in the first part of the proof of Lemma 5.3 it follows that
ρa = 0, ϕ = −1, µa
r
vr = −va, for all va
∈ Γ(ker χ). (6.4)
Now we are prepared to prove the theorem:
Proof of Theorem 6.3. If (M, c) is induced by a projective structure, the stated properties
hold according to Proposition 4.9 and previous observations. For the converse direction, it
remains to show that K2 = idT
, which is equivalently characterised by the identities (6.2).
Then the properties (a)–(d) of Proposition 4.9 will be satisfied and the result will follow from
the characterisation Theorem 4.14.
The expansion of the prolonged conformal Killing equation (2.13) for k gives
Dakb = µab + gab, (6.5)
Daµbr = −2Pa[bkr] − Wbrasks
, (6.6)
A Projective-to-Conformal Fefferman-Type Construction 31
according to (2.10) and (2.11). Next, from (6.4) we especially have µb
rkr = −kb. Applying Da
to both sides of this equality and using (6.5) we obtain
Daµbr kr
+ µb
r
µar + µba = −(µab + gab).
From (6.6), (W) and (6.3) we have Daµbr kr = 0, hence the previous display shows µa
rµrb = gab.
This together with (6.4) implies that all identities from (6.2) are satisfied, hence K2 = idT
.
A Explicit matrix realisations
Here we provide explicit realisations of the Lie algebras introduced in Section 3.2 in terms of
matrices. We will consider the inner product h and the involution K on Rn+1,n+1 given by the
block matrices
h :=
0 In+1
In+1 0
and K :=
In+1 0
0 −In+1
with respect to the standard basis (e1, . . . , e2n+2). Then E = e1, . . . , en+1 and F = en+2, . . . ,
e2n+2 and the decomposition (3.4) can be written as
˜g = Λ2
(E ⊕ F) =
E ⊗ F Λ2E
Λ2F E ⊗ F
.
For ˜v := e1 + e2n+2, the Lie algebra ˜p of the parabolic subgroup P ⊆ G is of the following form
˜p =








a Ut w 0 −Wt −b
X B V W C −X
0 Y t c b Xt 0
0 −Y t −d −a −Xt 0
Y D −Z −U −Bt −Y
d Zt 0 −w −V t −c








, (A.1)
where a, b, c, d, w ∈ R with a − b = d − c, U, V, W, X, Y, Z ∈ Rn−1, B ∈ gl(n − 1) and C, D ∈
so(n − 1). The nilradical ˜p+ = ˜p⊥ is then of the form
˜p+ =








a Ut w 0 −V t −a
0 0 V V 0 0
0 0 a a 0 0
0 0 −a −a 0 0
0 0 −U −U 0 0
a Ut 0 −w −V t −a








.
A choice of Levi subalgebra ˜g0 ⊆ ˜p determines a grading ˜g = ˜g− ⊕ ˜g0 ⊕ ˜p+. We shall choose
˜g0 = ˜p ∩ ˜pop, where ˜pop ⊆ ˜g is the stabiliser of the light-like vector en+2. Explicitly,
˜g0 =








a 0 0 0 0 0
X B V 0 C −X
0 Y t c 0 Xt 0
0 −Y t −a − c −a −Xt 0
Y D −Z 0 −Bt −Y
a + c Zt 0 0 −V t −c








.
32 M. Hammerl, K. Sagerschnig, J. ˇSilhan, A. Taghavi-Chabert and V. ˇZ´adn´ık
The embedding i : g → ˜g of Lie algebras has the form A → A 0
0 −At . The subgroup Q =
i−1(P) is contained in P, the stabiliser in G of v = (˜v)E = e1; the inclusion of corresponding
Lie algebras is
q = g ∩ ˜p =


a Ut w
0 A V
0 0 −a

 ⊆


a Ut w
0 B V
0 Xt c

 = p,
where tr(A) = 0 and a + tr(B) + c = 0. The standard projective grading g = g− ⊕ g0 ⊕ p+,
g− =


0 0 0
X 0 0
y 0 0

 , g0 =


a 0 0
0 B V
0 Xt c

 , p+ =


0 Ut w
0 0 0
0 0 0

 ,
is compatible with the previous conformal grading so that the reduced Lie subalgebra q0 := q∩g0
coincides with the intersection of g0 ∩ ˜g0. Explicitly,
q0 =


a 0 0
0 A V
0 0 −a

 , (A.2)
where tr(A) = 0.
Acknowledgements
The authors express special thanks to Maciej Dunajski for motivating the study of this construction
and for a number of enlightening discussions on this and adjacent topics. KS would also like
to thank Pawel Nurowski for drawing her interest to the subject and for many useful conversations.
MH gratefully acknowledges support by project P23244-N13 of the Austrian Science Fund
(FWF) and by ‘Forschungsnetzwerk Ost’ of the University of Greifswald. KS gratefully acknowledges
support from grant J3071-N13 of the Austrian Science Fund (FWF). JˇS was supported
by the Czech science foundation (GAˇCR) under grant P201/12/G028. AT-C was funded by
GAˇCR post-doctoral grant GP14-27885P. VˇZ was supported by GAˇCR grant GA201/08/0397.
Finally, the authors would like to thank the anonymous referees for their helpful comments and
recommendations.
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132 Attachments
Bull. London Math. Soc. 00 (2018) 1–5 Ce2018 London Mathematical Society
doi:10.1112/blms.12136
Feﬀerman–Graham ambient metrics of Patterson–Walker metrics
Matthias Hammerl, Katja Sagerschnig, Josef ˇSilhan, Arman Taghavi-Chabert and
Vojtˇech ˇZ´adn´ık
Abstract
Given an n-dimensional manifold N with an aﬃne connection D, we show that the associated
Patterson–Walker metric g on T∗
N admits a global and explicit Feﬀerman–Graham ambient
metric. This provides a new and large class of conformal structures which are generically not
conformally Einstein but for which the ambient metric exists to all orders and can be realised in
a natural and explicit way. In particular, it follows that Patterson–Walker metrics have vanishing
Feﬀerman–Graham obstruction tensors. As an application of the concrete ambient metric
realisation we show in addition that Patterson–Walker metrics have vanishing Q-curvature. We
further show that the relationship between the geometric constructions mentioned above is very
close: the explicit Feﬀerman–Graham ambient metric is itself a Patterson–Walker metric.
1. Introduction and main result
Given a signature (p, q) conformal structure [g] on an m = p + q dimensional manifold M,
it was shown in seminal work by Feﬀerman and Graham (see [7, 8]) that under speciﬁc
conditions the conformal structure can be encoded equivalently as a signature (p + 1, q + 1)
pseudo-Riemannian metric (M, g) with vanishing Ricci curvature. This description has been
fundamental in constructing and classifying conformal invariants (see, for example, [3, 7]) and
for constructing and studying conformally invariant diﬀerential operators (see [11, 12]).
To build the Feﬀerman–Graham ambient metric for given local coordinates x on M, one
ﬁrst considers the ray bundle of metrics in the conformal class [g], written as R+ × Rm
with
coordinates (t, x). The ambient space M is obtained by adding a new transversal coordinate
ρ ∈ R, and then an ansatz for the Feﬀerman–Graham ambient metric g is
g = t2
gij(x, ρ)dxi
dxj
+ 2ρdt dt + 2tdt dρ, (1)
where g = gij(x, 0)dxi
dxj
is a representative metric in the conformal class. It is directly visible
from the formula that g is homogeneous of degree 2 with respect to the Euler ﬁeld t∂t on M.
To show existence of a Feﬀerman–Graham ambient metric g for given g, the ansatz (1)
determines an iterative procedure to determine gij(x, ρ) as a Taylor series in ρ satisfying
Ric(g) = 0 to inﬁnite order at ρ = 0. For m odd, the existence (and a natural version of
uniqueness) of g as an inﬁnity-order series expansion in ρ is guaranteed for general gij(x). For
m = 2n even, the existence of an inﬁnity-order jet for gij(x, ρ) with Ric(g) = 0 asymptotically
at ρ = 0 is obstructed at order n, and is controlled by the vanishing of the conformally invariant
Feﬀerman–Graham tensor O. Moreover, existence does not in general guarantee uniqueness.
Results which provide global Feﬀerman–Graham ambient metrics, where g can then be constructed
in a natural way from g and satisﬁes Ric(g) = 0 globally and not just asymptotically
at ρ = 0, are rare, in both even and odd dimensions. A special instance, where global ambient
Received 2 March 2017; revised 1 December 2017.
2010 Mathematics Subject Classiﬁcation 53A30 (primary), 53A55, 53B30 (secondary).
The second author acknowledges support by INdAM via project FIR 2013 — Geometria delle equazioni differenziali.
The third author was supported by the Czech science foundation (GAˇCR) under grant P201/12/G028,
and the ﬁfth author was supported by the same foundation under grant GA17-01171S.
2 HAMMERL, SAGERSCHNIG, ˇSILHAN, TAGHAVI-CHABERT AND ˇZ´ADN´IK
metrics can at least be shown to exist, occurs for g real-analytic and m either being odd or m
even with the obstruction tensor O of g vanishing. The simplest case of geometric origin for
which one has global ambient metrics consists of locally conformally ﬂat structures (M, [g]),
where (M, g) exists and is unique up to diﬀeomorphisms (see [8, Chapter 7]). Another wellknown
geometric case are conformal structures (M, [g]) which contain an Einstein metric g: If
Ric(g) = 2λ(m − 1)g, then g on R+ × M × R can be written directly in terms of g as
g = t2
(1 + λρ)2
g + 2ρdt dt + 2tdt dρ. (2)
In work by Thomas Leistner and Pawel Nurowski it was shown that the so-called pp-waves
admit global ambient metrics in the odd-dimensional case and under speciﬁc assumptions in the
even-dimensional case, see [14]. Concrete and explicit ambient metrics for speciﬁc examples of
families of conformal structures induced by generic 2-distributions on 5-manifolds and generic
3-distributions on 6-manifolds have been constructed in [2, 15, 16, 18]. Recent progress in
obtaining classes of conformal structures for which the Feﬀerman–Graham ambient metric
equations become linear and have explicit solutions was obtained in [1].
The present article expands the class of metrics for which canonical ambient metrics exist
globally to Patterson–Walker metrics: Given an aﬃne connection D on an n-manifold N with
n 2, which is supposed to be torsion-free and to preserve a volume form, the Patterson–
Walker metric g is a natural split-signature (n, n) metric on the total space of the co-tangent
bundle T∗
N, see [13] for historical background on Patterson–Walker metrics, references and a
modern treatment. Our main result is as follows.
Theorem 1. Let D be a torsion-free aﬃne connection on N which preserves a volume
form. Denote local coordinates on N by xA
and the induced canonical ﬁbre coordinates on
T∗
N by pA. Let ΓA
C
B and RicAB denote the Christoﬀel symbols and the Ricci curvature of
D, respectively. Let
g = 2 dxA
dpA − 2 ΓA
C
B pC dxA
dxB
(3)
be the Patterson–Walker metric induced on T∗
N by D. Then
g = 2ρdt dt + 2tdt dρ
+ t2
2dxA
dpA − 2pCΓA
C
BdxA
dxB
+
2ρ
n − 1
RicABdxA
dxB
(4)
is a globally Ricci-ﬂat Feﬀerman–Graham ambient metric for the conformal class [g].
For generic D, the resulting Patterson–Walker metric g is not conformally Einstein, see
[13, Theorem 2]. In particular, Theorem 1 provides a large class of conformal structures which
are not conformally Einstein but which admit globally Ricci-ﬂat and explicit ambient metrics.
As an immediate consequence of the existence of the ambient metric for (M, [g]), the
conformally invariant Feﬀerman–Graham obstruction tensor O associated to [g] vanishes.
It is not diﬃcult to check Ricci-ﬂatness of (4) directly: Speciﬁcally, one employs [8, formula
(3.17)], which is applicable to any ambient metric in the normal form (1). The computation is
then based on the following key facts: The Ricci curvature of the Patterson–Walker metric g
is up to a constant multiple just the pullback of the Ricci curvature of D and this tensor and
its covariant derivative are totally isotropic, see [13]. We note that formula (4) for g says that
the ambient metric is in fact linear in ρ and the iterative procedure determining the ambient
metric stops after the ﬁrst step.
A geometric proof of vanishing Ricci curvature is presented in the next section. This is based
on a combination of the well-known Patterson–Walker and Thomas cone constructions, both
AMBIENT METRICS OF PATTERSON–WALKER METRICS 3
of which we recall. It follows directly from the geometric construction outlined below that the
Feﬀerman–Graham ambient metric of a Patterson–Walker metric is again a Patterson–Walker
metric, namely, the one associated to the Thomas cone connection. This nice and interesting
fact is elaborated in Theorem 2.
2. Geometric construction of the ambient metric
The association D g generalises to a natural association from projective to conformal
structures. Recall that two aﬃne connections D, D on N are called projectively related or
equivalent if they have the same geodesics as unparameterised curves, which is the case if and
only if there exists a 1-form Υ ∈ Ω1
(N) with
DX Y = DX Y + Υ(X)Y + Υ(Y )X (5)
for all X, Y ∈ X(N). It is suﬃcient to restrict ourselves to special connections in a projective
class, that is, to those that preserve some volume form. For projective structures it is useful
to employ suitably scaled projective density bundles, deﬁned as E(w) := (∧n
TN)−w/(n+1)
, for
arbitrary weight w. Then a section s : N → E+(1) corresponds to a choice of a special aﬃne
connection D in the projective equivalence class [D], and any s = ef
s corresponds to D
projectively related to D via (5) with Υ = df. We deﬁne M := T∗
N ⊗ E(2) the (projectively)
weighted co-tangent bundle of N. Then, as was shown in [13], two projectively related aﬃne
connections D, D on N induce two conformally related metrics g, g on M, and we therefore
have a natural association (N, [D]) (M, [g]).
The cone C := E+(1) carries the canonical and well-known Ricci-ﬂat Thomas cone connection
∇ associated to the projective class [D], see [5, 17]. (The speciﬁc weight 1, which is diﬀerent
from the one in [5], is more convenient for our computations.) We will need a local formula
for ∇: Let s : N → C be the scale corresponding to an aﬃne connection D ∈ [D], providing
a trivialisation C ∼= R+ × N via (x0
, x) → s(x)x0
. In this trivialisation the Thomas cone
connection is given by
∇X Y = DX Y −
1
n − 1
Ric(X, Y )Z, ∇X Z = X, (6)
where X, Y ∈ X(N) and Z = x0
∂x0 is the Euler ﬁeld on C. It is in fact easy to see directly from
formula (6) that the thus deﬁned aﬃne connection ∇ on the Thomas cone C is independent of
the choice of scale and it is Ricci-ﬂat.
Employing a local coordinate patch on N which induces coordinates xA
, yA on the co-tangent
bundle T∗
N and coordinates x0
, xA
, yA, y0 on T∗
C ∼= R+ × T∗
N × R, the Patterson–Walker
metric g associated to ∇ is
g = 2dxA
dyA + 2dx0
dy0 −
4
x0
yBdx0
dxB
− 2yCΓA
C
BdxA
dxB
+ 2
x0
y0
n − 1
RicABdxA
dxB
.
(7)
Ricci-ﬂatness of g follows directly from Ricci-ﬂatness of ∇, see [13, Theorem 2]. Via the change
of coordinates t = x0
, ρ = y0/x0
, pA = yA/(x0
)2
the metric g transforms to (4), which is the
form of a Feﬀerman–Graham ambient metric (1). In particular, this shows Theorem 1.
We conclude this section by summarising the construction:
4 HAMMERL, SAGERSCHNIG, ˇSILHAN, TAGHAVI-CHABERT AND ˇZ´ADN´IK
Theorem 2. Given a projective structure (N, [D]) on an n-dimensional manifold N, the
geometric constructions indicated in the following diagram commute:
In particular, the induced conformal structure [g] admits a globally Ricci-ﬂat Feﬀerman–
Graham ambient metric g which is itself a Patterson–Walker metric.
Remark. As a Patterson–Walker metric, (M, g) carries a naturally induced homothety k of
degree 2, which takes the form 2pA∂pA
+ 2ρ∂ρ. According to [13, Lemma 5.1] the inﬁnitesimal
aﬃne symmetry Z of ∇ lifts to a Killing ﬁeld, which one computes as t∂t − 2pA∂pA
− 2ρ∂ρ.
In particular it follows that the Euler ﬁeld t∂t of the Feﬀerman–Graham ambient metric g
can be written as the sum of this Killing ﬁeld and the homothety k. The tangent bundle TM
carries the maximally isotropic (n + 1)-dimensional subspace spanned by {∂pA
, ∂ρ} which is
preserved by ∇. This subspace can be equivalently described by a ∇-parallel pure spinor s
on M. The ambient Killing ﬁeld k and the ambient parallel pure spinor s correspond to a
homothety k of g and a parallel pure spinor χ on M that belong to the characterising objects
of the Patterson–Walker metric g, see [13, Theorem 1].
3. Vanishing ‘Q-curvature’
The Q-curvature Qg of a given metric g is a Riemannian scalar invariant with a particularly
simple transformation law with respect to conformal change of metric. It has been introduced
by Branson in [4] and has been the subject of intense research in recent years, see, for example,
[6] for an overview. Computation of Q-curvature is notoriously diﬃcult, see, for example, [10].
An explicit form of the Feﬀerman–Graham ambient metric g for a given metric g allows a
computation of Qg. Using the fact that our g is actually a Patterson–Walker metric, this
computation is particularly simple.
Theorem 3. The Patterson–Walker metric g associated to a volume-preserving, torsion-free
aﬃne connection D has vanishing Q-curvature Qg.
Proof. We follow the computation method for Qg from [9]: For this it is necessary to compute
−Δn
log(t), where Δ is the ambient Laplacian on M = R+ × T∗
N × R associated to g and
t : M → R+ is the ﬁrst coordinate. Restricting −Δn
log(t) to the cone R+ × T∗
N × {0} and
evaluating at t = 1 yields Qg. To show that the Q-curvature vanishes for g, it is in particular
suﬃcient to show that Δ log(t) = 0.
However, the function t : M → R+ is horizontal since it is just the pullback of the coordinate
function x0
: C → R+ on the Thomas cone C ∼= R+ × N via the canonical projection T∗
C → C.
It follows from the explicit formula for the Christoﬀel symbols of a Patterson–Walker
metric that Δ vanishes on any horizontal function, see [13, Section 2.1]. Thus in particular
Δ log(t) = 0, and then also Qg = 0.
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Matthias Hammerl
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna
Austria
matthias.hammerl@univie.ac.at
Katja Sagerschnig
Dipartimento di Scienze Matematiche
INdAM-Politecnico di Torino
Corso Duca degli Abruzzi 24
10129 Torino
Italy
katja.sagerschnig@univie.ac.at
Josef ˇSilhan
Faculty of Science
Masaryk University
Kotl´aˇrsk´a 2
61137 Brno
Czech Republic
silhan@math.muni.cz
Arman Taghavi-Chabert
Dipartimento di Matematica “G. Peano”
Universit`a di Torino
Via Carlo Alberto 10
101012 Torino
Italy
ataghavi@unito.it
Vojtˇech ˇZ´adn´ık
Faculty of Education
Masaryk University
Poˇr´ıˇc´ı 31
60300 Brno
Czech Republic
zadnik@mail.muni.cz
138 Attachments
J. Math. Anal. Appl. 473 (2019) 112–140
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
Conformal theory of curves with tractors
Josef Šilhan a,∗
, Vojtěch Žádník b
a
Institute of Mathematics and Statistics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
b
Faculty of Education, Masaryk University, Poříčí 31, 60300 Brno, Czech Republic
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 July 2018
Available online 22 December 2018
Submitted by J. Lenells
Keywords:
Conformal geometry
Tractor calculus
Curves
Invariants
We present the general theory of curves in conformal geometry using tractor
calculus. This primarily involves a speciﬁcation of distinguished parametrizations
and relative and absolute conformal invariants of generic curves. The absolute
conformal invariants are deﬁned via a tractor analogue of the classical Frenet frame
construction and then expressed in terms of relative ones. Our approach applies
likewise to conformal structures of any signature; in the case of indeﬁnite signature
we focus especially on the null curves. It also provides a conceptual tool for handling
distinguished families of curves (conformal circles and conformal null helices) and
conserved quantities along them.
© 2018 Elsevier Inc. All rights reserved.
1. Introduction
The local geometry of curves is a classical subject, nowadays developed for various geometric structures
using various views and methods. The study generally starts in ﬂat spaces, the passage to general curved
cases demands new ideas and attitudes. We are going to evolve an approach based on conformal tractor
calculus. In this section, we present main sources of inspiration, summarize main results and introduce main
tools needed later.
1.1. Main sources and methods
To our knowledge, the ﬁrst general treatment of curves in conformal Riemannian manifolds is due to
Fialkow [9], which is therefore used as a basic reference for comparisons. In that paper, a conformally
invariant Frenet-like approach is developed. This primarily requires a natural distinguished parametrization
of the curve (in the place of the arc-length parameter in the Riemannian setting), a natural starting object
(in the place of the unit tangent vector) and a notion of derivative along the curve (in the place of the
* Corresponding author.
E-mail addresses: silhan@math.muni.cz (J. Šilhan), zadnik@mail.muni.cz (V. Žádník).
https://doi.org/10.1016/j.jmaa.2018.12.038
0022-247X/© 2018 Elsevier Inc. All rights reserved.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 113
restricted Levi-Civita connection). Having all these instruments, it is easy to derive a conformal analogue
of Fernet formulas with a distinguished set of conformal invariants, the conformal curvatures of the curve.
For generic curves in an n-dimensional manifold, this yields n − 1 conformal curvatures, one of which has
an exceptional ﬂavour.
The Frenet frame can be seen as an example of a more general concept of moving frame by Cartan.
An application of the latter method for immersed submanifolds, especially curves, in the homogeneous
Möbius space is presented by Schiemangk and Sulanke in [16] and [20], which we adduce as important
sources of inspiration. That way, the curve in the homogeneous space is covered by a curve in the principal
group so that the restriction of the Maurer–Cartan form to the lift reveals the generating set of invariants.
The lift can be identiﬁed with, and practically is constructed as, a curve of (pseudo-)orthonormal frames in
the ambient Minkowski space, whose dimension is two more higher than the dimension of the Möbius space.
Both the homogeneous principal bundle with the Maurer–Cartan form and the associated homogeneous
vector bundle, whose ﬁbre is the ambient Minkowski space with the induced linear connection, has a counterpart
over general conformal manifolds: the former leads to the notion of conformal Cartan connection,
the latter to the Thomas standard tractor bundle with its linear connection and parallel bundle metric.
These two approaches are basically equivalent, in this paper we exploit the latter one. One of its main
advantages is that everything can be set oﬀ pretty directly in terms of underlying data, while it still has
very conceptual ﬂavour. This, on the one hand, allows one to easily adapt key ideas from the homogeneous
setting to the tractorial one. On the other hand, expanding any tractorial formula yields a very concrete
tensorial expression that allows comparisons to results obtained by that means. We refer to the essential
work [2] by Bailey, Eastwood and Gover both for the generalities on tractor calculus and for an initial step
in the study of (distinguished) curves in this manner.
Besides the just cited main sources, there is a wide literature discussing curves in conformal and other
geometries from various aspects. A short review of the development of the topic for conformal structures
can be found in the introduction of [5]. Another references that are close to our purposes are [3], [4] and [13].
An exhibition of tensorial techniques in the study of curves in related geometries can be found in [14]. For
typical subtleties in dealing with null curves in pseudo-Riemannian geometry, see e.g. [7]. For an application
of the Cartan’s method of moving frame to curves in a broad class of homogeneous spaces, see [6].
1.2. Aim, structure and results
We study the local diﬀerential geometry of curves in general conformal manifolds of general signature
using the tractor calculus. Although the subject is indeed classic, the systematic tractorial approach is novel.
This approach has not only obvious formal beneﬁts, but also a potential for discovering new results and
relations. In particular, the discussion for null curves in the case of indeﬁnite signature (for which almost
nothing is known) is very parallel to the one in positive deﬁnite case and leads to results that we, at least,
would never obtain by other means.
The rough structure of the paper is as follows: In sections 2 and 4 we develop the theory for conformal
Riemannian structures, in section 5 we discuss its analogies in indeﬁnite signature. We primarily deal with
generic curves, whereas special cases are brieﬂy mentioned in accompanying remarks. However, the most
special case—conformal circles—is dealt individually in section 3.
In section 2 we follow the setting of [2], where one already ﬁnds the tractorial determination of the preferred
class of projective parametrizations on any curve as well as the projectively parametrized conformal
circles. Continuing further with tractors of higher order, we build a natural reservoir of relative conformal
invariants associated to any curve (subsection 2.3). The simplest of these leads to the notion of conformal
arc-length, the distinguished conformally invariant parametrization of the curve (subsection 2.4). This invariant
vanishes identically along the curve if and only if the curve is an arbitrarily parametrized conformal
circle (Proposition 3.3). As a demonstration of the ease of use of tractors we describe some conserved quan-
114 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
tities along projectively parametrized conformal circles on manifolds admitting an almost Einstein scale,
respectively normal conformal Killing ﬁeld (subsection 3.2).
In section 4 we launch the Frenet-like procedure to absolute conformal invariants of curve:
• construct a pseudo-orthonormal tractor Frenet frame along the given curve,
• diﬀerentiate with respect to the conformal arc-length and extract the tractor Frenet formulas,
• the coeﬃcients of that system determine the generating set of invariants.
For the curve in an n-dimensional manifold, we have n − 1 conformal curvatures, one of which has an
exceptional ﬂavour. The vanishing of the exceptional curvature has an immediate interpretation relating the
above mentioned parametrizations (Proposition 4.3). By construction, all conformal curvatures are expressed
via the tractors from the tractor Frenet frame, i.e. with respect to the conformal arc-length parametrization.
Alternative expressions in terms of initial tractors are also possible. In particular, for all nonexceptional
curvatures, we have very simple formulas using the previously deﬁned relative conformal invariants, i.e. with
respect to an arbitrary parametrization (Theorem 4.6). For a given scale, we also indicate how to express
conformal curvatures in terms of the Riemannian ones (subsection 4.3). Besides the pure pleasure, this eﬀort
allows a comparison of our invariants with those in the literature, especially in [9].
In section 5 we adapt the previous scheme to conformal manifolds of indeﬁnite signature. In that case
we distinguish space-, time- and light-like curves according to the type of their tangent vectors. A full
type classiﬁcation of curves (via associated tractors) becomes very rich, depending on the dimension and
signature. The construction of the tractor Frenet frame has to be adapted to the respective type, which makes
any attempt on its universal description impossible. Avoiding the complicated branching of the discussion,
we only point out the main features (Remark 4.4(3)) and focus on the light-like curves. Among these
curves we identify an appropriate analogue of conformal circles, the conformal null helices. Notably, in their
characterization another family of relative conformal invariants of Wilczynski type appears (Theorem 5.8).
More details on the construction of the tractor Frenet frame and expressions of the corresponding conformal
curvatures are discussed in the case of Lorentzian signature (subsection 5.4).
1.3. Notation and conventions
Most of the following conventions is taken from [2]. A conformal structure of signature (p, q) on a smooth
manifold M of dimension n = p+q is a class of pseudo-Riemannian metrics of signature (p, q) that diﬀer by
a multiple of an everywhere positive function. For all tensorial objects on M we use the standard abstract
index notation. Thus, the symbol μa
and μa refers to a section of the tangent and cotangent bundle, which
is denoted as Ea
:= TM and Ea := T∗
M, respectively, multiple indices denote tensor products, e.g. μa
b
is a section of Ea
b
:= T∗
M ⊗ TM etc. Round brackets denote symmetrization and square brackets denote
skew symmetrization of enclosed indices, e.g. sections of E[ab] = T∗
M ∧ T∗
M are 2-forms on M. By E[w]
we denote the density bundle of conformal weight w, which is just the bundle of ordinary (−w
n )-densities.
Tensor products with another bundles are denoted as Ea
[w] := Ea
⊗ E[w] etc. In what follows, the notation
as μa
∈ Ea
[w] always means that μa
is a section (and not an element) of Ea
[w], global or local according to
the context.
Conformal structure on M can be described by the conformal metric gab which is a global section of
E(ab)[2]. Any raising and lowering of indices is provided by the conformal metric, e.g. for μa
∈ Ea
[w] we have
μa = gabμb
∈ Ea[w + 2]. A conformal scale is an everywhere positive section of E[1]. The choice of scale
σ ∈ E[1] corresponds to the choice of metric gab ∈ E(ab) from the conformal class so that gab = σ−2
gab. The
corresponding Levi-Civita connection is denoted as ∇. The Schouten tensor, which is a trace modiﬁcation of
the Ricci tensor, is denoted as Pab. Transformations of quantities under the change of scale will be denoted
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 115
by hats. In particular, for σ = fσ, Υa = f−1
∇af and any μa
∈ Ea
, the Levi-Civita connection and the
Schouten tensor change as
∇aμb
= ∇aμb
+ Υaμb
− μaΥb
+ μc
Υcδa
b
, (1)
Pab = Pab − ∇aΥb + ΥaΥb − 1
2 ΥcΥc
gab. (2)
Let Γ ⊂ M be a curve with a parametrization t : Γ → R and the tangent vector Ua
so that Ua
∇at = 1.
Regular curve is called space-like, time-like, respectively light-like (or null), in accord with the type of its
tangent vector Ua
, i.e. if UcUc
> 0, UcUc
< 0, respectively UcUc
= 0. Of course, the curve may change the
type of its tangent vectors. Since the whole study that follows is of very local nature, we restrict ourselves
only to the (segments of) curves of ﬁxed type. In the ﬁrst two cases, the density
u := |UcUc| ∈ E[1] (3)
is nowhere vanishing and will be employed later. Henceforth we always assume Γ is smooth. Along the curve
we use notation d
dt := Uc
∇c and
U a
:= d
dt Ua
, U a
:= d2
dt2 Ua
, . . . , U(i)a
:= di
dti Ua
(4)
for the derived vectors. By abuse of notation we often write Ua
∈ Ea
which should be read as Ua
∈ Ea
|Γ, and
similarly for any other quantities deﬁned only along the curve Γ. Note that the vector U(i)a
has order i + 1
(with respect to Γ). Obviously, none of vectors in (4) is conformally invariant. For instance, the acceleration
vector transforms according to (1) as
U a
= U a
− Uc
Uc Υa
+ 2Uc
Υc Ua
. (5)
From this it follows that, for space- and time-like curves (but not for null curves), a metric in the conformal
class may be chosen so that U a
= 0, i.e. the curve is an aﬃnely parametrized geodesic of the corresponding
Levi-Civita connection. This indicates the problem with a conformally invariant notion of osculating
subspaces. Instead, we are going to make use of tractors.
The conformal standard tractor bundle T over the conformal manifold M of signature (p, q), where
p + q = n = dim M, is the tractor bundle corresponding to the standard representation Rp+1,q+1
of the
conformal principal group O(p + 1, q + 1). Speciﬁcally, T has rank n + 2 and for any choice of scale, it is
identiﬁed with the direct sum
T = E[1] ⊕ Ea
[−1] ⊕ E[−1]
whose components change under the conformal rescaling as
σ
μa
ρ
=
⎛
⎝
σ
μa
+ Υa
σ
ρ − Υcμc
− 1
2 ΥcΥc
σ
⎞
⎠ . (6)
Here Υa is the 1-form corresponding to the change of scale as before. Note that the projecting (or primary)
slot is the top one. The bundle T is endowed with the standard tractor connection ∇, the linear connection
that is given by
∇a
σ
μb
ρ
=
∇aσ − μa
∇aμb
+ δa
b
ρ + Pa
b
σ
∇aρ − Pacμc
.
116 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
It follows this deﬁnition is indeed conformally invariant. The bundle T also carries the standard tractor
metric, the bundle metric of signature (p + 1, q + 1) that is schematically represented as
0 0 1
0 gab 0
1 0 0
,
i.e., for any sections U = (σ, μa
, ρ) and V = (τ, νa
, π) of T ,
U · V = μaνa
+ σπ + ρτ.
It follows the standard tractor metric is parallel with respect to the standard tractor connection.
2. Relative conformal invariants
Throughout this section we consider conformal structures of positive deﬁnite signature. The discussion
for space- and time-like curves in indeﬁnite signature shows only minor diﬀerences, the null case is more
involved, see section 5. In the ﬁrst two subsections we only slightly extend the setting of [2, section 2.8].
Then we introduce a natural family of relative conformal invariants and the notion of conformal arc-length
parameter of curve.
2.1. Canonical lift and initial relations
In the positive deﬁnite signature, any vector is space-like. Hence, for a regular smooth curve Γ ⊂ M with
a ﬁxed parametrization t, the density (3) becomes u =
√
UcUc and it is nowhere vanishing. This provides
a lift of Γ to the standard tractor bundle T that is given by
T :=
0
0
u−1
. (7)
Along the curve we use the notation d
dt := Uc
∇c and
U := d
dt T , U := d2
dt2 T , . . . , U(i)
:= di+1
dti+1 T (8)
for the derived tractors. The notation is chosen so that the highest order term in the middle slot of U(i)
is
a multiple of U(i)a
. Note that the tractor U(i)
has order i + 2 (with respect to Γ). By construction, both
the tractor lift T and all tractors in (8) are conformally invariant objects. Explicitly, the ﬁrst two derived
tractors are
U =
⎛
⎝
0
u−1
Ua
−u−3
UcU c
⎞
⎠ , (9)
U =
⎛
⎝
−u
u−1
U a
− 2u−3
UcU c
Ua
−u−3
UcU c
− u−3
UcU c
+ 3u−5
(UcU c
)2
− u−1
PcdUc
Ud
⎞
⎠ . (10)
The tractors T , U, U are linearly independent, and satisfy
T · T = 0, T · U = 0, T · U = −1,
U · U = 1, U · U = 0.
(11)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 117
The ﬁrst nontrivial identity is
U · U = 3u−2
UcU c
+ 2u−2
UcU c
− 6u−4
(UcU c
)2
+ 2PcdUc
Ud
. (12)
In order to simplify expressions, we will use the following notation
α := U · U , β := U · U , γ := U · U , etc.
Initial relations above and their consequences may be schematically indicated by the Gram matrix which,
for the sequence (T , U, U , U , U ), has the form
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 0 −1 0 α
0 1 0 −α −3
2 α
−1 0 α 1
2 α 1
2 α − β
0 −α 1
2 α β 1
2 β
α −3
2 α 1
2 α − β 1
2 β γ
⎞
⎟
⎟
⎟
⎟
⎟
⎠
. (13)
Clearly, the generating rule may be written as
U(i)
· U(i+j+1)
= U(i)
· U(i+j)
− U(i+1)
· U(i+j)
.
Later we will need some details on the next derived tractor U . As a consequence of T · U = 0, the
projecting slot of U must vanish. With the help of (12), the middle slot of U may be expressed as
u−1
U a
− 3u−3
UcU c
U a
+ 3
2 u−3
UcU c
− 3
2 u−1
(U · U ) + 2u−1
Uc
Ud
Pcd Ua
− uUc
Pc
a
.
2.2. Reparametrizations
Let ˜t = g(t) be a reparametrization of the curve Γ. All objects related to the new parameter ˜t will be
denoted by tildes in accord with d
d˜t
= g −1 d
dt , where g = dg
dt . In particular, Ua
= g −1
Ua
, u = g −1
u and
T = g T . Using just the chain rule and the Leibniz rule, one easily veriﬁes that
U = U + g −1
g T ,
U = g −1
U + g −2
g U + (g −2
g − g −3
g 2
)T ,
U = g −2
U + 2g −2
S(g)U + g −2
S(g) T ,
(14)
where S is the Schwarzian derivative,
S(g) :=
g
g
−
3
2
g
g
2
= (ln g ) −
1
2
(ln g )
2
.
According to the starting relations (11), we have
U · U = g −2
U · U − 2S(g) ,
U · U = g −4
U · U − 4S(g)U · U + 4S(g)2
.
(15)
Hence vanishing of U · U determines a natural projective structure on any curve, cf. [2, Proposition
2.11]:
118 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Proposition 2.1 ([2]). The equation U · U = 0, regarded as a condition on the parametrization of a curve,
determines a preferred family of parametrizations with freedom given by the projective group of the line.
Any parameter from this family will be called projective.
From (15) it further follows that
U · U − (U · U )2
= g −4
U · U − (U · U )2
.
Hence the function
Φ := U · U − (U · U )2
= β − α2
(16)
is a relative conformal invariant of the curve.
2.3. Relative conformal invariants
In general, a relative conformal invariant of weight k of the curve Γ is a conformally invariant function
I : Γ → R that transforms under a reparametrization ˜t = g(t) of the curve as
I = g −k
I.
In particular, vanishing, respectively nonvanishing, of any relative invariant is independent on reparametrizations.
Conformal invariants of weight 0 are the absolute invariants.
Let us denote
Δi := det Gram T , U, U , . . . , U(i−2)
, (17)
the determinant of the Gram matrix corresponding to the ﬁrst i tractors from the derived sequence
T , U, U , . . . . From (11) it is obvious that Δ1 = Δ2 = 0 and Δ3 = −1 independently of the curve and
its parametrization. These three determinants are thus absolute, but trivial conformal invariants. The ﬁrst
nontrivial invariant is Δ4. From (13) it follows that
Δ4 = −β + α2
= −Φ. (18)
In general, we have
Lemma 2.2. For i = 4, . . . , n + 2, the Gram determinant Δi is a relative conformal invariant of weight
i(i − 3).
Proof. As a generalization of (14) we have
U
(j)
= g −j
U(j)
mod T , U, U , . . . , U(j−1)
,
for any j = 1, . . . , n. From this and properties of the determinant it follows that (17) changes under the
reparametrization as
Δi = g −2·2
· · · g −2(i−2)
Δi = g −i(i−3)
Δi,
for any i = 4, . . . , n + 2. 2
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 119
For i = 4, . . . , n + 2, the tractor metric restricted to T , U, . . . , U(i−2)
is nondegenerate, thus vanishing
of Δi is equivalent to the fact that the determining tractors are linearly dependent. In particular, for i ≥ 4,
vanishing of Δi implies vanishing of Δi+1. Note also that the weight of any Δi is even, thus its sign is
independent of all reparametrizations.
2.4. Conformal arc-length
By (16), respectively (18), we deﬁned a nontrivial relative conformal invariant of the lowest possible
weight and order. It is actually nonnegative:
Lemma 2.3. Φ ≥ 0.
Proof. For a parametrization belonging to the projective family of Proposition 2.1 we have Φ = U · U .
Since the top slot of U vanishes, the previous expression equals to the norm squared of the middle slot
of U and so it is nonnegative. Since the weight of Φ is even, it is a nonnegative function independently of
the parametrization of the curve. 2
Suppose that Γ is a curve with nowhere vanishing Φ. Then
ds := 4
Φ(t) dt
is a well-deﬁned conformally invariant 1-form along the curve whose integration yields a distinguished
parametrization of the curve; it is given uniquely up to an additive constant. Such parameter is called the
conformal arc-length. Note that if s is the conformal arc-length then Φ(s) = 1.
It will be clear from later alternative expressions that this is the same distinguished parameter as one
ﬁnds in literature, e.g. in [9] or [5]. In the later reference, one also ﬁnds a notion of vertex, which is the
point of curve where Φ = 0. The vertices of curves are clearly invariant under conformal transformations.
Generic curves are vertex-free, the opposite extreme is discussed in the next section.
Remark 2.4. For space- and time-like curves in the case of general indeﬁnite signature, the relative conformal
invariant Φ deﬁned by (16), respectively (18), is not necessarily nonnegative as in Lemma 2.3. It may even
happen that it vanishes although the corresponding tractors T, U, U , U are linearly independent. The
notion of vertex in such cases would rather be deﬁned by the linear dependence of these tractors than by
vanishing of Φ, cf. Remark 3.4(2).
3. Conformal circles and conserved quantities
We continue the exposition with the assumption of positive deﬁnite signature. The only diﬀerence in
indeﬁnite signature concerns the notion of conformal circles that are deﬁned just for the space- or time-like
directions. Reasonable analogies for null directions are discussed in subsection 5.3.
3.1. Conformal circles
In this subsection we consider the curves for which Φ vanishes identically, i.e. the curves consisting only of
vertices. The conformal arc-length is not deﬁned for such curves and they are excluded from the discussion
in the main part of this paper as the most degenerate cases. However, these curves form a very distinguished
family of curves that coincide with the so-called conformal circles, the curves which are in some sense the
closest conformal analogues of usual geodesics on Riemannian manifolds. In the ﬂat case, they are just the
circles, respectively straight lines, i.e. the conformal images of Euclidean geodesics.
120 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Conformal circles are well known and studied (under various nicknames1
) in the literature. A quick survey
with an explanation of the relationship of the deﬁnition below to distinguished curves of Cartan’s conformal
connection can be found in [13, section 5.1]. Conformal circles on a general conformal manifold can also be
related to ordinary circles in the ﬂat model via the notion of Cartan’s development of curves.
Here we adapt the notation of [1], where conformal circles are deﬁned as the solutions to the system of
conformally invariant third order ODE’s
U a
= 3u−2
UcU c
U a
− 3
2 u−2
UcU c
Ua
+ u2
Uc
Pc
a
− 2Uc
Ud
Pcd Ua
. (19)
Contracting, respectively skew symmetrizing (19), with Ua
yields the following pair of equations equivalent
to (19):
UaU a
= 3u−2
(UcU c
)2
− 3
2 UcU c
− u2
Uc
Ud
Pcd, (20)
U[aUb] = 3u−2
UcU c
U[aUb] + u2
Uc
Pc[bUa]. (21)
From (5) we know that, for any curve with an arbitrary parametrization, a metric in the conformal class
may be chosen so that it is an aﬃnely parametrized geodesic of the corresponding Levi-Civita connection,
i.e. U a
= 0. It is then easy to see from (20) and (21) that
Proposition 3.1 ([1]). The curve is a conformal circle if and only if there is a metric in the conformal class
such that the curve is an aﬃnely parametrized geodesic and Uc
Pca = 0.
From the explicit expressions in subsection 2.1 we may read the following tractorial interpretations
of the equations above: The equation (19) is equivalent to vanishing of the middle slot of the tractor
U + 3
2 (U · U ) U (since the primary slot is zero, this is indeed a conformally invariant condition). This
implies that the middle slots of tractors U and U are collinear, which is just the condition (21). In
particular, this condition is independent of any reparametrization, i.e. solutions to (21) are the conformal
circles parametrized by an arbitrary parameter. The equation (20) is equivalent to U ·U = 0. In particular,
solutions to (20) correspond to the projective parameters of the curve. Considering further the injecting slot
of the tractor U , it is shown in [2, Proposition 2.12] that
Proposition 3.2 ([2]). The curve is a projectively parametrized conformal circle if and only if it obeys
U · U = 0 and U = 0. (22)
We may easily generalize this statement as follows:
Proposition 3.3. The following conditions are equivalent:
(a) the curve is a conformal circle (with and arbitrary parametrization),
(b) all its points are vertices, i.e.
Φ = U · U − (U · U )2
= 0,
(c) the rank 3 subbundle T , U, U ⊂ T is parallel along the curve.
1
E.g. they are called conformal null curves in [9] or conformal geodesics in [10] and [13]. Note that the conformal geodesics
studied in [8] or [15] form a diﬀerent, more general, class of curves.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 121
Proof. From subsection 2.1 we know that the tractors T , U, U are linearly independent for any curve and
its parametrization. From subsection 2.3 we know that Φ = 0 is equivalent to the fact that the tractors
T , U, U , U are linearly dependent, i.e. U belongs to the subbundle T , U, U ⊂ T which is therefore
parallel. Thus (b) and (c) are equivalent.
Among tractors T , U, U , U , only U has nonzero projecting slot. Hence the previous condition means
that U is a linear combination of T and U. Since the middle slot of T vanishes, the previous is equivalent
to the middle slots of U and U being collinear, which is just the condition (21). Thus (c) and (a) are
equivalent. 2
Remarks 3.4.
(1) Note the two conditions in (22) are not independent: if U = 0 then U · U = 0.
(2) Conformal circles in the case of indeﬁnite signature exist only for space- or time-like directions. They
have all of the above stated properties except the condition (b) of Proposition 3.3, which is no more
equivalent to the others, cf. Remark 2.4.
3.2. Conserved quantities
As an application of the current approach we describe some conserved quantities (or ﬁrst integrals) along
conformal circles for some speciﬁc conformal structures. Here we consider conformal structures admitting
almost Einstein scales or conformal Killing ﬁelds with some additional property. The description of the
conserved quantities is easily given due to the tractorial characterization of these additional data, which
can be found e.g. in [11].
An almost Einstein scale is a section σ ∈ E[1] satisfying the conformally invariant condition that the
trace-free part of ∇a∇bσ + Pabσ ∈ Eab[1] vanishes. Then σ is nonvanishing on an open dense set where
the metric gab = σ−2
gab is Einstein. The tractorial characterization of this fact is that the tractor ﬁeld
LT
(σ) ∈ T is parallel, where LT
: E[1] → T denotes the BGG splitting operator.
Proposition 3.5. Let Γ be a conformal circle with a projective parameter t. Assume the conformal manifold M
admits an almost Einstein scale σ ∈ E[1] and let S = LT
(σ) ∈ T be the corresponding parallel tractor.
Then the function s := U · S is a conserved quantity along Γ, i.e. d
dt s = 0.
Proof. A projectively parametrized circle satisﬁes U = 0, the tractor S corresponding to an almost Einstein
scale satisﬁes S = 0. Thus d
dt (U · S) = 0. 2
A conformal Killing ﬁeld ka
∈ Ea
is an inﬁnitesimal symmetry of the conformal structure. This is
equivalent to the vanishing of the trace-free part of ∇(akb) ∈ E(ab)[2]. The tractorial characterization of this
fact is that the tractor ﬁeld K := LΛ2
T
(ka
) ∈ Λ2
T satisﬁes
∇aK = kc
Ωca, (23)
where LΛ2
T
: Ea
→ Λ2
T denotes the BGG splitting operator and Ωcd ∈ E[cd] ⊗ Λ2
T is the curvature of the
tractor connection ∇. Further, along the given curve, we introduce the 1-form
a := Ub
Ωab(U, U ). (24)
Proposition 3.6. Let Γ be a conformal circle with a projective parameter t. Assume the conformal manifold M
admits a conformal Killing ﬁeld ka
∈ Ea
satisfying ka
a = 0, and let K := LΛ2
T
(ka
) ∈ Λ2
T be the
corresponding tractor ﬁeld. Then the function k := K(U, U ) is a conserved quantity along Γ, i.e. d
dt k = 0.
122 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Proof. A projectively parametrized circle satisﬁes U = 0. The tractor K corresponding to ka
satisﬁes (23)
and hence ( d
dt K)(U, U ) = ka
a = 0. Also, K is skew, thus d
dt (K(U, U )) = 0. 2
Note that the assumption ka
a = 0 in the Proposition is automatically satisﬁed for normal conformal
Killing ﬁelds which are characterized by the vanishing of (23).
Remarks 3.7.
(1) It is somewhat tedious to expand tractorial formulas for the conserved quantities s and k. To proceed,
one needs the respective splitting operators and some manipulation. From the explicit expressions it in
particular follows that the considered quantities are nontrivial. All that can be found already in [19].
Note also that both s and k are quantities of order 2, while the conformal circle equation is of order 3.
Further related and interesting results can be found in recent article [12].
(2) Expanding (24) yields
a = Ub
WabcdUc
U d
− 2u2
Ub
∇[aPb]cUc
,
where Wab
c
d is the conformal Weyl tensor. Remarkably, the condition a = 0 plays a role in the
variational approach to conformal circles, see [1, p. 218].
4. Absolute conformal invariants
For a generic curve, the relative conformal invariants Δi from section 2 are all nontrivial, provided that
4 ≤ i ≤ n + 2. One may therefore easily build a galaxy of absolute conformal invariants. In this section we
construct a minimal set of such invariants, which is done in a natural if not canonical way. In order to make
the construction complete, the assumption of positive deﬁnite signature is important here. In particular,
the standard tractor metric has signature (n + 1, 1).
4.1. Tractor Frenet formulas and conformal curvatures
Let Γ be a generic curve, let s be its conformal arc-length parameter and let T , U = d
ds T , U = d
ds U, . . .
be the corresponding tractors as above. The restriction of the tractor metric to the subbundle T , U, U
is nondegenerate and has signature (2, 1). Thus its orthogonal subbundle in T is complementary and has
positive deﬁnite signature. We are going to transform the initial tractor frame (T, U, U , . . . ) into a natural
pseudo-orthonormal one. Firstly, under the transformation
U0 := T , U1 := U, U2 := −U − 1
2 (U · U ) T , (25)
the initial relations (11) transform to
U0 · U0 = 0, U0 · U1 = 0, U0 · U2 = 1,
U1 · U1 = 1, U1 · U2 = 0,
U2 · U2 = 0.
(26)
Secondly, by the standard orthonormalization process we may complete this triple to a tractor frame
(U0, U1, U2; U3, . . . , Un+1) (27)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 123
such that Ui belongs to the span T , U, . . . , U(i−1)
, for all admissible i, and the corresponding Gram
matrix is
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
1
1
...
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (28)
More concretely, for any i = 3, . . . , n, the frame tractors are
Ui = V i/ V i · V i, where
V i = U(i−1)
− (U(i−1)
· Ui−1) Ui−1 − · · ·
· · · − (U(i−1)
· U0) U2 − (U(i−1)
· U1) U1 − (U(i−1)
· U2) U0.
(29)
The last tractor Un+1 is determined by the orthocomplement in T to the codimension one subbundle
T , . . . , U(n−1)
= U0, . . . , Un up to orientation. We do not need to ﬁx the orientation now, it is done
only later. The frame (27) is called the tractor Frenet frame associated to the curve Γ.
Now, diﬀerentiating the constituents of the tractor Frenet frame with respect to the conformal arc-length
s and expressing the result within that frame yields Frenet-like identities. As in the classical situation,
it follows that the coeﬃcients of that system determine the generating set of absolute invariants of the
curve. The pseudo-orthonormality of the tractor Frenet frame implies a lot of symmetries among these
coeﬃcients.
Just from (25) we have
U0 = U1, (30)
U1 = K1U0 − U2, where K1 := −1
2 U · U (31)
is the ﬁrst nontrivial coeﬃcient, the ﬁrst conformal curvature of the curve. In general, for any admissible i,
the derivative Ui is a linear combination of tractors U0, U1, . . . , Ui+1, namely,
Ui = (Ui · U2) U0 + (Ui · U1) U1 + (Ui · U0) U2 + (Ui · U3) U3 + · · · + (Ui · Ui+1) Ui+1,
where the coeﬃcients satisfy Ui ·Uj = −Ui ·Uj and they have to vanish if i = j or j ≥ i+2. In particular,
from (30) we may read U0 · U1 = 1 and U0 · Uj = 0, for j = 0, 2, . . . , n + 1. Similarly, from (31) we further
read U1 · U2 = K1 and U1 · Uj = 0, for j = 1, 3, . . . , n + 1. Substituting these facts into the previous
display, for i = 2, we see that the only unknown coeﬃcient is the one of U3, i.e. U2 · U3. It however follows
that this coeﬃcient is constant so that we have
Lemma 4.1.
U2 = −K1U1 − U3. (32)
Proof. From (29), according to (25) and (11), we obtain
V 3 = U − (U · U) U + (U · U ) T .
124 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Using U · U = −U · U , which is a consequence of U · U = 0, we obtain
V 3 · V 3 = U · U − (U · U )2
= Φ = 1
and thus U3 = V 3. From (25) and (29) we know that U2 equals to −U mod T , U , respectively
−V 3 mod U0, U1 . Hence U2 · U3 = −V 3 · U3 = −1. 2
Following the same ideas as before, one gradually and easily obtains
U3 = U0 + K2U4,
Ui = −Ki−2Ui−1 + Ki−1Ui+1, for i = 4, . . . , n,
Un+1 = −Kn−1Un,
(33)
where K2, . . . , Kn−1 are the higher conformal curvatures of the curve. The bunch of equations (30), (31),
(32) and (33) form the tractor Frenet equations associated to the curve. Schematically they may be written
as
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
K1 −1
−K1 −1
1 K2
−K2
...
...
Kn−1
−Kn−1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (34)
Altogether, we summarize as follows:
Proposition 4.2. The system of tractor Frenet equations determines a generating set of absolute conformal
invariants of the curve. On a conformal Riemannian manifold of dimension n, it consists of n−1 conformal
curvatures that are expressed, with respect to the conformal arc-length parametrization, as
Ki =
U1 · U2, for i = 1,
Ui+1 · Ui+2, for i = 2, . . . , n − 1.
(35)
In (31) we have an alternative expression of the ﬁrst conformal curvature in terms of initial tractors.
Thus, as a consequence of Proposition 2.1 and the ensuing deﬁnition, we obtain an interpretation of that
invariant:
Proposition 4.3. The conformal arc-length parameter belongs to the projective family of parameters if and
only if K1 = 0.
Remarks 4.4.
(1) The matrix in (34) is skew with respect to the inner product corresponding to (28). A change of basis
leads to a diﬀerent matrix realization, e.g., just a permutation of the frame tractors leads to following
rewriting of (34):
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 125
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U3
...
Un
Un+1
U2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
K1 −1
1 K2
−K2
...
...
Kn−1
−Kn−1
−K1 −1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U3
...
Un
Un+1
U2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
This block decomposition reﬂects the standard grading of the Lie algebra so(n+1, 1) with the parabolic
subalgebra corresponding to the block lower triangular matrices. This choice is very close to the ones
in [20] or [5].
(2) For generic curves, the (n + 2)-tuple of derived tractors is linearly independent. For speciﬁc curves, this
is not the case and we cannot build the full tractor Frenet frame and all tractor Frenet formulas. One
may however mimic the previous procedure as long as the tractors are independent:
If, say, the tractors T , U, U , . . . , U(i)
are linearly independent and U(i+1)
belongs to their span,
then they form a parallel subbundle in T of rank i + 2 with the corresponding orthonormal tractors
U0, U1, U2, . . . , Ui+1. This yields a part of the tractor Frenet formulas above, in which the ﬁrst
i−1 conformal curvatures occur; the remaining ones may be considered to be zero. (In this vein, conformal
circles may be considered as the most degenerate curves whose all conformal curvatures vanish, cf.
Proposition 3.3.) In the ﬂat case, this means that the curve itself is contained in an i-sphere, respectively
i-plane, i.e. in the conformal image of an i-dimensional Euclidean subspace. See also [9, section 12] for
further details.
(3) For space- and time-like curves in the case of indeﬁnite signature, we remark the following: The same
transformations as in (25) lead to the relations (26) where the type of curve is reﬂected just in the sign
of the middle element, hence also in the signature of the subbundle T , U, U in T . The orthogonal
complement of this subbundle in T has in general indeﬁnite signature. Thus, the tractors V i from (29)
may have any sign, including zero. If the sign is negative, we just adjust the corresponding deﬁnition
of Ui and, accordingly, one sign in the bottom-right block of (28) changes. If the tractor V i is isotropic
then we cannot satisfy both the diagonal form of the bottom-right block of (28) and the condition
Ui ∈ T , U, . . . , U(i−1)
. Of course, one can always transform given tractors to a pseudo-orthonormal
tractor frame, but only with some additional choices. The rest remains basically the same, only the
tractor Frenet formulas may look diﬀerent and some of the resulting invariants has to be related to
the respective choices. (Nevertheless, it is clear that the exceptional conformal curvature K1 and its
interpretation as in Proposition 4.3 are not inﬂuenced by these issues.) These observations lead to a ﬁner
type characterization of space- and time-like curves. Its complete branching structure is manageable
only in a concrete dimension and signature.
4.2. Conformal curvatures revised
In the deﬁnition above, the orientation of the last tractor from the tractor Frenet frame was a matter of
choice. In what follows we assume the orientation of Un+1 is chosen so that it belongs to the same half-space
as U(n)
with respect to the hyperplane U0, . . . , Un . Thus, according to (29), we have
U(i)
= V i+1 mod U0, U1, . . . , Ui , (36)
126 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
for any i = 3, . . . , n. Also, from deﬁnitions and the tractor Frenet formulas, one progressively obtains
T = U0, U = U1, U = −U2 + K1U0, U = U3 + 2K1U1 + K1U0, etc. (37)
In particular, it holds U = U3 mod U0, U1, U2 . Similarly, it follows that U = K2U4 mod
U0, U1, U2, U3 and, inductively,
U(i)
= K2 · · · Ki−1Ui+1 mod U0, U1, . . . , Ui , (38)
for i = 3, . . . , n. Altogether, from (36), (38) and Ui = V i/
√
V i · V i, we obtain
K2 · · · Ki−1 = V i+1 · V i+1,
which leads to the following conclusion:
Proposition 4.5. With respect to the conformal arc-length parametrization, the higher conformal curvatures
can be expressed as
Ki =
V i+2 · V i+2
V i+1 · V i+1
, for i = 2, . . . , n − 1. (39)
In particular, all these curvatures are positive. The ﬁrst conformal curvature K1 does not ﬁt to this
uniform description and its sign may be arbitrary, including zero.
An expansion of (39), according to (25) and (29), provides expressions of higher conformal curvatures in
terms of initial tractors. Instead of exploiting this demanding procedure, we are going to use the relative
conformal invariants from subsection 2.3. From the construction of the tractor Frenet frame it follows
that Δi+1 = Δi(V i · V i), for all admissible i. Since Δ3 = −1, we see that all admissible determinants
are negative. Hence we can express V i · V i as a quotient of determinants, whose substitution into (39)
yields
Ki =
Δi+1Δi+3
−Δi+2
. (40)
Up to this point, everything has been related to the parametrization by the conformal arc-length. Now
we consider an arbitrary parametrization of the curve.
Theorem 4.6. With respect to an arbitrary parametrization of the curve, the higher conformal curvatures
can be expressed as
Ki =
Δi+1Δi+3
−Δi+2
4
√
−Δ4
, for i = 2, . . . , n − 1. (41)
Proof. According to Lemma 2.2, the weight of the right hand side of (41) is
1
2 (i + 1)(i − 2) + 1
2 (i + 3)i − (i + 2)(i − 1) − 1 = 0,
thus it deﬁnes an absolute conformal invariant. With respect to the conformal arc-length parametrization,
(41) coincides with (40), hence the statement follows. 2
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 127
Alternatively, the statement is deducible from (40) by the reparametrization according to subsection 2.2
and Lemma 2.2, where g = Φ
1
4 . That way, we may obtain also an expression of the very ﬁrst conformal
invariant, K1, which does not ﬁt to the just given description: Its expression with respect to the conformal
arc-length is given in (31), which transforms under reparametrizations according to (15). The substitution
of g = Φ
1
4 and a small computation reveals that
Proposition 4.7. With respect to an arbitrary parametrization of the curve, the ﬁrst conformal curvature can
be expressed as
K1 = −1
2 Φ− 1
2 U · U − 1
2 (ln Φ) + 1
16 (ln Φ) 2
=
= −1
2 Φ− 5
2 Φ2
U · U − 1
2 ΦΦ + 9
16 Φ 2
.
(42)
Remarks 4.8.
(1) Not only the conformal curvatures, but the tractor Frenet frame itself can be built with respect to
an arbitrary parametrization. E.g. expressions of the ﬁrst three tractors (25) transform, under the
reparametrization according to (14) with g = Φ
1
4 , so that
U0 = Φ
1
4 T ,
U1 = U + 1
4 (ln Φ) T ,
U2 = −Φ− 1
4 U + 1
4 (ln Φ) U + 1
2 U · U + 1
32 (ln Φ) 2
T .
(43)
(2) Our exceptional invariant K1 corresponds to a similarly exceptional invariant Jn−1 of [9, section 5] as
follows. For any relative conformal invariant Q of weight 1, an appropriate combination of Q and its
derivatives leads to an appropriate transformation of the new quantity under reparametrizations of the
curve. Namely, with the same conventions as yet,
2QQ − 3Q 2
= g −4
2QQ − 3Q 2
− 2S(g)Q2
.
Combining with (15), it easily follows that the function
Q−2
U · U − 2Q−3
Q + 3Q−4
Q 2
is an absolute conformal invariant of the curve. Now, the substitution Q = Φ
1
4 leads to an invariant,
which diﬀers from our K1, respectively Fialkow’s Jn−1, just by a constant multiple. (For the comparison
with Jn−1, the formula (44) from the next subsection is needed.)
4.3. Conformal curvatures in terms of Riemannian ones
Expanding any of the above expressions for the conformal curvatures yields very concrete and very ugly
formulas in terms of the underlying derived vectors. As a compromise between the explicitness and the
ugliness we show how to rewrite the previous formulas in terms of the Riemannian curvatures of the curve.
This way we will also be able to compare our invariants with their more classical counterparts.
Within this paragraph we consider a generic curve parametrized by the Riemannian arc-length parameter,
according to a chosen scale. The corresponding Riemannian Frenet frame is denoted by (e1, . . . , en) and the
Riemannian Frenet formulas are2
2
Warning: in this subsection we relax the abstract indices, i.e. we write ei rather than ea
i etc. The raising and lowering indices
with respect to the chosen metric is denoted by and , respectively.
128 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
e1 = κ1e2,
ei = −κi−1ei−1 + κiei+1, for i = 2, . . . , n − 1,
en = −κn−1en−1,
where primes denote the derivative with respect to the Riemannian arc-length and κ1, . . . , κn−1 are the
Riemannian curvatures of the curve. Accordingly, the expressions (3) and (4) read as
u = 1, U = e1, U = κ1e2, U = −κ2
1e1 + κ1e2 + κ1κ2e3, etc.
Following the development of subsection 2.1, the ﬁrst tractors are
T =
0
0
1
, U =
0
e1
0
, U =
−1
κ1e2
−P(e1, e1)
,
U =
0
−κ2
1e1 + κ1e2 + κ1κ2e3 − P(e1, e1)e1 − P(e1, −)
∗
.
Hence we obtain
U · U = κ2
1 + 2P(e1, e1),
Φ = U · U − (U · U )2
= κ 2
1 + κ2
1κ2
2 + terms involving P.
(44)
Substitution of (44) into (42) leads to an expression of the ﬁrst conformal curvature K1 in terms of ﬁrst
two Riemannian curvatures κ1, κ2, some terms involving the Schouten tensor and their derivatives. The full
expansion leads to a huge formula even in the ﬂat case.
Formula enthusiasts may continue further in this spirit and express the higher derived tractors. We add
some details on the Gram matrix corresponding to the ﬁrst ﬁve tractors. It is displayed in (13) where,
modulo terms involving P,
α = κ2
1, β = κ 2
1 + κ4
1 + κ2
1κ2
2,
γ = 9κ2
1κ 2
1 + (κ1 − κ3
1 − κ1κ2
2)2
+ (2κ1κ2 + κ1κ2)2
+ κ2
1κ2
2κ2
3.
Of course, the term κ3 is nontrivial only if n ≥ 4. From this and (41) one could deduce an expression for
the second conformal curvature K2 in terms of κ1, κ2, κ3, P and their derivatives. Although the result is
expected to be messy, it can be condensed into the following neat form:
K2 = κ 2
1 + κ2
1κ2
2
− 5
4
(2κ 2
1 κ2 + κ2
1κ3
2 + κ1κ1κ2 − κ1κ1 κ2)2
+ (κ 2
1 + κ2
1κ2
2)κ2
1κ2
2κ2
3
1
2
,
modulo terms involving P.
Remark 4.9. In the ﬂat case, for n = 3, the previous display agrees precisely with the invariant which is
called conformal torsion in [5] (the other invariant there corresponds to our K1). Note that vanishing of K2
in such case is equivalent to κ2 = 0, or κ2 = 0 and κ2
κ1
=
κ1
κ2
1κ2
. The former condition means that the
curve is planar, the latter one means the curve is spherical. By virtue of Remark 4.4(2), this is an expected
behaviour.
For another comparisons let us also describe ﬁrst few tractors from the tractor Frenet frame constructed
in subsection 4.1. According to (43), the easy tractors are
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 129
U0 = Φ
1
4
0
0
1
, U1 =
⎛
⎝
0
e1
1
4 (ln Φ)
⎞
⎠ , U2 = −Φ− 1
4
⎛
⎜
⎝
−1
1
4 (ln Φ) e1 + κ1e2
1
32 (ln Φ) 2
+ 1
2 κ2
1
⎞
⎟
⎠ ,
where Φ is given in (44). The ﬁrst nontrivial step concerns an expression of U3, which one calculates as
U3 = Φ− 1
2
⎛
⎝
0
κ1e2 + κ1κ2e3 + P(e1, e1)e1 − P(e1, −)
κ1κ1
⎞
⎠ . (45)
Remarks 4.10.
(1) Incidentally, the middle slot of the tractor in (45) corresponds to the starting quantity in Fialkow’s
approach, the so-called ﬁrst conformal normal of the curve, cf. [9, equation (4.21)]. Using this and an
invariantly deﬁned derivative along the curve (with respect to the conformal arc-length parameter), the
conformal Frenet identities and the corresponding n − 2 conformal curvatures are deduced in [9, section
4]. The last Fialkow’s conformal curvature is constructed ad hoc in [9, section 5], see the discussion
in Remark 4.8(2).
(2) At this stage we can easily count the orders of individual invariants although some estimates could be
done earlier. The relative conformal invariant Φ is of order 3, cf. (44). From this and (42) we see that
the ﬁrst conformal curvature K1 is of order 5. For i ≥ 2, an estimate based on (41) yields the order of
ith conformal curvature Ki is at most i + 3, but the current expressions show it is actually i + 2.
5. Null curves in general signature
The next step is to consider possible generalizations of the previous treatment to conformal manifolds
of arbitrary signature. Thus, in the following we suppose that M is a conformal manifold of dimension
n = p + q and signature (p, q) and T is the standard tractor bundle with the tractor metric of signature
(p + 1, q + 1). Generally, we should distinguish curves according to the type of tangent vectors/tractors
and also according to the type of further tractors in the tractor Frenet frame. This leads to a diversity of
possible cases, cf. Remark 4.4(3). The typical situation we need to understand is when the tangent vector
together with its several derivatives generate a totally isotropic subspace of the tangent space, cf. the notion
of r-null curves below. One can view this as a counterpart of the Riemannian setting discussed so far. In
the last subsection we discuss the Lorentzian signature in more detail.
5.1. r-null curves
For null curves the density (3) vanishes identically and cannot be used for the lift to T . Depending on
the signature, we may consider curves that are more and more isotropic, without being degenerate. On a
conformal manifold of signature (p, q), for any r ≤ min{p, q}, a curve is called the r-null curve if the vectors
Ua
, . . . , U(r−1)a
are linearly independent and null, whereas U(r)a
is not null. Consequently, it holds
U(i)
c U(j)c
=
0, for i + j < 2r,
(−1)r−i
U
(r)
c U(r)c
, for i + j = 2r.
(46)
Note that the notion of r-null curves is well deﬁned as it does not depend on the chosen scale. Indeed, one
can easily verify by induction using (1) that
U(i)a
= U(i)a
mod Ua
, . . . , U(i−1)a
,
130 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
for all 1 ≤ i ≤ 2r. In this vein, space- and time-like curves may be termed 0-null curves. As an opposite
extreme, we use the notation ∞-null curves for curves, whose all vectors U(i)a
are isotropic; such curves
are necessarily degenerate. The most prominent—and the most degenerate—instance of such curves are the
null geodesics.
For an r-null curve, it follows from (1) that the norm squared of U(r)c
is a nowhere vanishing density of
conformal weight 2. Thus,
u := |U
(r)
c U(r)c| ∈ E[1]
provides a lift T ∈ T as in (7), to which we add the derived tractors U, U etc. as before. Explicit expressions
are very analogous to those in (9), (10) etc. up to some shift. For i ≤ 2r, we have
U(i)
=
⎛
⎜
⎝
0
u−1
U(i)a
+ · · ·
∗
⎞
⎟
⎠ ,
where the zero in the primary slot appears as −u−1
UcU(i−1)c
= 0 and the dots in the middle slot denote
terms of lower order. In particular, we have T · T = · · · = U(r−1)
· U(r−1)
= 0 and
:= U(r)
· U(r)
= ±1. (47)
The ﬁrst nontrivial quantity is U(r+1)
· U(r+1)
. In order to simplify expressions, we will use the notation
α := U(r+1)
· U(r+1)
, β := U(r+2)
· U(r+2)
, γ := U(r+3)
· U(r+3)
, etc. (48)
The pattern of the initial relations remains the same as in (13) up to the signs concerning and a shift in
the south-east direction. After some computation, one shows that the ﬁrst few antidiagonals of the Gram
matrix read as
U(i)
· U(j)
=
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, for i + j < 2r,
(−1)r−i
, for i + j = 2r,
0, for i + j = 2r + 1,
(−1)r+1−i
α, for i + j = 2r + 2,
(−1)r+1−i 2(r−i)+3
2 α , for i + j = 2r + 3,
(−1)r−i
β + (−1)r+1−i (r−i+2)2
2 α , for i + j = 2r + 4,
(49)
where we use the convention U(−1)
:= T .
Under the reparametrization ˜t = g(t) as in subsection 2.2, one easily veriﬁes that T = g r+1
T , from
which one deduces the transformations of higher order tractors. Considering U(i)
:= 0, for i < −1, it turns
out that
U
(k)
= g r−k
U(k)
+ Akg −1
g U(k−1)
+ (Bkg −1
g − Ckg −2
g 2
)U(k−2)
+ · · · , (50)
for k = 0, 1, . . . , where the coeﬃcients Ak, Bk, Ck are given by the recurrence relations
Ak+1 = Ak + r − k,
Bk+1 = Bk + Ak,
Ck+1 = Ck − (r − k − 1)Ak,
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 131
with the initial conditions A0 = r + 1, B0 = 0 and C0 = 0. It is an elementary, but tedious, exercise to solve
and tidy up this system. For k = r + 1, it turns out that
Ar+1 = 1
2 (r + 1)(r + 2),
Br+1 = 1
6 (r + 1)(r + 2)(2r + 3),
Cr+1 = −1
8 (r + 1)(r + 2)(r2
− r − 4).
From this and the initial relations it follows that
α = g −2
α − 2 Br+1g −1
g + (A2
r+1 + 2Cr+1)g −2
g 2
.
Since A2
r+1 + 2Cr+1 = 3Br+1, the previous simpliﬁes to
α = g −2
(α − 2 Br+1S(g)) . (51)
Thus, the equation α = 0, regarded as a condition on the parametrization of an r-null curve, determines a
preferred family of parametrizations with freedom the projective group of the line. This is a generalization
of Proposition 2.1 which just corresponds to r = 0. Altogether we may conclude with
Proposition 5.1. For any admissible r, any r-null curve carries a preferred family of projective parameters.
It is well known that null geodesics carry a preferred family of aﬃne parameters.
Further, the Gram determinants are deﬁned and denoted just the same as in (17). Initial relations for
an r-null curve lead to Δi = 0, for i = 1, . . . , 2r + 2, and Δ2r+3 = − . The ﬁrst potentially nontrivial
determinant is Δ2r+4. Analogously to Lemma 2.2 we have
Lemma 5.2. Along any r-null curve, the Gram determinant Δi, for i = 2r + 4, . . . , n + 2, is a relative
conformal invariant of weight i(i − 2r − 3).
For any i = 2r + 4, . . . , n + 2, the vanishing of Δi is equivalent to the fact that the determining tractors
are linearly dependent. The weight of any Δi is even, hence its sign is not changed by reparametrizations. In
contrast to Lemma 2.3 (corresponding to r = 0), we cannot say anything about the particular sign of Δ2r+4.
For an r-null curve with nowhere vanishing Δ2r+4, we deﬁne the conformal pseudo-arc-length parameter by
integration of the 1-form
ds := 2r+4
|Δ2r+4(t)| dt (52)
along the curve; it is given uniquely up to an additive constant. The invariant Δ2r+4 (hence also all higher
ones) may vanish. In particular, this happens automatically if 2r + 4 > n + 2, i.e. if n < 2r + 2. From the
introductory counts we know that r and n are related by n ≥ 2r, therefore the critical dimensions are n = 2r
and 2r + 1. In such cases, none of the just considered invariants can be used to deﬁne a natural parameter
of the curve. However, there is another couple of relative conformal invariants as shown in subsection 5.2
below; see also Remark 5.15 for further comments.
At this stage, we sketch how to adapt the construction of the tractor Frenet frame from subsection 4.1
for r-null curves with the pseudo-arc-length deﬁned. The smallest nondegenerate subbundle in T gradually
built from the derived tractors is T , U, . . . , U(2r+1)
. The restriction of the tractor metric to this subbundle
132 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
has signature (r + 2, r + 1), respectively (r + 1, r + 2), thus its orthogonal complement in T has signature
(p − r − 1, q − r), respectively (p − r, q − r − 1). We may always start with the prescription
U0 := T , U1 := U, . . . , Ur+1 := U(r)
. (53)
Then we have to consider transformations determining Ui ∈ T , U, . . . , U(i−1)
, for i = r + 2, . . . , 2r + 2,
so that the Gram matrix corresponding to the tractors (U0, . . . , U2r+2) is antidiagonal with values ±1
(this substitutes the upper-left block in (28)). Note that in contrast to the case r = 0, the deﬁnition for
Ur+2, . . . , U2r+2 is not unique. This freedom is already visible from the ﬁrst tractor of this subsequence
which may be given as
Ur+2 = − U(r+1)
− 1
2 αU(r−1)
+ · · · , (54)
where the dots stay for any linear combination of lower order tractors U(r−2)
, . . . , T . To accomplish the full
tractor frame, i.e. to determine (U2r+3, . . . , Un+1), we continue by an orthonormalization process. Going
along, we may meet an isotropic tractor: in such case we face the same problem as discussed in Remark 4.4(3)
and some additional choices are inevitable.
Having constructed the tractor Frenet frame, we diﬀerentiate with respect to the conformal pseudo-arclength
to obtain the tractor Frenet equations with the generating set of absolute conformal invariants. Just
from (53) and (54) we easily deduce ﬁrst few Frenet-like identities:
U0 = U1, . . . , Ur = Ur+1,
Ur+1 = K1Ur − Ur+2, where K1 := −1
2 α.
Clearly, the ﬁrst conformal curvature is not aﬀected by the freedom in the construction of the tractor
Frenet frame. From the expression of K1 and the remark preceding Proposition 5.1 we immediately have
the following generalization of Proposition 4.3.
Proposition 5.3. The conformal pseudo-arc-length parameter belongs to the projective family of parameters
of an r-null curve if and only if K1 = 0.
The freedom in the construction of (Ur+2, . . . , U2r+2) inﬂuences the corresponding part of tractor
Frenet equations and so the corresponding conformal curvatures, namely, K2, . . . , Kr+1. The respective
freedom in the construction of (U2r+3, . . . , Un+1) inﬂuences some of the remaining conformal curvatures
Kr+2, . . . , Kn−r−1. Notably, they are n − r − 1 in total.
Various expressions analogous to (35) and (39) (with respect to the conformal pseudo-arc-length parameter),
or (41) and (42) (with respect to an arbitrary parameter) are deducible. We provide more details only
in subsection 5.4 for the case of Lorentzian signature.
5.2. Invariants of Wilczynski type
For an r-null curve, vanishing of Δ2r+4 means that the tractors T , U, . . . , U(2r+2)
are linearly dependent.
Since the ﬁrst 2r +3 tractors from this sequence are linearly independent and U(k)
= T (k+1)
, this condition
may be interpreted as a tractor linear ODE,
T (2r+3)
+ q2r+2T (2r+2)
+ · · · + q1T + q0T = 0, (55)
where q0, . . . , q2r+2 are functions expressible in terms of pairings of the initial tractors. Following [21, ch. II,
§4], we may consider the so-called Wilczynski invariants of this equation, i.e. the set of 2r + 1 essential
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 133
invariants denoted as Θ3, . . . , Θ2r+3 (the indices indicate weights). Of course, we may associate Wilczynski
invariants to the linear diﬀerential operator on the left hand side of (55), i.e. also to general r-null curves
with nonzero Δ2r+4. As a matter of fact, these invariants are relative conformal invariants of the curve.
Assuming q2r+2 = 0, i.e. the so-called semi-canonical form, an explicit form of ﬁrst two of these invariants
is given by the following Lemma.
Lemma 5.4 ([17]). Let y(k)
+ qk−2y(k−2)
+ · · · + q0y = 0 be a linear ODE of kth order in the semi-canonical
form (qk−1 = 0). Then the ﬁrst two Wilczynski invariants are
Θ3 = qk−3 −
k − 2
2
qk−2,
Θ4 = qk−4 −
k − 3
2
qk−3 +
(k − 2)(k − 3)
10
qk−2 −
(5k + 7)(k − 2)(k − 3)
10k(k + 1)(k − 1)
q2
k−2.
The formula for Θ3 and Θ4 was known already to Laguerre and Schlesinger, respectively, both predating
the work of Wilczynski. Accurate references and modern treatment of this classical subject can be found in
[18, section 4.1].
Now we need to relate coeﬃcients qi from the equation (55) to quantities α, β, γ etc. from (48). By pairing
both sides of (55) with T , it follows from (49) that q2r+2 = 0, i.e. the equation is in the semi-canonical
form. Considering similarly the pairing of both sides of (55) with T = U, T = U and with T = U ,
one computes
q2r+2 = 0, q2r+1 = α, q2r =
2r + 1
2
α , q2r−1 = − β +
r2
2
α + α2
. (56)
Combining this with Lemma 5.4, we obtain the resulting form of Θ4:
Proposition 5.5. Let , α and β be the quantities associated to an r-null curves (with respect to an arbitrary
parametrization) as in (47) and (48). Then Θ3 = Θ5 = 0 and
Θ4 = − β −
r(r + 3)
10
α +
(r + 3)(2r + 5)(5r + 4)
10(r + 1)(r + 2)(2r + 3)
α2
is a relative conformal invariant of the weight 4.
Proof. Both vanishing of Θ3 and the form of Θ4 follows from (56) and Lemma 5.4, where k = 2r + 3. The
particular weight of Θ4 follows from the deﬁnitions (48) and the exponent of g in (50).
To show Θ5 = 0, we shall for simplicity assume a projective parametrization of the curve, i.e. α = 0. In
particular, the equation is in the canonical Laguerre–Forsyth form for which the expressions of Wilczynski
invariants are well known, see [21, ch. II, eqn. (48)]. Analogously to the computation leading to (56), we
obtain
U(i)
· U(2r+5−i)
= (−1)r−i 2(r − i) + 5
2
β and q2r−2 = −
2r − 1
2
β .
Now substituting coeﬃcients q2r−1 and q2r−2 into the just referred formula, one easily veriﬁes that
Θ5 = 0. 2
Observe that Θ4 recovers Δ4 from (18) for r = 0.
134 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Remarks 5.6.
(1) The Laguerre–Forsyth form of the equation is preserved by transformations with freedom the projective
group of the line, which corresponds precisely to the condition α = 0. Also, it follows that the linear
equation is equivalent to the trivial equation if and only if all Wilczynski invariants vanish. In our case,
this is equivalent to the vanishing of α, β, etc. from (48) up to U(2r+2)
· U(2r+2)
.
(2) Experiments with speciﬁc r’s suggest the equation (55) could be self-adjoint, i.e. all odd Wilczynski
invariants vanish. This would mean that we have just r (rather than 2r) new and potentially nontrivial
invariants Θ4, . . . , Θ2r+2.
5.3. Conformal null helices
For any space- or time-like curve with an arbitrary parametrization, a metric in the conformal class may
be chosen so that U a
= 0, cf. (5). For null curves, this condition holds only if the curve is null geodesic, i.e.
the null curve whose acceleration vector U a
is proportional to Ua
. Consequently, all higher order vectors
are proportional to Ua
. These curves, although very important, cannot be lifted to the standard tractor
bundle in the sense considered above. That is why they are excluded from our considerations.
For an r-null curve, we see from (46) and the subsequent discussion that the vectors Ua
, . . . , U(2r)a
are
linearly independent for any choice of scale. Therefore we cannot achieve any of the conditions U a
= 0,
. . . , U(2r)a
= 0 by a conformal change of metric. The simplest conceivable statement is
Lemma 5.7. For an r-null curve with an arbitrary parametrization, a metric in the conformal class may be
chosen so that U
(r)
c U(r)c
= ±1 and U(2r+1)a
= 0.
Proof. This can be proved by various means. For the sake of latter references, we employ the tractors
anyhow it may seem artiﬁcial. Firstly, we may always choose the metric so that U
(r)
c U(r)c
= ±1. With this
assumption, the tractors associated to the curve have the form
T =
0
0
1
, U =
0
Ua
0
, U =
0
U a
ρ1
, . . . ,
U(j)
=
⎛
⎝
εj
U(j)a
mod Ua
, . . . , U(j−2)a
ρj
⎞
⎠ ,
for j = 2, . . . , 2r + 1, where ρj are in general nonzero while all εj vanish except for ε2r+1 = (−1)r−1
.
Secondly, the choice of metric may be adjusted so that the bottom slot of U vanishes and the expressions
of previous tractors are unchanged: according to (6), the corresponding Υa has to satisfy ΥcU c
= ρ1 and
ΥcUc
= 0. Hence U =
0
U a
ρ2
, i.e. there is only the leading term in the middle slot. Inductively, we may
achieve a rescaling so that U(j)
=
0
U(j)a
0
, for all j ≤ 2r. Hence U(2r+1)
=
(−1)r−1
U(2r+1)a
∗
. Finally, we may
consider a rescaling so that the middle slot of the last tractor vanishes: according to (6), the corresponding
Υa
equals to (−1)r
U(2r+1)a
along the curve. Hence the statement follows. 2
Now we are ready to identify the closest relatives of conformal circles among r-null curves. We know
from preceding subsections that the ﬁrst 2r +3 derived tractors T , U, . . . , U(2r+1)
are linearly independent.
Therefore we can never achieve any of the conditions U = 0, . . . , U(2r+1)
= 0 for r-null curves. The
simplest conceivable condition appears in item (a) of the following Theorem.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 135
Theorem 5.8. For an r-null curve, the following conditions are equivalent:
(a) the curve with a projective parametrization (α = 0) obeys U(2r+2)
= 0,
(b) the curve (with and arbitrary parametrization) obeys Δ2r+4 = 0 and Θ4 = · · · = Θ2r+3 = 0,
(c) there is a metric in the conformal class such that U
(r)
c U(r)c
= ±1, U(2r+1)a
= 0 and Uc
Pca = 0.
Proof. Expressing the condition Δ2r+4 = 0 as (55), the equivalence of (a) and (b) follows from the general
theory of linear diﬀerential equations and the fact that Θ3 vanishes automatically, cf. Proposition 5.5 and
Remark 5.6(1).
For the rest we assume the scale is chosen as in the proof of Lemma 5.7. In such scale, the derived tractors
are
T =
0
0
1
, U =
0
Ua
0
, . . . , U(2r)
=
⎛
⎝
0
U(2r)a
0
⎞
⎠ , U(2r+1)
=
⎛
⎝
(−1)r−1
0
−Uc
U(2r)d
Pcd
⎞
⎠ .
The additional assumption Uc
Pca = 0 from (c) then yields U(2r+2)
= 0 and so (a) holds.
Conversely, from (a) it in particular follows U(2r+1)
· U(2r+1)
= 0, hence the bottom slot of U(2r+1)
has
to vanish. Then U(2r+2)
= 0 implies that Uc
Pca = 0 and so (c) holds. 2
Any curve satisfying (any of) the conditions (a)–(c) is called conformal r-null helix. The condition
Δ2r+4 = 0 is equivalent to the fact that the subbundle T , U, . . . , U(2r+1)
⊂ T of rank 2r + 3 is parallel
along the curve. In the ﬂat case, this means that the conformal r-null helix is contained in the conformal
image of a (2r + 1)-dimensional Euclidean subspace. With condition (c), it easily follows that these curves
are just the conformal images of so-called null Cartan helices, cf. [7]. Note that the conformal circles ﬁt to
the just given description for r = 0.
Remark 5.9. As in subsection 3.3, one may construct conserved quantities along conformal r-null helices
on speciﬁc conformal manifolds. The two examples mentioned in Propositions 3.5 and 3.6 have obvious
counterparts for general r, provided that the respective functions are replaced by
s := U(2r+1)
· S and k := K(U, U(2r+1)
).
The reasoning is the same, explicit expressions are analogous, but more complicated.
5.4. Remarks on Lorentzian signature
In this subsection we suppose an n-dimensional conformal manifold of Lorentzian signature (n − 1, 1),
hence the standard tractor bundle T with the bundle metric of signature (n, 2). Following the scheme of
previous subsections, a lot of things simplify as the maximal isotropic subspace in the tangent space has
dimension one.
The only admissible r-null curves in this signature correspond to r = 1. Therefore we may speak without
a risk of confusion just about null curves, instead of 1-null curves. The assumption UcUc
= 0 implies that
UcU c
= 0, thus the vector U a
must be space-like. The lift to T is provided by the density u = UcU c. The
smallest nondegenerate subbundle in T built from the derived tractors is of rank m5 and has signature (3, 2).
Hence its orthogonal complement is positive deﬁnite. The ﬁrst few starting relations are indicated in the
following Gram matrix for the sequence T , U, U , U , U , U , cf. (48) and (49):
136 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 0 0 0 1 0
0 0 0 −1 0 α
0 0 1 0 −α −3
2 α
0 −1 0 α 1
2 α 1
2 α − β
1 0 −α 1
2 α β 1
2 β
0 α −3
2 α 1
2 α − β 1
2 β γ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The projective parametrization of the null curve corresponds to α = U · U = 0.
The trivial Gram determinants are Δ1 = Δ2 = Δ3 = Δ4 = 0 and Δ5 = 1. The ﬁrst nontrivial one is
Δ6 = γ + α α − 2βα − 9
4 α 2
+ α3
,
provided that the dimension of conformal manifold is n ≥ 4, which is assumed in the rest of this subsection
(the special case n = 3 is dealt individually in Remark 5.15). This invariant is nonnegative:
Lemma 5.10. Δ6 ≥ 0.
Proof. For a projective parametrization of the null curve, α = 0, we have Δ6 = γ. According to this choice,
one easily veriﬁes that the tractor
V 5 := U − β U − 1
2 β T
belongs to the orthogonal complement of the subbundle T , U, . . . , U in T . This complement has the
positive deﬁnite signature, hence we conclude with γ = U · U = V 5 · V 5 ≥ 0. Since the weight of Δ6
is even, it is a nonnegative function independently of the parametrization of the curve. 2
Accordingly, the 1-form (52) leading to the deﬁnition of conformal pseudo-arc-length may be substituted
by
ds := 6
Δ6(t) dt. (57)
Clearly, vanishing of Δ6 is equivalent to vanishing of V 5, which yields exactly the equation (55), for
r = 1. Among the corresponding Wilczynski invariants, there is only one nontrivial, namely,
Θ4 = − 1
25 (25β + 10α − 21α2
), (58)
cf. Proposition 5.5. According to Theorem 5.8, we may conclude with
Proposition 5.11. Conformal null helices are characterized by the pair of equations
Δ6 = 0 and Θ4 = 0.
To build the tractor Frenet frame, we start with
U0 := T , U1 := U, U2 := U . (59)
Now, all admissible transformations determining U3 and U4, so that the 5-tuple (U0, . . . U4) has the
antidiagonal Gram matrix, are
U3 := −U − 1
2 α U + k T ,
U4 := U + α U + U + 1
2 (α2
− β) T ,
(60)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 137
where k + = 1
2 α . For the rest of this demonstration we choose
k = 0 and = 1
2 α . (61)
As mentioned above, the orthocomplement to the subbundle T , . . . , U = U0, . . . , U4 has positive
deﬁnite signature. Therefore, we easily complete the tractor Frenet frame
(U0, . . . , U4; U5, . . . , Un+1)
in the spirit of (29) so that Ui ∈ T , U, . . . , U(i−1)
, for all admissible i, and the corresponding Gram
matrix is
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
...
1
1
...
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Now, diﬀerentiating with respect to the conformal pseudo-arc-length we approach the identities with the
generating set of absolute conformal invariants. Just from (59) and (60), with respect to the choice (61), we
easily deduce ﬁrst four Frenet-like identities:
U0 = U1, (62)
U1 = U2, (63)
U2 = K1U1 − U3, where K1 := −1
2 α, (64)
U3 = K2U0 − K1U2 − U4, where K2 := 1
2 (α2
− β). (65)
The next step is subtle, but fully analogous to the one announced in Lemma 4.1: the only coeﬃcient in the
expression of U4, which cannot be deduced from the previous identities, is the one of U5, i.e. U4 · U5. It
however follows that this coeﬃcient is constant, namely,
U4 = −K2U1 + U5. (66)
(The crucial point in the argument is the expression of V 5 and the observation that V 5 · V 5 = Δ6. Since
we assume parametrization by the conformal pseudo-arc-length, this equals to 1, thus U5 = V 5.) The rest
is easy so that one quickly summarizes as
U5 = −U0 + K3U6,
Ui = −Ki−3Ui−1 + Ki−2Ui+1, for i = 6, . . . , n,
Un+1 = −Kn−2Un.
(67)
The cluster of equations from (62) through (66) to (67) forms the tractor Frenet equations associated to the
null curve determining its conformal curvatures. Schematically, the tractor Frenet equations may be written
as
138 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
U5
U6
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
K1 −1
K2 −K1 −1
−K2 1
−1 K3
−K3
...
...
Kn−2
−Kn−2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
U5
U6
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Altogether, we summarize as follows:
Proposition 5.12. The system of tractor Frenet equations determines a generating set of absolute conformal
invariants of the null curve. On a conformal manifold of Lorentzian signature and dimension n ≥ 4, it
consists of n − 2 conformal curvatures that are expressed, with respect to the conformal pseudo-arc-length
parametrization, as
Ki =
⎧
⎨
⎩
Ui+1 · Ui+2, for i = 1, 2
Ui+2 · Ui+3, for i = 3, . . . , n − 2.
Among these curvatures, only K2 depends on additional choices in the construction of U3 and U4.
As in subsection 4.2, there are alternative ways of expressing the conformal curvatures. The ﬁrst two, i.e.
the two exceptional, conformal curvatures K1 and K2 are given in terms of initial tractors with respect to the
conformal pseudo-arc-length parametrization in (64) and (65). One may obtain the expressions with respect
to an arbitrary reparametrization by a substitution according to (51), respectively (50), where g = 6
√
Δ6.
In the former case, one ends up with the formula very similar to the one in (42), the later case is more
ugly. For the higher conformal curvatures we may proceed as follows. Firstly, an analogue of Proposition 4.5
turns out to be
Proposition 5.13. With respect to the conformal arc-length parametrization, the higher conformal curvatures
can be expressed as
Ki =
V i+3 · V i+3
V i+2 · V i+2
, for i = 3, . . . , n − 2. (68)
Secondly, it holds Δi+1 = Δi(V i · V i), for all admissible i. Since Δ5 = 1, we in particular see that all
admissible determinants are positive. Now, expressing V i · V i via the determinants, substituting into (68)
and passing to an arbitrary parametrization, we obtain the following substitute of Theorem 4.6:
Theorem 5.14. With respect to an arbitrary parametrization, the higher conformal curvatures can be expressed
as
Ki =
Δi+2Δi+4
Δi+3
6
√
Δ6
, for i = 3, . . . , n − 2.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 139
Remark 5.15. As we already noticed, the study of null curves in dimension n = 3 requires an extra care.
The relative invariant Δ6 vanishes automatically in that case, therefore it cannot be used to deﬁne the
distinguished parametrization of the curve. However, we still have the Wilczynski invariant Θ4 given by (58).
Thus, for a generic null curve in 3-dimensional manifold, we deﬁne an alternative conformal pseudo-arc-length
parameter by integrating the 1-form
ds := 4
|Θ4(t)| dt,
instead of (57). Accordingly, we may follow the construction of conformal invariants as above with the
following conclusion: the whole tractor Frenet frame consists just from the ﬁrst ﬁve tractors displayed in
(59)–(60) and the corresponding tractor Frenet equations are just (62)–(66), with U5 = 0. In those equations,
two conformal invariants appear, namely, K1 and K2. It however follows from their expressions, and the
fact that Θ4 = 1 for the current conformal pseudo-arc-length parametrization, that K2 is expressible as a
function of K1 and its derivatives, namely,
K2 = −
2
5
K1 +
8
25
K2
1 +
1
2
.
Therefore we end up with only one signiﬁcant absolute conformal invariant, K1, as expected. An interpretation
of that invariant is still the same as in Propositions 5.3.
In the case that Θ4 vanishes identically, we recover the conformal null helices discussed above, cf. Proposition
5.11.
Acknowledgments
Authors thank to Boris Doubrov, Rod Gover and Igor Zelenko for useful discussions. Experiments, checks
and comparisons of some explicit expressions were done with the computational system Maple. JŠ was
supported by the Czech Science Foundation (GAČR) under grant P201/12/G028, VŽ was supported by the
same foundation under grant GA17-01171S.
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