MASARYK UNIVERSITY FACULTY OF SCIENCE HABILITATION THESIS Filtered manifolds with distinguished transformations and transformation groups Lenka Zalabová Brno 2019 Abstract This habilitation thesis discusses geometries with distinguished (local) transformations and (local) properties of the geometries following from the existence of these (local) transformations. We focus mostly on parabolic geometries, [CaS109], that form an important subclass of Cartan geometries, [Sha97]. Main part of the work is based on collaboration with Jan Gregorovič devoted to various generalizations of symmetric spaces. We combine several methods developed for parabolic geometries. My viewpoint is more geometric and is based on studying (local) Weyl connections, their relation to curvatures and their compatibility with (local) transformations, [Zal09, ZallOa] and [ZallOb] (Chapter 1). The viewpoint of Jan Gregorovič focuses on homogeneous geometries and their description via functorial constructions based on algebraic methods, [Grel2c, Grel2a, Grel3]. The combination of the methods allows to give complete description of (both local and global) geometric properties of generalized symmetric geometries and their classification, [GrZal6b, GrZal5b] and [GrZal7](Chapter 2). Generalized symmetries of almost CR structures, [GrZal8] (Chapter 3), serve as an example. The next part of the work deals with submaximally symmetric parabolic geometries. Complex submaximally symmetric parabolic geometries are completely described in [KrThl4], and it follows from their discussion that real submaximally symmetric parabolic geometries shall be studied case by case. In the collaboration with Boris Kruglikov and Henrik Winther we study submaximally symmetric almost quaternionic structures, which are real parabolic geometries. We give submaximal dimensions and models in [KrWiZal8] (Chapter 4). Finally, there are interesting applications in geometric control theory. In my collaboration with Jaroslav Hrdina, Aleš Návrat and Petr Vašík, we focus on control theory on Lie groups. For various non-holonomic mechanisms, their configuration spaces are filtered manifolds that often form parabolic geometries modeled on nilpotent Lie groups. We study controllability and optimal control of these mechanisms (with respect to a suitable sub-Riemannian metric) using properties of these geometries and their transformations, [HrZal9] (Chapter 5). We use the CAM system Maple and the package DifferentialGeometry by Ian Anderson to realize computations, [AnTol2]. Contents Abstract i Chapter 0: Introduction 1 1. Motivation: Affine geometries 2 1.1. Homotheties on affine geometries 2 1.2. Homotheties on affine geometries revisited 3 1.3. Symmetric and (sub)maximal affine geometries 3 2. Results: Cartan geometries 4 2.1. Automorphisms with natural tangent actions 4 2.2. Automorphisms with natural tangent actions on filtered manifolds 6 2.3. Existence and uniqueness of generalized symmetries 8 2.4. Invariant admissible connections 11 2.5. Example: Symmetries of almost CR structures 12 2.6. Submaximal models for almost quaternionic structures 16 3. Applications: Geometric control theory 17 References 22 Chapter 1: Symmetries of parabolic contact structures 25 Chapter 2: Local Generalized Symmetries and Locally Symmetric Parabolic Geometries 43 Chapter 3: On Symmetric CR Geometries of Hypersurface Type 77 Chapter 4: Submaximally Symmetric Almost Quaternionic Structures 98 Chapter 5: Local geometric control of a certain mechanism with the growth vector (4, 7) 115 Chapter 0: Introduction i 1. Motivation: Affine geometries 1.1. Homotheties on affine geometries. Affine spaces An and their transformations are the best known examples of geometrical objects, [KoNo63]. The affine plane A2 is already studied at secondary schools and affine transformations are described there as translations, scalings and homotheties, similarity transformations, reflections, rotations, shear mappings, and compositions of them in any combination and order. Affine manifolds generalize affine spaces to the curved setting. An affine manifold or affine geometry is a smooth manifold M together with a linear connection V. A (local) affine transformation f is a (local) smooth map / : M —> M such that /*V = V. There always are two basic invariants of affine manifolds preserved by all affine transformations, the torsion T(£,r]) = V^ry — V^^ — [£,77] and the curvature i?(£, tJ){v) = V^V^z^ — V^V^ — V^flp for £, 77, v G TM. The space An is (globally) the only affine manifold equipped with a connection with vanishing curvature and torsion. It is a general principle in geometry that the existence of non-trivial curvature or torsion implies restrictions on (the existence of) transformations. Conversely, the existence of a non-trivial transformation can restrict the curvature and the torsion of the geometry. Let us demonstrate this principle on homotheties. The affine space An carries homotheties fx^ of any ratio k 7^ 0 with the center at any point x. Indeed, since fx,k{x) = x, we can identify An with the vector space Vn with the origin x. Then the homothety is given in each basis of Vn by multiplying the coordinates by the matrix To define a (local) homothety of the ratio k 7^ 0 at some point x of a general (curved) affine manifold (M, V), we identify TXM with Vn. Thus we consider a (local) transformation such that fx,k(x) = x and TxfXtk-£, = k£ for all £ G TXM. Let us point out that (local) homotheties fx^ are (local) affine transformations with natural tangent action at x which means that • the (local) morphism preserves the point x, and • for all bases of TXM, the coordinate descriptions of its differential (viewed linear endomorphism) are equal. Then, at the center x of the (local) homothety fx,k, the torsion T equals to /* kT = kT, because T is an invariant tensor of type (2,1), and the curvature i? equals to fxkR = k2R, because R is an invariant tensor of type (3,1). Thus there cannot exist (local) non-trivial homotheties at points with non-vanishing torsion. The only (local) homotheties that can exist on torsion-free but curved affine manifolds are • the identity for k = 1, and • the (local) point symmetry for k = — 1. Moreover, if there is a (local) point symmetry fx-\ =: fx, then, at the center x, Vi? equals to fxS7R = — Vi?, because Vi? is an invariant tensor of type (4,1). Thus if there is a (local) point symmetry at x, then T and Vi? vanish at x. Altogether, there can exist (local) point symmetries on affine geometries (M, V) only if T = 0 and Vi? = 0, while there are no (local) homotheties for k 7^ ±1. Conversely, there always exist (local) point symmetries at each point of (M, V) if T = 0 and Vi? = 0, [KoNo69, Chapter XI] or [HelOl]. They are given by (local) geodesic symmetries of the geodesies of V, or equivalently, by symmetries in normal geodesic coordinates of V (on a normal neighborhood of x). In particular, they are 2 given uniquely and they change smoothly from point to point, i.e., form a unique smooth system of symmetries. 1.2. Homotheties on affine geometries revisited. Let us consider the first order frame bundle p : V1 M —> M over an n-dimensional manifold M, which is a principal bundle with the structure group GL(n,R), the general linear group. The fiber over each x G M consists of all bases of TXM and elements of GL(n,R) are transition matrices acting as changes of bases. There is the canonical form 9 G Sl1(V1M,Wl) such that 6(u)(£) are coordinates of the projection Tp.£ G Tp{u)M at the basis u. Then we have TM ~ V1MxGL^nfL)Rn, where the pair [u, X] G Tp(u)M represents the tangent vector at p(u) with coordinates X in the basis u. Indeed, = holds for t G GL(n,R), which reflects the fact that the change of the basis gives the change of coordinates for the transition matrix t. Altogether, (J^M —>• M,ff) is a first order G-structure with the structure group GL(n,R), [KoMiS193]. There is the well known 1-1 correspondence between affine connections V on M and principal connections 7 G £l1(TjlM,gl(n,R)), where gl(n,M) is the general linear Lie algebra. The correspondence is given by inducing the connection on the associated bundle TM. Each (local) map / : M —>• M induces a (local) bundle map V1/ : TjlM —>• TjlM preserving 9, and / is a (local) affine transformation of V if and only if V1f preserves 7. It is now illustrative to study (local) affine transformations / with natural tangent action at x from this viewpoint. If f(x) = x, then V1 f(u) = us for u G p^1(x) and some s G GL(n, R). Then the tangent map Tf satisfies at the point x (1) Txf.[u,X] = ipifiulX} = {us,X} = {u,sX]. The change of u to ut for some transition t G GL(n,R) gives Tjlf(ut) = (ut)t_1st. Then the coordinate description of Txf in all frames is equal if t~xstX = sX for all X G Rn and t G GL(n,R). Thus maps with natural tangent actions correspond to elements s of the center Z(GL(n,R)) of GL(n,R). In fact, V1/ is constant along the fiber over x for s G Z(GL(n, R)). Direct computation gives (2) Z(GL(n,R)) = J ^: : ■" :j : k G R>< J = ^ 'idR" : k G RX^' where Rx := R \ {0}. Thus the only possible (local) morphisms / with natural tangent action at x satisfy Txf.fu, X} = [u, kX] for k G Rx and we are back in the classical setting from Section 1.1. In particular, homotheties are the only possible (local) morphisms of affine manifolds with natural tangent actions. Altogether, we have got a simple algebraic restriction on possible transformations with natural tangent action. Contrary to the classical viewpoint, this concept can be generalized to many types of geometric structures that can be described as Cartan and parabolic geometries, [CaS109]. 1.3. Symmetric and (sub)maximal affine geometries. The above concept leads to so-called (locally) symmetric spaces. A manifold M with an affine connection V is locally symmetric if T = 0 and Vi? = 0, [KoNo69, HelOl]. Then there are point symmetries defined locally at each point of M. If the symmetries are globally defined, then the manifold M is homogeneous and the group generated by symmetries acts transitively on M. Then we call the pair (M, V) symmetric space, [KoNo69, HelOl]. Let us remind that there is also algebraic definition of a symmetric space as a space M together with a map S : M x M —>• M such that • S(x, x) = x, • S(x,S(x,y)) = y, 3 • S(x, S(y, z)) = S(S(x, y), S(x, z)), and • i is an isolated fixed point of S(x,—) for all x, y, z G M, [Kow80, Loo67]. In the case m is a smooth manifold and S is a smooth system of affine transformations, we recover exactly the classical setting. In particular, (local) symmetric spaces are examples of affine manifolds carrying many (local) affine transformations. However, there are other interesting affine geometries with many (local) transformations. It is well known, [Sha97], that affine transformations of an affine manifold form a finite-dimensional Lie group. In the case of n-dimensional affine manifold, the biggest possible dimension of this group is n2 + n. This happens in the case of the affine space An, where the group is exactly the group A(n, R) of affine transformations X i ^ AX + b for the regular matrices A of order n and vectors b G Rn. There is a natural question what are the possible smaller dimensions. Similarly to symmetric spaces, this question is closely related to the existence of a curvature and a torsion. It is proved in [Ego67] that if the dimension of the group of affine transformations is not maximal, than it cannot be bigger than n2. In particular, for an affine manifold (M, V) with non-zero curvature, the maximal possible dimension of the transformation group is n2 and the group acts transitively on m. Their realizations can be found in [Ego67, Section 3.A]. It is also proved there that such manifold is projectively fiat. Let us note that the third possible dimension is n2 — 1 and its realization can be found in [Ego67, Section 3.B]. Consider the action of A(n,R) on An. The stabilizer of a point x G An is GL(n,R) C A(n,R), so the group of affine transformations of A% := An \ {x} has dimension n2. However, A„ is locally equivalent to An and its transformation group is locally maximal. Moreover, local transformations do not extend generally to global ones and we are mostly interested in the local viewpoint. To avoid these problems, it is convenient to swap to infinitesimal level and to study corresponding Lie algebras instead of the transformation groups. Thus we study infinitesimal transformations, i.e., vector fields whose flows are affine transformations at all times, [Ego67, Section 2]. This leads to the so-called gap phenomenon studied for many geometric structures with finite-dimensional transformation groups, [KrThl4, Krl6, KrMaThl6, KrMal4, DoThl4]. The authors ask the following question: • What is the maximal possible dimension of Lie algebra of infinitesimal transformations among all manifolds that are not everywhere flat? The dimension is then called submaximal. In addition to [Ego67], let us mention here paper [KrMal4] where authors study submaximal metric affine structures. Let us remark that the submaximal models discovered in [Ego67] are not metric. 2. Results: Cartan geometries 2.1. Automorphisms with natural tangent actions. The concept of Cartan geometries provides a natural generalization of affine geometries and G-structures which allows to study wide class of geometries in a uniform way, [Sha97, CaS109]. Let G be a Lie group, p C G its closed subgroup and peg their Lie algebras. A Cartan geometry of type (G, p) is a principal P-bundle p : q —> m together with a 1-form uj G 511(^,g) such that (1) (rp)*uj = Adp-i o uj for all p G p, where Ad denotes (the restriction of) the Adjoint action of G on g, (2) uj(Cx(u)) = X for all X G p, where are the fundamental vector fields, (3) uj{u) : Tuq —> q is a linear isomorphism for all u G q. We are interested in local properties, so we can assume m connected and locally connected. A (local) automorphism of the Cartan geometry is a (local) principal 4 P-bundle morphism ip : Q —> Q such that p*uj = uj. Each (local) automorphism (p determines a (local) underlying morphism / : M —> M, and we assume that the groups P C G are chosen in such a way that there is 1-1 correspondence between (local) automorphisms p and the (local) underlying morphisms /, [CaS109, Section 1.5.3]. Automorphisms of Cartan geometries always form a finite-dimensional Lie group Aut((?,w), [CaS109, Section 1.5]. In fact, a choice of u G Q provides an inclusion Aut(Q, uj) Q given by ip <—> p(u). As an example, let us consider G to be the affine group A(n, R) = GL(n, R) xi Rn and P = GL(n,R). Then the frame bundle p : VXM —>• M together with the form uj := 0 + 7 for a fixed principal connection form 7 forms an affine Cartan geometry. (Local) automorphisms of affine Cartan geometries are (local) automorphisms of the bundle VXM preserving the soldering form 9 and the connection form 7. Their (local) underlying morphisms are (local) affine transformations of (M, V) for V corresponding to the principal connection 7. Affine Cartan geometries are typical examples of reductive Cartan geometries, i.e., the quotient g/p may be identified with an Ad(_ff)-invariant subspace of g. In general, Cartan geometries are not reductive. Cartan bundles can be viewed as abstract analogs of frame bundles. The points of Q are (higher-order) frames on which P acts by transitions, while a; is a straight generalization of the affine connections, [Sha97, CaS109]. In general, it is not reasonable to speak about homotheties for general Cartan geometries. Nevertheless, affine Cartan geometries suggest a way to generalize the concept of automorphisms with natural tangent action to general Cartan geometries. In fact, automorphisms with natural tangent action give the only reasonable generalization of homotheties on affine spaces. Let us consider (local) automorphisms ip with underlying (local) maps / such that f(x) = x and let u G p^1(x). The action of each such (local) automorphism p is described by the transition s G P between u and p{u) = us at x. It holds TM ~ q Xp q/p via factorizing of [u,X] Tp.iJj^1(u)(X) by the action of P and we get (3) Txf.[u, X + pj = Mu),X + p] = [us, X + p] = fu, MS(X + p)] on TXM, where Ad is the truncated Adjoint action. In fact, if we view u G Q as frame, then X+p plays the role of coordinates in TXM ~ g/p in the basis u and Ads realizes the change of coordinates for the change of the basis from u to us (viewed as an endomorphism of g/p). Let ut, t G P, be a different frame from the fiber over x. If ip is represented by the element s at u G p^1(x), then the same ip is represented by the element t~1st at ut G p^1(x). So we get a direct analog of behavior from the affine case. Motivated by homotheties and affine transformations with natural tangent actions in the affine geometries, we are interested in the following (local) morphisms. Definition 1. We call a (local) automorphism ip of a Cartan geometry an automorphism with natural tangent action at x if for the (local) underlying morphism / : M —> M induced by the automorphism ip it holds • for the point x G M we have f(x) = x, and • for all frames u G p^1(x) C Q, the coordinate descriptions of the endomorphism Txf : TXM —> TXM (viewed as an endomorphism of g/p) are equal. It follows from (3) that (local) automorphisms with natural tangent actions at x are exactly (local) automorphisms represented (at arbitrary frame u G p^1(x)) by elements s G P such that Ads = Adt_ist on g/p for all t G P. Thus they correspond to elements of the center Z(Ad(P)) of the image of P in GL(g/p) for Ad. 5 The curvature is the basic invariant of any Cartan geometry (with respect to all automorphisms). It is given by The curvature n vanishes if and only if the Cartan geometry is locally isomorphic with the flat model (G —>• G/P,ujc)- Flat models are the simplest examples of homogeneous Cartan geometries which are geometries {Q —> M, lS) such that A.ut{Q,uj) is transitive on M (by means of underlying morphisms). Let us emphasize that local automorphisms of a flat model can be always globally extended, the automorphism group of the flat model coincides with G and has maximal possible dimension among all Cartan geometries of type (G,P), [CaS109, Section 1.4.]. In general, the existence of special (local) automorphisms like automorphisms with natural tangent actions can induce restrictions on the curvature of Cartan geometry. Let us demonstrate it on automorphisms with natural tangent actions on affine Cartan geometries. Let us write elements of A(n, R) as (1, n)-block matrices (1%), where d e R" and A G GL(n,R), and GL(n, R) C A(n, R) consists of elements satisfying 6 = 0. Then the Lie algebra a(n,M) consists of (l,n)-block matrices (^,q), where w G Rn and C G gl(n,M), and the subalgebra gl(n,M) consists of elements with w = 0. We have According to the reductivity, the Ad-action of GL(n, R) on a(n, R) decomposes into the action on Rn = a(n, R)/qI(ti, R) and the action on gl(n, R). Then Ad = Ad|j{n is the map given by B (X >->• BX) for all X G Rn. Computation gives as in (2). Thus (local) automorphisms of affine Cartan geometries with natural tangent actions are (local) automorphisms represented by elements Z(GL(n,R)), in accordance with Section 1.2. Let us emphasize that there is 1-1 correspondence between Z(Ad(P)) and Z(GL(n,R)), which can be viewed as the preimage of the center of the image of the Ad-action in P. In the affine case, k : V1M —>• A2Rn* (Rn © gl(n, R)) decomposes (with respect to the action of GL(n, R)) into the torsion valued in A2Rn* Rn and the curvature valued in A2Rn* qI(ti, R). We get k(u) = k(us) = s.k(u) for an automorphism corresponding to s G Z(GL(n,R)). Assume s = k ■ idgn. Then the restriction of k(u) to A2Rn* M,lS) of type (G,P) for a semisimple Lie group G and its parabolic subgroup P, [CaS109, Section 3.1]. We assume M is connected and simply connected and G is simple1. We consider g to be |/j|-graded simple Lie algebra g = g_fe © • • • © g0 ® • • • ® Qk such that p = g0 © • • • © Qk- We fix the Levi decomposition P = Go xi exp(p+), where p+ := gi © • • • ffig/c, and Go is the Lie group of grading-preserving elements of P. We write g_ := g_fc © • • • © g_i. There always is the P-invariant filtration of g given by q1 = Qi S) ■ ■ ■ S) Qk which induces a filtration TlM ~ Q xP Ql/p of TM. All results can be formulated with slightly weaker assumptions, [GrZal5b]. k:G^A2(b/p)*®q, k(u)(X,Y) = [X,Y] u([u-1(X),u-1(Y)])(u). Z(Ad(P)) = {k ■ idRn : k G Rx} = Z(GL(n,R)) 6 We always consider regular parabolic geometries, i.e., the curvature n only has components of positive homogeneity, [CaS109, Section 3.1.7.]. Each such geometry is equivalent to the underlying structure on the manifold which consists of • a filtration T%M of TM such that the associated graded bundle gr(TM) = gr_fe(TM) © ■ ■ ■ © gr_1(TM) of TM, where gr,(TM) = TlM/Tl+1M, satisfies gr(Ta,.M) ~ g_ for all x G M, and • a reduction of the structure group of gr(TM) with respect to Ad : Go —>• Autgr(g_). This correspondence is 1-1 assuming that the parabolic geometry is normal, i.e., d* o k = 0 for the codifferential d* in the standard complex computing Lie algebra homology of p+ with coefficients in g, [CaS109, Section 3.1.10.]. Then the basic invariant of regular normal parabolic geometries is the harmonic curvature kh defined as the projection of n to ker(<9*)/im(<9*). The curvature n can be computed explicitly from kh using a differential operator, [CaSlSoOl]. The existence of the canonical filtration {T*M} on parabolic geometries (p : Q —>• M, bS) implies that it is not reasonable to 'fix ratios of homotheties' on the whole TM, [Zal06, GrZal3] and [ZallOb] (Chapter 1). Let us remind that (local) mor-phisms of Cartan geometries with natural tangent actions correspond to elements s G P such that Ads = Adf-i„f on g/p for all t G P. For parabolic geometries, s = go exp Z and t = hg exp V, go, hg G Go and Z, V G p+, and rxst = h^goho exp(-AdgohoV) exp(AdhoZ) exp V. In particular, we have AdSo = Adh-igghg on g_i ~ g_1/p and p+ acts trivially on g_i. Thus to study the action on T_1M, we can focus on elements of the center Z(Gq). Let us emphasize that viewing g_i as Go-module, each element of Z(Gq) acts on each irreducible component of g_i as multiplication by a single eigenvalue. Thus if X, Y G g_i such that AdgoX = kX and AdgoY = IY for g0 G Z(G0), then Adgo[X,Y] = [AdgoX, AdgoY] = kl ■ [X,Y]. In particular, we still have 'multiplication by eigenvalue' as for homotheties, but possible eigenvalues are different on different components. In general, we can prescribe the tangent action only on the distinguished distribution T_1M ~ Q Xp g~Vp in TM, because underlying morphisms for parabolic geometries respect Lie brackets. The above computation shows that on the level of the associated grading gr(TM), underlying morphisms with natural tangent actions are determined uniquely by their actions on T_1M. However, it is generally difficult to study such morphisms on TM because of the action of exp(p+). The above observations motivate to study (local) automorphisms of parabolic geometries with natural tangent actions on T_1M, [GrZal5b, GrZal7, GrZal6b]. Definition 2. A (local) automorphism ip of a parabolic geometry has a natural tangent action at x on T~1M if • the (local) underlying morphism / of ip satisfies f(x) = x, and • the restricted tangent action Txf : T~XM —>• TaT1M of / is such that for all frames u G p^1(x), the coordinate descriptions of the endomorphism Txf : T~XM —>• TaT1M (viewed as an endomorphism of g_1/p — 0-i) are equal. Thus we get from above that the (local) automorphisms ip with natural tangent actions at x on T_1M enjoy the following two equivalent descriptions: (1) they are represented by elements s such that Ads = Adt-ist on g_1/p for all t G P, (2) they correspond to elements of the center Z(Ad\s-i/p(P)) of the image of Ad:-P->GL(0-Vp). 7 The preimage of Z(Ad|3-i/p(P)) in P has the form Z(Gq) xiexp(p+), where elements of Z(Go) determine the action on g_i ~ 0_1/P while exp(p+) acts trivially on 0_1/p. The existence of exp(p+) implies that elements having the same action on 0_1/P are not unique. Let us remind that there is 1-1 correspondence between elements of the center of the Ad-action and their preimages in P in the case of reductive Cartan geometries, while exp(p+) makes the situation more complicated for parabolic geometries. We can also describe (local) automorphisms ip with natural tangent actions at x on T_1M corresponding to sexp(Z) G Z(Gq) xi exp(p+) locally on a neighborhood of x. Let us consider the normal coordinate system at u G p~1(x) given by flows of constant fields oj~1(X) for X in some neighborhood of 0 in g_. Then we have ip : Fli"1(x)(u) i y Y\f1{X) (us exp(Z)) over the normal neighborhood of x. Then s provides a linear change of coordinates of X in g_ while exp(p+) changes the normal coordinate system itself and its action is non-linear. (Local) automorphisms represented by elements s G Z(Gq) at some u then form a distinguished class of automorphisms which are given by a geodesic transformation in suitable normal coordinates. Let us emphasize that there is just one normal coordinate system in the case of affine Cartan geometries which reflects the non-existence of exp(p+) in the affine case. Definition 3. We call a (local) automorphism ip with natural tangent action at x on T_1M (local) s-symmetry for s G Z(Gq) or (local) generalized symmetry at x, if there is a frame u G p~1(x) such that G/P,uj), then there are infinitely many symmetries at eP represented by s at arbitrary frame g G P, and transitivity gives symmetries at each point of G/P. In particular, for G and s G Z(Gq), the flat model G/P is s-symmetric and there is infinite number of s-symmetries at each point, [Zal06]. We proved 8 in our work [GrZal7, GrZal6b, Zal09, ZallOb, GrZal3] that uniqueness of (local) s-symmetries also often occurs for non-fiat parabolic geometries of various types (G, P) for various elements s G Z(Gq). An important feature of (regular, normal) parabolic geometries is that we can study the actions of generalized symmetries on components of the harmonic curvature kh instead of the whole curvature n. Let us stress that the harmonic curvature kh '■ g —> ker(<9*)/im(<9*) is P-equivariant and exp(p+) acts trivially on ker(<9*)/im(<9*). Thus k^j descends to a Go-equivariant function valued in ker(<9*)/im(<9*) ~ H2(q_,q), which is completely reducible Go-module, [CaS109, Section 3.1.12]. Thus kh decomposes according to the decomposition of H2(q-,q) into irreducible components p and there is an algorithmic way to describe its components using the Kostant's theorem, [CaS109, Section 3.3.]. Since elements of Z(Gq) act on each irreducible component of g by a multiple, the same holds for components of _ff2(g_, g) and we can firstly mimic naively ideas from the affine case. Assume there is a (local) s-symmetry ip for s G Z(Go) at x and a component kh : Q —> p C g* A g*2 Qaa of kh (where p is irreducible as a Go-submodule of H2(g_,g)). The element s acts (in the frame u G p~1(x)) by multiples ki on corresponding components of gai. At the point x then (4) ip*(na)(u) = kn(ip(u)) = ku(us) = s.ku(u) = • ku(u), where . denotes the action on corresponding tensor product. This gives the rough restriction ^3(^1^2) 1 = 1 on the possible multiples ki. Thus the curvature kh can have values only in the components of H2(g_,g) satisfying these restrictions for given s. In this way, we get the following trivial but the crucial result, which is the first step in restricting possible generalized symmetries and components of harmonic curvature, [Zal09],[ZallOb](Chapter 1) and [GrZal7](Chapter 2). Theorem 1. If s G Z(Gq) is such that \i is not contained in an l-eigenspace of the action of s, then there is no s-symmetry at the points with non-zero kh valued in p. However, it is often not sufficient to study only the Go-action because of the possible action of exp(p+). At this point, it is reasonable to swap to the setting of Lie algebras. For G p C H2(g_,g), let us denote by (5) ann(4>) := {A G g0 : A.

)l = {Z G Qi : ad(Xi).. .ad(Xj)(Z) G ann() for all Xlt.. .Xl G g_i}. Then we define the crucial property of the geometry generalizing classical prolongation rigidity, [KrThl4, Yam93]. Definition 4. For arbitrary fixed s G Z(Gq) and irreducible component p C H2(q_,q) (viewed as a component of kjj), we say that the triple (g,p,/i) is prolongation rigid outside of the l-eigenspace of s, if for all (p G p, all prolongations pr(4>)i are contained in the l-eigenspace of s, [GrZal7, Definition 1.5.]. Assume there are two (local) s-symmetries ipi,if2 at x for s G Z(Gq). If u G p^1(x) is such that Aut(Q,uj) given by i 4 ipx and the corresponding system / : M —>• Diff(M) of (local) underlying morphisms fx on M. Contrary to the affine symmetric spaces, such systems are not necessarily smooth in general. One can easily find non-smooth systems on fiat models. We constructed examples of non-homogeneous locally-flat s-symmetric geometries that do not carry any smooth system of s-symmetries in [Zall4, GrZal5a], motivated by [Pod89], and examples of such non-fiat geometries can be found in [GrZal6a]. However, we give the following crucial characterization of smooth systems on non-fiat geometries, [GrZal7, Theorem 1.8] (Chapter 2). Theorem 3. Suppose s G Z(Gq) is such that (g,p,/i) is prolongation rigid outside of the l-eigenspace of s. Suppose that the parabolic geometry {Q —> M,uj) of type (G, P) has got an everywhere non-zero component of the harmonic curvature in /i. (1) The geometry is (locally) s-symmetric if and only if there is a smooth system S of (local) s-symmetries. (2) In such case, the system S is unique. Remark 2. For AHS and parabolic contact geometries in the case of s such that Ads = —id on q—i, the result and some further consequences were already shown in [ZallOa, GrZal3]. The result can be strengthen in the case of homogeneous parabolic geometries, because the existence of an s-symmetry at one point implies the existence of a system of s-symmetries, [GrZal6b]. The general result for homogeneous geometries 10 can be found in [GrZal6b, Proposition 7.1]. As an example, let us present here the following result for homogeneous AHS geometries proved also by more direct methods in [Zal09, ZallOa]. Corollary 1. Let us consider an AHS structure of type (G,P). Assume k =/= 0 and assume there is a non-trivial s-symmetry at a point of a homogeneous parabolic geometry of type (G,P). Then M is either (locally) symmetric or (locally) Z3-symmetric space (i.e., s3 equals the identity). 2.4. Invariant admissible connections. In the case of affine symmetric spaces, there is a distinguished admissible affine connection which is invariant with respect to all symmetries, [KoNo69, HelOl]. There is a class of Weyl connections on each parabolic geometry playing a significant role in the theory of parabolic geometries, [CaS109, Section 5.1.] and [CaS103]. These are admissible affine connections satisfying a certain normalization condition on their curvature and torsion. They form an affine space modeled over the space of one-forms on M and there are explicit formulas for transformations of Weyl connection in [CaS103, Proposition 3.9]. Formulas are generally complicated and one shortly writes V = V + T for T G fi1(M). Analogously to (local) point symmetries in the affine case, there are (local) geodesic transformations l^ of geodesies of Weyl connections V with a fixed point x, which are simply given in the normal coordinate system for the Weyl connection V as (local) diffeomorphisms with coordinates Ads G GL(g_) for s G Z(Go) at x. Clearly, these (local) transformations can be different for different Weyl connection. It is natural to question the invariance of Weyl connections with respect to (local) generalized symmetries. Thus, at a point x we ask whether there is a Weyl connection allowing (local) s-symmetries as its (local) geodesic transformation. If there is a (local) s-symmetry at each point, then one seeks for a Weyl connection (or at least a class of Weyl connections) such that the above holds for the (local) s-symmetries at all points. Assume there is a (local) s-symmetry ip at some point x. The simplest situation appears if there is an invariant Weyl connection V, i.e., /*V = V holds on a neighborhood of x for the underlying morphism / of ip. The following result, motivated by the affine case, was achieved in [GrZal7, Theorem 1.8] (Chapter 2). Theorem 4. Suppose s G Z(Gq) is such that (g,p,/i) is prolongation rigid outside of the l-eigenspace of s. Suppose that a (locally) s-symmetric parabolic geometry (Q —> M, lS) of type (G, P) has everywhere non-zero component of the harmonic curvature valued in /i. If Ad(s) G GL(g_) has no eigenvalue 1, then there is exactly one Weyl connection invariant with respect to the (local) s-symmetries at all points. Remark 3. Special cases for AHS and parabolic contact geometries were studied earlier in [Zal09, ZallOa, ZallOb, GrZal5a, GrZal3], based on different direct methods. In general, there need not exist invariant Weyl connections. However, there can still exist an invariant class of Weyl connections. It turns out that the correct concept is the following, [GrZal7, Definition 1.7] (Chapter 2). Definition 5. Let [V] be the maximal subclass of the class of Weyl connections such that • for all connections V G [V], the corresponding geodesic transformations Z J share the same tangent action TxLj at each x G M, and • all connections from [V] restrict to the same partial connection on all smooth subbundles of TM for all eigenvalues of Txl^ different from 1. Let / be a smooth map assigning to each x G M the (local) diffeomorphism / J for some Weyl connection V (depending on x) in [V]. 11 (1) The class [V] is f-invariant if /(x)*V g [V] and /(x)*V(x) = V(x) for each Weyl connection V g [V]. (2) The Weyl connections V in the /-invariant class [V] are called almost f -invariant Weyl connections. (3) An almost /-invariant Weyl connection V is invariant at x if /(x)*V = V. (4) An almost /-invariant Weyl connection V is invariant if /(x)*V = V for all x g M. Remark 4. Details about invariant and almost invariant Weyl connections can be also found in [GrZal6b, Definition 5] for homogeneous case, and [Zal09, ZallOa, GrZal3] for AHS and parabolic contact geometries. We summarize the results of [GrZal7](Chapter 2) concerning almost invariant Weyl connections in the following statement. Theorem 5. Suppose s g Z(Gq) is such that (g,p,/i) is prolongation rigid outside of the l-eigenspace of s. Suppose that the parabolic geometry (Q —> M,uS) of type (G, P) has everywhere non-zero component of the harmonic curvature in p. The geometry is (locally) s-symmetric if and only if there is a non-trivial f-invariant class of Weyl connections. This statement clearly generalizes the Theorem 4 in which the class consists of exactly one connection which is invariant. The complete description of (locally) s-symmetric geometries with invariant (classes of) Weyl connections and the description of their geometric properties was given in [GrZal7, Section 5]. These properties depend on the specific type of the triple which is prolongation rigid outside of the l-eigenspace of s and on the properties of (almost) invariant Weyl connections. In particular, they cannot be summarized easily. In the case of homogeneous s-symmetric geometries, we can describe geometries with invariant (classes of) Weyl connections and their properties nicely in the language of holonomy reductions for parabolic geometries, [CaGoHal4]. For homogeneous geometries, we can describe explicitly the group generated by all (global) generalized symmetries and we consider holonomy reductions with respect to this group, [GrZal6b]. We give two methods of construction of examples of s-symmetric homogeneous parabolic geometries in [GrZal6b, GrZal5b]. Many examples and partial classifications can be also found in [Grel2a, Grel3, Grel2b]. 2.5. Example: Symmetries of almost CR structures. Non-degenerate partially integrable almost CR structures of hypersurface type are the best known examples of parabolic contact structures, [CaS109, Section 4.2.]. We study s-symmetries for s such that Ads = —id on 0_i, i.e., we consider s-symmetries inducing —id on the contact distribution. These symmetries on almost CR geometries were firstly studied in [KaZaiOO, AltMeNalO], where authors however involve a compatible metric on the distribution. We get the existence of a compatible metric as a new result. An almost CR structure (of hypersurface type) is a smooth manifold M of dimension 2n + 1 together with a distribution % C TM of dimension 2n and an almost complex structure J on %, i.e., J : % —> % is an endomorphism with the property J2 = —id. We assume n > 1. The almost CR structure is non-degenerate if H is completely non-integrable and thus defines a contact structure on M. The endomorphism J extends by complex linearity to an endomorphism of the com-plexification CH of H. It decomposes as CH = H1'0 ® H0'1 into holomorphic and anti-holomorphic bundles which are exactly eigenbundles for eigenvalues i and — i of J. An almost CR structure is partially integrable if [H1'0, H1'0] C H1'0 ® H0'1, 12 or equivalently, if [£, rj] — [ g_i and the harmonic curvature valued in q*^1 Ag^ ®go- The harmonic torsion coincides (up to a multiple) with the Nijenhuis tensor N and the harmonic curvature is known as the Chern-Moser tensor or Weyl tensor W. The existence of (local) symmetries gives the following restriction on curvature, [ZallOb, GrZal8] (Chapters 1 and 3). Theorem 7. Following facts hold for an almost CR structure {M,%,J). (1) // there is a (local) symmetry at x G M, then N{x) = 0. (2) // there is a non-involutive (local) symmetry at x, then W(x) = 0. (3) There is at most one (local) symmetry at each x with W(x) =/= 0. (4) The (local) symmetry at x is involutive if and only if there is an invariant Weyl connection V at x defined locally on a neighborhood of x. (5) It holds \7^W(x) = 0 for each invariant Weyl connection at x and £ G %. In particular, Txsx is involutive and TXM decomposes into ±l-eigenspaces, where Hx is the — 1-eigenspace and there is the one-dimensional 1-eigenspace T+M complementary to %. Remark 5. The results were proved in [ZallOb] using methods analogous to the affine case. We also confirmed the results using prolongation rigidity outside of the l-eigenspace of s of the triple (su(p + l,q+ l),p,W) in [GrZal7, Section 5.4]. Finally, we improved the results and their proofs in [GrZal8]. Altogether, (locally) symmetric almost CR structures are integrable. Moreover, they admit only systems of involutive (local) symmetries if W is not vanishing. Let us now swap to global symmetries and globally symmetric CR geometries. The local version of most of the following result is also available and can be found in [GrZal7, Section 5.4] (Chapter 2). The results are more difficult to formulate because of technicalities but they are analogous to global results. We show in [GrZal7, Section 5.4] that the triple (su(p +1, q+1), p, W) is prolongation rigid outside of the 1-eigenspace of s. The results summarized in previous two sections then imply that if the geometry is symmetric and W =/= 0, then the uniquely given symmetries form a smooth system. Let us remark that we have many smooth as well as non-smooth systems of symmetries on fiat geometries (which particularly satisfy N = 0, W = 0). We describe the smooth system by a smooth map S : M x M —>• M given by S(x, y) = sx(y) for the symmetry sx at x G M. In [GrZal8, Proposition 2] (Chapter 3), we prove an important characterization of symmetric CR structures. The following object plays an important role there. A reflexion space (in the sense of [Loo67]) is a space M together with a map S : M x M —>• M satisfying the following conditions • S(x, x) = x, • S(x, S(x, y)) = y, and • S(x,S(y,z)) = S(S(x,y),S(x, z)) for all x,y,z G M. Let us emphasize that contrary to the definition of symmetric spaces, reflexion spaces do not assume that x is an isolated fixed point of S(x, —). Theorem 8. Suppose that {M,%,J) is a symmetric CR geometry. Then either 14 (1) W = 0 and the CR geometry is locally equivalent to the flat model PSU(p + l,q+l)/P, or (2) W =/= 0 and the group generated by symmetries is a Lie group that acts transitively on M, i.e., the CR geometry is homogeneous and (M,S) is a homogeneous reflexion space, where S is the smooth system of uniquely given symmetries. Remark 6. The result was firstly proved under stronger assumptions in [GrZal7] and partially follows from results given in Section 2.3. In [GrZal8] we give a complete proof of this statement. We study properties of parabolic contact structures with smooth systems of symmetries in detail in [GrZal3]. In particular, we get the following more detailed description of symmetric CR geometries. Theorem 9. Let S be a smooth system of involutive symmetries on (M, %, J). The following statements are equivalent: (1) The pair (M, S) is a reflexion space. (2) There exists an almost S-invariant Weyl connection. (3) All symmetries from the system S preserve the subbundle T+M formed by the subspaces TxM. (4) The smooth system S induces a structure of a symmetric space on the leaf space N of the foliation for the involutive distribution T+ M. In particular, if (M, %, J) is a CR geometry with a smooth system of symmetries S such that (M, S) is a reflexion space, then • M = K/H, where K is the Lie group generated by symmetries from S and H is the stabilizer of a point, • the stabilizer L of the leaf F going through eH is a closed subgroup of K, • N = K/L and F = L/H. We discuss in [GrZal3, Section 3] conditions under which the underlying symmetric space is a pseudo-Hermitean symmetric space. Let us finally focus on pseudo-Riemannian metrics compatible with symmetries on non-fiat CR geometries (M,H,J). The following crucial result studies distinguished compatible Weyl connections and pseudo-Riemannian submetrics and metrics, [GrZal8, Theorem 16] (Chapter 3). Theorem 10. Let K be the Lie group generated by all symmetries of a non-flat symmetric CR geometry (M, %, J). Suppose that Ad(_ff°)|q/fj = Ad(i?)|q/f,, where H° denotes connected component of identity of the stabilizer H C K of a point and q is the l-eigenspace of s in I. There exist • a distinguished Weyl connection V preserving the corresponding Reeb field, • a K-invariant pseudo-Riemannian metric g on %, and • a K-invariant Webster metric g on TM, such that (1) Vff = 0,Vff = 0; (2) g\ii = g and the Reeb field of V is orthogonal to % and has length \, (3) choosing the Reeb field of V as a trivialization of TM/% ® C, the pseudo-Riemannian metric g on% coincides with the real part of the Levi form up to a constant multiple, (4) the symmetry at x is linear in geodesic coordinates of V at x, reverses the directions of %x and preserves the direction of the Reeb field o/V at x. 15 2.6. Submaximal models for almost quaternionic structures. An infinitesimal transformation of a parabolic geometry {Q —> M, uj) is a P-equivariant vector field £ on Q such that C^ui = 0. The crucial fact is that flows of £ at all times then are automorphisms of the parabolic geometry. For all parabolic geometries of type (G,P), the maximal possible dimension of the Lie algebra of infinitesimal transformations is dim(g). This is realized for geometries that are locally isomorphic to the fiat model (G —>• G/P,ujc)- Let us remark that the maximal possible dimension of transformation group is dim(g), too, and this is realized for the flat model. However, it can be smaller for geometries that are only locally flat. One can e.g. remove a point from the flat model to get a manifold with the dimension of the Lie algebra of transformations being dim(g) and the dimension of the Lie group of transformations being dim(p), see Section 1.3. There is a natural question on submaximal dimension of the Lie algebra of infinitesimal transformations and realizations of corresponding geometries, because submaximal geometries are other interesting examples of geometries with many symmetries. Thus we ask the following question: • What is the maximal possible dimension of the Lie algebra of infinitesimal transformations among all (regular normal) parabolic geometries of type (G, P) that are not everywhere fiat? Let us emphasize that regularity and normality are technical assumptions from this viewpoint, because one is always interested in (infinitesimal) transformations of underlying structures and among all parabolic geometries that induce the same underlying structure, there always is a regular normal one, [CaS109, Section 3.1.]. In [KrThl4] authors study submaximal dimensions and corresponding models for (regular normal) complex parabolic geometries. They use the theory of Tanaka prolongations, [Tan70, Tan79], prolongation rigidity, [Yam93], and properties of minimal orbits, [OnVi90], to give a complete classification of submaximal dimensions for complex parabolic geometries, [KrThl4, Appendix C]. They also show that their restriction is sharp which means that they can always realize explicit examples of submaximal geometries. Indeed, they give a method of construction of these examples in [KrThl4, Section 4]. The method of construction of submaximal models is based on deformations of fiat models. In fact, the method is closely related to one of methods of the construction of symmetric parabolic geometries, [GrZal5b]. Indeed, submaximal models are homogeneous and we show in [GrZal5b, Zal09] that they are always symmetric for suitable s G Z(Gq). It turns out that the theory of [KrThl4] cannot be applied in the case of real parabolic geometries of non-split type, although main ideas still work, and they must be treated case by case. There are known results e.g. for CR geometries and c-projective geometries, [Krl6, KrMaThl6]. There also are results in [DoThl4] for conformal parabolic geometries developed by slightly different method. We give results for almost quaternionic structures in [KrWiZal8] (Chapter 4). Classically, an almost quaternionic structure on a manifold M is a rank three subbundle Q C End(TM) such that locally (in a neighborhood of each point) we can find a basis /, J,K of Q with I2 = J2 = K2 = —id and IJ = K. A manifold M with a fixed almost quaternionic structure Q is an almost quaternionic manifold. A (local) automorphism of (M, Q) is a (local) diffeomorphism of M that preserves Q, [AleMar96]. An almost quaternionic manifold (M, Q) can be described as a normal parabolic geometry (Q —>• M, ui) of type (PGL(n+l, H), P), where P is the stabilizer of a quaternionic line in H™+1, [CaS109, Section 4.1.8.]. The fiat model then is the quaternionic projective space HP™ which is the set of quaternionic lines in H™+1. The group PGL(n+l, H) acts transitively on HP™ as automorphisms of the natural quaternionic structure. In particular, maximal possible dimension of a Lie algebra of 16 infinitesimal transformations for quaternionic manifolds of quaternionic dimension n is dim(sl(n + 1), H) = 4(n + l)2 - 1. In the case of general (curved) almost quaternionic structures, the harmonic curvature kh has two components: • the structure torsion k\ of homogeneity 1 valued in A 0-i, and • the quaternionic Weyl curvature k2 of homogeneity 2 valued in q*^1 /\q*^1 00- Let us sketch the idea of finding the submaximal dimension: Each choice of u G Q provides an inclusion Axit{Q, uj) <—$■ Q. On the level of corresponding algebra of infinitesimal transformations inf((?,u;) we get the inclusion vnf(C7, cj) ^->- 0, £ *->• cj(£(w)). The image f(u) together with the new bracket [X, ^]f(«) := [X, Y] — k(u)(X,Y) for X, Y G f(u) forms a filtered Lie algebra. We consider its associated grading s(u) = gr(f(w)) = S-i(u) ®s0(u) ®si(u). It holds dim(inf(G,uj)) = dim(s(u)). In particular, sq(u) C ann(kh(u)) C 0o- Moreover, the annihilator is of maximal dimension if and only if the Go-orbit through the component of kh has minimal dimension in the projectivization of ff2(0-i,0). Since almost quaternionic structures are prolongation rigid, [KrThl4, Corollary 3.4.8], the component Si(u) does not appear. Thus s(u) C 0_i © ann(kH(u)) C 0_i © q0. Finally, there are two possibilities for k = kh = k\ and k = kh = «2 giving the possible maximal bound. In [KrWiZal8] (Chapter 4), we prove the following statement. Theorem 11. The maximal dimension of the Lie algebra of infinitesimal transformations for almost quaternionic structures (M, Q) with dim(M) = 4n > 4 and kh = ki + «2 7^ 0 is An2 — An + 9. This is realized in both cases, when k = kh = «i and k = kh = «2 ■ We exclude the case n = 1 because it is equivalent to a conformal structure and it is studied in [DoThl4]. In this case, the submaximal dimension is 8. We give the proof of the submaximal dimension in [KrWiZal8, Section 4] (Chapter 4). In [KrWiZal8, Section 5], we show that the dimension is realizable in both cases and we give a complete description of the models. It turns out that the model for K2 7^ 0 is locally an affine symmetric space which is not Riemannian. Among all Weyl connections, there is the distinguished Ricci-fiat connection with vanishing torsion and parallel curvature, so this is the connection compatible with the symmetries. The model for the case k\ ^ 0 is locally representable as a Lie group. 3. Applications: Geometric control theory Transformations of Cartan geometries find interesting applications in geometric control theory and robotics. We focus on (local) control theory of various mechanical systems, [HrZal9] (Chapter 5). Instead of reminding the whole theory in detail, we just review the general concepts and we demonstrate them on the example of vertical rolling disc, which is one of the simplest mechanical systems, [Blol5, Bro82]. It is a natural question in robotics to discuss the controllability and find local extremal trajectories of various mechanisms that move in the plane. Examples of such mechanisms are snake robots, [HrNaVal6], or trident snake robots, [Ish04, PiThol4, HrNaMaVal7]. For example, the 3-link snake consists of thee rigid links of constant length interconnected by rotating joints and to each link, there is a wheel attached at the center of the link. The classical trident snake consists of a 17 body in the shape of an equilateral triangle and three rigid links of constant length connected to the vertices of the body by rotating joints. Configuration spaces of such mechanisms locally coincide with Rn for suitable n. For example, the configuration space of the 3-link snake is locally W5 and local coordinates can be chosen as [x, y, 8, ipi, (p2\ according to Figure 1. The configuration space of classical trident snake coincides locally with W6, where coordinates can be chosen as [x, y, 8, ipi, if2, ^3] according to Figure 1. (a) 3—link snake (b) classical trident snake Figure 1. Snake robots We can also generalize these classical mechanisms by changing their configuration spaces. For example, we consider a generalized trident snake which consists of a body in the shape of an equilateral triangle and three prismatic links connected to the vertices of the body, where two of them are attached by fixed joints and one of them by rotating joint according to Figure 2. Its configuration space locally coincides with W7 and coordinates can be chosen as [x, y, 8, ip2, ii, £2, ^3]- I Figure 2. Generalized trident snake Moreover, there are natural non-holonomic constraints coming from natural restrictions on the movement of mechanisms given by the fact that wheels cannot move in directions perpendicular to their velocities. These restrictions give additional (local) structure on the configuration space. Indeed, such condition on each wheel determines one pfaffian form and altogether, we get a pfaffian system which determines (locally, at regular points) a distribution on Rn. If this distribution is 18 bracket generating at some point of the configuration space, then the mechanical system is controllable in a neighborhood of this point, i.e., we can reach each point in this neighborhood, [AgBaBol9]. In such case, the distribution determines (locally, at these points) a filtration of the configuration space and often carries additional geometric structure that can be described as a Cartan or even parabolic geometry. For example, the 3-link snake corresponds to the famous (2, 3, 5)-geometry of Cartan, [Carl894, CarlO], and the classical trident snake corresponds to the famous (3, 6)-geometry of Bryant, [Bryl6]. The generalized trident snake carries the so-called generalized path geometry in dimension 7, [CaS109, Section 4.4.3]. Example 1. Let us consider a vertical disc rolling in the plane according to Figure 3. Its configuration space is (locally but not globally) W3 with coordinates [x,y,6] Figure 3. Vertical rolling disc given by the touch point Po = [x,y] G M2 and the velocity orientation 9 G [0,27r]. The constraint given by the fact that the disc cannot move in the direction perpendicular to its velocity gives the pfaffian — s'm(9)dx + cos(9)dy, and its kernel is (locally) the bracket generating 2-distribution on W3 generated by vector fields i>i := J| and i>2 := cos(#)J^ +sin(#)J^. In fact, these two fields correspond to two natural movements of the disc which are • the rotation around an axis parallel with z at ?o, and • the move in the plane in the direction (cos(0), sin(0)) from P0. Their Lie bracket [i>i,t>2] = := — sin(#)J^ + cos(#)J^ is not contained in the distribution. Thus we have (locally) a bracket-generating distribution giving a filtration (2, 3) and the system is everywhere controllable. Let us remark that together with [i>i, = —1>2 these two brackets are the only non-trivial brackets. Let us emphasize that the above Cartan and parabolic geometries are defined locally and they can generally have non-trivial invariants (i.e., the harmonic curvature in the case of parabolic geometries). The usual way to avoid this problem in robotics is to swap to nilpotent approximation at a suitable point, [AgSar87, Her86]. Geometrically, the nilpotent approximation is a suitable representative of the associated grading and it can be viewed as the maximally symmetric model of the geometry (i.e., flat model for Cartan and parabolic geometries). Example 2. The nilpotent approximation at the origin [0,0,0] of configuration space of the vertical rolling disc is given by a distribution generated by fields ni := W> 712 := ~§x + ®~5y' Their bracket equals to [ni,n2] = := J^. This is the only non-trivial bracket and all remaining brackets are trivial. We get a contact distribution giving a filtration (2, 3) and the vector fields model the three-dimensional Heisenberg algebra. 19 Methods of control theory can be used efficiently to study extremal trajectories and local minimizers for optimal control problems that are related to plane mechanisms, [AgBaBol9, AgSac04]. Example 3. Let us start with the original mechanical system. Configuration space of the vertical rolling disc can be (locally) viewed as a Lie group S with solvable Lie algebra generated by the elements e\, e2, e3 corresponding to the vector fields vii v2, v3. The Lie algebra is given by [ei, e2] = e3, [e2, £3] = e\. So we can use the concepts of control theory on Lie groups. To study optimal movements, we consider the canonical control metric gg on the distribution D$ = (ei,e2), i-e-; eii e2 are orthogonal and both have the length 1. Let us emphasize that D$ and gs are invariant with respect to the action given by multiplication in S and we have an invariant control system (S, D$, gs)- We would like to study the following optimal control problem for ccS and the control u = (iii, u2) G M2 with the boundary condition c(0) = s1; c(to) = 82 for arbitrary fixed s\,s2 G S, where we minimize ^ fg°(ui + u2)dt. We use the Hamiltonian viewpoint and the Pontryagin's maximum principle to find a system whose solutions are extremal trajectories, [AgBaBol9, Section 7]. Thus on T* S we get the Pontryagin's system h'1(t) = -h3(t)h2(t), h2(t) = h3(t)h1(t), h'3(t) = -h1(t)h2(t), x'(t) = h2(t)cos(9(t)), y'(t) = h2(t)sm(9(t)), ff{t)=h1{t), where (x,y,8) are base coordinates and hi = (A, Vi), A G T*S are (fiber) coordinate functions. The system can be solved explicitly but the solution is given using Jacobi functions, [SaMolO], and it is not possible to use it in practical applications. Let us thus swap to the nilpotent approximation. It corresponds to the Lie group N with Heisenberg algebra, i.e., the nilpotent algebra given by elements f\,f2,f3 corresponding to ni,n2,n3 such that f2] = f3. We again consider the canonical control metric g^ on the distribution Dn = f2), i.e., /1, f2 wee orthogonal and both have the length 1. Thus we have an invariant control system (N, Dn, g^) on the nilpotent Lie group N. Thus we study the following optimal control problem c(t) = Ul (?) +«2 (g) for c G N and the control u = (1*1,1*2) with the boundary condition c(0) = p\, c(t0) = p2 for arbitrary fixed pi,p2 G N, where we minimize | f0°(ui + u^dt. In this case, we get on T*N the Pontryagin's system h[(t) =-h3(t)h2(t), h'2(t)=h3(t)h1(t), h'3(t) = 0, x'{t) = h2(t), y'2{t) = h2(t)0(t), 9'{t) = h!(t), where (x, y, 8) are base coordinates and hi = (A, rii), A G T*N wee (fiber) coordinate functions. This system can be solved explicitly. Its solutions are extremal trajectories and these project to local minimizers (x(i),y(i),6(i)). The fiber (vertical) system has (in the generic case h3 7^ 0) the solutions of the form hx = C2 sin(Cii) + C*3 cos(Cii) h2 = C*3 sin(Cii) - C*2 cos(Cii) h3 = C1 20 for constants 62,63 and C*i 7^ 0. Then the base (horizontal) system has solutions satisfying the initial condition x(0) = y(0) = 9(0) = 0 of the form = 7^3 - C2sin(Cit) - C3cos(Cit)) w = 77^2 (2Ci (Ci + - ^263 cos(Cit) + 2C2C3 cos(2C1t) (7) 4Ci - 4C2 sin(Cit) + (C2 - C*32) sin(2Cit) + 26*36*3) = 7^2 - 62cos(6it) + C3sin(Cit)) w for constants C*i, C2, C3 as above. These curves are arc-length parametrized if Cf + Cf = 1. Let us point out that it is sufficient to consider solutions with x(0) = y(0) = 9(0) = 0 because solutions with different starting point can be found using multiplication in N. Finally, let us look at explicit local minimizers in order to see how the solutions of the nilpotent system approximate the solutions of the original system, in a neighborhood of the origin. We can solve the original system numerically in Maple. Then we can compare the numeric solutions with solutions of the nilpotent system with the same initial condition. On the figures, we provide two examples of local minimizers c(t),t e [0, 2ir] with the initial conditions x(0) = y(0) = 9(0) = 0, hi(0) = |, ^2(0) = ^ and for two choices of /13(C)), namely /13(C)) = 2 and /13(C)) = 20. Here the dot line denotes the numeric solution of the original system and the solid line denotes the analytic solution of the nilpotent system. (a) Choice h3(0) = 2. (b) Choice h3(0) = 20. Figure 4. Comparison of solutions By a transformation of a control system we consider a (local) diffeomorphism of the configuration space preserving the control distribution and the control metric. As usual, the infinitesimal transformations are vector fields whose flows are transformations of a control system at each time. In many cases, they simply are (local) infinitesimal transformations of the corresponding parabolic geometry that in addition preserve the control metric. In the case of nilpotent control systems (e.g. nilpotent approximations determined by mechanical systems), infinitesimal transformations form a Lie algebra which is a subalgebra of q due to the flatness, see Section 2.1. In particular, there can be transformations preserving the origin, 21 i.e., the isotropy subalgebra is often non-trivial. For example, in the case of the trident snake, infinitesimal transformations of the corresponding nilpotent control system form a 9-dimensional subalgebra of so (3, 4), where 6 fields reflect the brackets generating (3, 6)-structure and the isotropy subalgebra is isomorphic to so(3), [Mya02]. Infinitesimal transformations of corresponding nilpotent control system of generalized trident snake with filtration (4, 7) form a 10-dimensional subalgebra of sl(5,M), where 7 fields reflect the brackets generating (4, 7)-structure and the isotropy subalgebra is isomorphic to so(3), [HrZal9] (Chapter 5). Let us emphasize that in the (3, 6)-case, the isotropy subalgebra is simply the orthogonal algebra of the metric on the distribution at the origin while in the (4, 7)-case, it is the restriction of the orthogonal algebra to a 3-dimensional subspace of the distribution at the origin given canonically by the generalized path geometry, [CaS109]. Infinitesimal transformations of nilpotent control systems are usually easy to find and they give a tool to study points at which local minimizers c(t) stop to be optimal. Consider a trajectory c(t) starting at the origin o of the system (N, Dm,9n) to some p e N and assume there is a transformation / of (N, Dm , qm) such that /(o) = o and f(p) = p. If the path c(t) is not contained in the fixed point set of /, then c(t) := / o c(t) gives another local minimizer of the same length such that c(o) = o and c(p) = p. In this way, we construct two different paths from o to p of the same length, so the local minimizer cannot be optimal after the point p. This is done by [Mya02, MonMorl7] for the (3, 6)-case. We study infinitesimal transformations of nilpotent system for the mechanism with the growth vector (4, 7) and their possible action on local minimizers in [HrZal9]. Example 4. In the case of the nilpotent control system (N, Dm,9n) f°r the vertical rolling disc form, the Lie algebra of infinitesimal transformations is generated by fields tl = JL,t2 = Xfy + JL,t3 = fy and U = - x& + e-^fy. The transformations t\,t2,t% satisfy [^1,^2] = £3 and generate the Heisenberg algebra. The only isotropy transformation is £4. The fixed point set of £4 is of the form B = {[0, b, 0] : b G M.}. One can verify that each local minimizer (7) intersects with B at the point [0, ^ ^fi ^, 0] at the time t = This is the first point with this property and there is a one-parametric family of such minimizers. In particular, each arc-length parametrized local minimizer (7) intersects with B at the point [0, ^,0] at the time t = ^. Let us finally remark that the analytic curves (solid-line) in Figure 4 are pictured exactly to the points where they stop to be optimal. References [AgBaBol9] A. Agrachev, D. Barilari, U. Boscain, A Comprehesive Introduction to sub-Riemannian Geometry (from Hamiltonian viewpoint). (2019). [AgSac04] A. A. Agrachev, Y. L. Sachkov, Control theory from the Geometric Viewpoint. Encyclopedia of Mathematics Sciences 84, Springer, 2004. [AgSar87] A. A. Agrachev and A. V. Sarychev, Filtrations of a Lie algebra of vector fields and the nilpotent approximation of controllable systems. (Russian) Dokl. Akad. Nauk SSSR 295 (1987), no. 4, 777-781; translation in Soviet Math. Dokl. 36 (1988), no. 1, 104-108. [AleMar96] D. V. Alekseevsky, S. Marchiafava, Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. (4) 171 (1996), 205-273. [AltMeNalO] A. Altomani, C. Medori, M. Nacinovich, On homogeneous and symmetric CR manifolds. Boll. Unione Mat. Ital. (9) 3 (2010), no. 2, 221-265. [AnTol2] I. M. Anderson, C. G. Torre, New symbolic tools for differential geometry, gravitation, and field theory. J. Math. Phys. 53 (2012), no. 1, 013511, 12 pp. [Bes87] A. L. Besse, Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008. 22 [Blol5] A. M. Bloch, An Introduction to Aspects of Geometric Control Theory. In: Krishnaprasad P., Murray R. (eds) Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics, vol 24. Springer, New York, 2015. [Bro82] R. W. Brockett, Control theory and singular Riemannian geometry. New directions in applied mathematics (Cleveland, Ohio, 1980), pp. 11—27, Springer, New York-Berlin, 1982. [Bryl6] R. L. Bryant, Conformal geometry and 3-plane fields on 6-manifolds. Developments of Cartan Geometry and Related Mathematical Problems, RIMS Symposium Proceedings, vol. 1502 (July, 2006), pp. 1-15, Kyoto University [CaSlSoOl] A. Cap, J. Slovák, V. Souček, Bernstein—Gelfand—Gelfand sequences. Ann. of Math. (2) 154 (2001), no. 1, 97-113. [CaS103] A. Cap, J. Slovák, Weyl structures for parabolic geometries. Math. Scand. 93 (2003), no. 1, 53-90. [CaS109] A. Cap, J. Slovák, Parabolic geometries. I. Background and general theory. Mathematical Surveys and Monographs, 154. American Mathematical Society, Providence, RI, 2009. [CaGoHal4] A. Cap, A. R. Gover, M. Hammerl, Holonomy reductions of Cartan geometries and curved orbit decompositions. Duke Math. J. 163 (2014), no. 5, 1035—1070. [Carl894] E. Cartan, Sur la structure des groupes de transformations finis et continus. These, Paris, 1894. [CarlO] E. Cartan, Les systemes de Pfaff a cinq variables et les equations aux deriees partielles du second ordre. Ann. Ec. Normále 27 (1910), 109-192. [DoTlil4] B. Doubrov, D. The, Maximally degenerate Weyl tensors in Riemannian and Lorentzian signatures. Differential Geom. Appl. 34 (2014), 25-44. [Ego67] I. P. Egorov, Motions in generalized differential-geometric spaces. (Russian) 1967 Algebra, Topology, Geometry, 1965 (Russian) pp. 375-428 Akad. Nauk SSSR Inst. Naučn. Tehn. Informácii, Moscou [FuHa04] W. Fulton, J. Harris, Representation theory. A first course. Graduate Texts in Mathematics. 129. Readings in Mathematics. Springer-Verlag, New York, 1991. [Grel2a] J. Gregorovič, General construction of symmetric parabolic geometries. Differential Geom. Appl. 30 (2012), no. 5, 450-476. [Grel2b] J. Gregorovič, Geometric structures invariant to symmetries, FOLIA Mathematica 18, Masaryk University, 2012. [Grel2c] J. Gregorovič, Local reflexion spaces. Arch. Math. (Brno) 48 (2012), no. 5, 323—332. [Grel3] J. Gregorovič, Classification of invariant AHS-structures on semisimple locally symmetric spaces. Cent. Eur. J. Math. 11 (2013), no. 12, 2062-2075. [GrZal3] J. Gregorovič, L. Zalabová, Symmetric parabolic contact geometries and symmetric spaces. Transform. Groups 18 (2013), no. 3, 711-737. [GrZal5a] J. Gregorovič, L. Zalabová, Notes on symmetric conformal geometries. Arch. Math. (Brno) 51 (2015), no. 5, 287-296. [GrZal5b] J. Gregorovič, L. Zalabová, On automorphisms with natural tangent actions on homogeneous parabolic geometries. J. Lie Theory 25 (2015), no. 3, 677—715. [GrZal6a] J. Gregorovič, L. Zalabová, A construction of non-flat non-homogeneous symmetric parabolic geometries. Arch. Math. (Brno) 52 (2016), no. 5, 291—302. [GrZal6b] J. Gregorovič, L. Zalabová, Geometric properties of homogeneous parabolic geometries with generalized symmetries. Differential Geom. Appl. 49 (2016), 388—422. [GrZal7] J. Gregorovič, L. Zalabová, Local generalized symmetries and locally symmetric parabolic geometries. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 032, 33 pp. [GrZal8] J. Gregorovič, L. Zalabová, On symmetric CR geometries of hypersurface type. Journal of Geometric Analysis, DOI:10.1007/sl2220-018-00110-l [HelOl] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001. [HrNáMaVal7] J. Hrdina, A. Návrat, R. Matoušek, P. Vašík, Geometric control of the trident snake robot based on CGA. Adv. Appl. Clifford Algebr. 27 (2017), no. 1, 633-645. [HrNáVal6] J. Hrdina, A. Návrat, P. Vašík, Control of 3-link robotic snake based on conformal geometric algebra. Adv. Appl. Clifford Algebr. 26 (2016), no. 3, 1069-1080. [HrZal9] J. Hrdina, L. Zalabová, Local geometric control of a certain mechanism with the growth vector (4, 7). arXiv: 1802.08480 [Her86] H. Hermes. Nilpotent approximations of control systems and distributions. SIAM J. Control Optim. 24 (1986), no. 4, 731-736. [Ish04] M. Ishikawa, Trident snake robot: locomotion analysis and control. IFAC Proceedings Volumes 37 (2004), no. 13, 895-900. 23 [KaZaiOO] W. Kaup, D. Zaitsev, On symmetric Cauchy-Riemann manifolds. Adv. Math. 149 (2000), no. 2, 145-181. [KoNo63] S. Kobyashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1, John Wiley and Sons, 1963. [KoNo69] S. Kobyashi, K. Nomizu, Foundations of Differential Geometry, Vol. 2, John Wiley and Sons, 1969. [KoMiS193] I. Kolář, P.W. Michor, J. Slovák, Natural Operations in Differential Geometry, Berlin-Heidelberg-New York, Springer-Verlag, 1993. [Kow80] O. Kowalski, Generalized Symmetric spaces. Lecture Notes in Mathematics 805, Springer-Verlag, 1980. [KrMal4] B. Kruglikov, V. Matveev, Submaximal metric projective and metric affine structures. Differential Geom. Appl. 33 (2014), suppl., 70-80. [KrThl4] B. Kruglikov, D. The, The gap phenomenon in parabolic geometries. J. Reine Angew. Math. 723 (2017), 153-215. [Krl6] B. Kruglikov, Submaximally symmetric CR-structures. J. Geom. Anal. 26 (2016), no. 4, 3090-3097. [KrMaThl6] B. Kruglikov, V. Matveev, D. The, Submaximally symmetric c-projective structures. Internat. J. Math. 27 (2016), no. 3, 1650022, 34 pp. [KrWiZal8] B. Kruglikov, H. Winther, L. Zalabová, Submaximally symmetric almost quater- nionic structures. Transform. Groups 23 (2018), no. 3, 723—741. [Loo67] O. Loos, Spiegelungsräume und homogene symmetrische Räume. (German) Math. Z. 99 1967 141-170. [Loo72] O. Loos, An intrinsic characterization of fibre bundles associated with homogeneous spaces defined by Lie group automorphisms. Abh. Math. Sem. Univ. Hamburg 37 (1972), 160—179. [MonMorl7] A. Montanari, D. Morbidelli, On the subRiemannian cut locus in a model of free two-step Carnot group. Calc. Var. Partial Differential Equations 56 (2017), no. 2, Art. 36, 26 pp. [Mya02] O. Myasnichenko, Nilpotent (3, 6) sub-Riemannian problem. J. Dynam. Control Systems 8 (2002), no. 4, 573-597. [OnVi90] A.L. Onischik, E.B. Vinberg, Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. [PiThol4] Z. Pietrowska, K. Thon, Dynamics and Motion Planning of Trident Snake Robot. J Intell. Robot Syst. 75 (2014), no.l, 17-28. [Pod89] F. Podesta, A class of symmetric spaces. Bull. Soc. Math. France 117 (1989), no. 3, 343-360. [SaMolO] Y. Sachkov, I. Moiseev, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calc. Var. 16 (2010), no. 2, 380-399. [Sha97] R.W. Sharpe, Differential geometry. Cartan's generalization of Klein's Erlangen program. With a foreword by S. S. Chern. Graduate Texts in Mathematics, 166. Springer-Verlag, New York, 1997. [Tan70] N. Tanaka, On differential systems, graded Lie algebras and pseudogroups. J. Math. Kyoto Univ. 10 1970 1-82. [Tan79] N. Tanaka, On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J. 8 (1979), no. 1, 23-84. [Yam93] K. Yamaguchi, Differential systems associated with simple graded Lie algebras. Progress in differential geometry, 413-494, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993. [Zal06] L. Zalabová, Remarks on symmetries of parabolic geometries. Arch. Math. (Brno) 42 (2006), suppl., 357-368. [Zal09] Zalabová L., Symmetries of parabolic geometries. Differential Geom. Appl. 27 (2009), no. 5, 605-622. [ZallOa] L. Zalabová, Parabolic symmetric spaces. Ann. Global Anal. Geom. 37 (2010), no. 2, 125-141. [ZallOb] L. Zalabová, Symmetries of parabolic contact structures. J. Geom. Phys. 60 (2010), no. 11, 1698-1709. [Zall4] L. Zalabová, A non-homogeneous, symmetric contact projective structure. Cent. Eur. J. Math. 12 (2014), no. 6, 879-886. 24 Chapter 1: Symmetries of parabolic contact structures article by Lenka Zalabova, published in Journal of Geometry and Physics 60 (2010), no. 11, 1698-1709. arXiv: 1003.5443 SYMMETRIES OF PARABOLIC CONTACT STRUCTURES LENKA ZALABOVA Abstract. We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally fiat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on number of different symmetries which can exist at such a point. Affine locally symmetric spaces are well known and studied objects in differential geometry. The classical definition says that a local symmetry at x on a manifold M with an affine connection V is a locally defined affine transformation such that sx(x) = x and Txsx = —id on TXM. We can understand V as a geometric structure on M such that the symmetry sx preserves this structure. There is a natural generalization of this concept: For a manifold with an arbitrary geometric structure, one can define a local symmetry as an automorphism of this geometric structure, which satisfies the two above conditions. Best known examples of this concept are Riemannian symmetric spaces, which play an important role in theoretical physics, see [1, 2], and projective symmetric spaces, see [3]. This concept also generalizes nicely to geometric structures which can be described as |l|-graded parabolic geometries, see [4, 5]. In this article, we are interested in symmetries of contact manifolds endowed with some additional structures which can be described as parabolic geometries, the so-called parabolic contact structures, see [6]. Discussion of the Levi bracket implies that we cannot define a symmetry in the classical sense, see [4]. Motivated by the definition of a symmetry for Cauchy-Riemann structures from [7], we define a symmetry at x as a morphism of the contact geometry such that sx(x) = x and Txsx = —id on the contact distribution at x. This definition works nicely for all parabolic contact structures. Then, to study symmetries on parabolic contact structures, we can use general techniques known from the theory of parabolic geometries, see [8, 6]. In this article, we discuss the curvature of symmetric parabolic contact geometries in detail. The theory of harmonic curvature for parabolic geometries allows us to prove that symmetric normal parabolic contact geometries must be torsion-free. Moreover, some types of them have to be locally flat, if they are symmetric. Contrary to the classical case, symmetries of parabolic contact structures do not satisfy sx o sx = id in general, i.e. they are not necessarily involutive. We use the theory of Weyl structures to study this question. More precisely, we show that only locally flat geometries can carry non-involutive symmetries at each point. Moreover, for each involutive symmetry on a parabolic contact geometry, there exists an admissible affine connection which is invariant with respect to the symmetry. Finally we show that in many cases, there can exist at most one symmetry at points with non-zero curvature. The author would like to mention very useful discussions with Andreas Cap during the work on this paper and the support of Eduard Cecil Center for Algebra and Geometry for her participation in the conference The 30th Winter School Geometry and Physics, 2010. 1 2 lenka zalabová 1. Parabolic contact structures We recall here basic definitions and summarize the properties of parabolic geometries. We discuss parabolic contact structures in detail. We also introduce Weyl structures, which will be our main tool to deal with parabolic contact geometries. We follow the concepts and notation of [6, 8] and the reader can find all the details and proofs therein. 1.1. Contact structures and parabolic geometries. Consider a manifold M endowed with a distribution H of TM of corank one. Then H C TM forms a filtration and, on the graded bundle gr(TM) = H © TM/H, there is the Levi bracket £ : H x H —> TM/H which is a bilinear bundle map induced by the Lie bracket of vector fields. The well known definition says that H C TM forms a contact structure on M if the Levi bracket is non-degenerate at each point. The subbundle H is then called contact distribution. We will discuss here contact manifolds endowed with some additional structures which can be described as parabolic geometries. Let us recall that, for a semisimple Lie group G and its parabolic subgroup P, a parabolic geometry of type (G, P) is a pair (p : Q —> M, uj) consisting of a principal P-bundle Q —> M and a 1-form ijJ G f21(^,g), called the Cartan connection, which is P-equivariant, reproduces generators of fundamental vector fields and induces a linear isomorphism TUQ = q for each u G q. The Lie algebra q of G is then equipped (up to the choice of Levi factor go m p) with a grading of the form 0_fc © • • • © 0o © • • • © 0fc such that the Lie algebra of P is exactly p = 0o © • • • © 0fe- There is the usual notation such that 0_ := 0_fe © • • • © 0_1; p+ := 0i © • • • © 0fe and P+ C P is the subgroup corresponding to p+. By Go we denote the subgroup in P, with Lie algebra q$, consisting of all elements whose Ad-action preserves the grading of 0. Each element g of P can be uniquely written as go exp Z\ ■ ■ ■ exp Zk for suitable go G Go and Zi (z Qi, thus exp Z\ ■ ■ ■ exp Zk G P+. Let us recall that for each parabolic geometry, there is an element E G 0o with the property [E, X] = iX for each I e g„ the so-called grading element. To study contact structures, we have to focus on a special case of |2|-grading: A contact grading of a simple Lie algebra 0 is a grading 0-2 © 0-i © 0o © 0i © 02 such that 0_2 is one-dimensional and the Lie bracket [—,—] : 0_i x 0_i —>• 0_2 is non-degenerate. Let us remark that for each contact grading, the subspace [0-2,02] coincides with the subspace generated by E. It is well known that the Cartan connection uj provides an identification TM ~ Q Xp 0/p. Suppose we have a parabolic geometry corresponding to a contact grading. Because each contact grading of 0 induces P-invariant filtration of the form 0 = 0~2 D 0_1 D 0° D 01 D 02 = 02, the subspace 0_1/p C 0/p defines a sub-bundle T_1M := Q Xp 0~Vp of corank one in TM. There is the Levi bracket on gr(TM) = T_1M © TM/T'^M and the geometry is called regular if the Levi bracket corresponds to the Lie bracket [—, —] : 0_i x 0_i —>• 0_2 under the above identification. Then, for regular parabolic geometries corresponding to contact gradings, the underlying filtration T~1M C TM defines a contact structure on M, and each such geometry is called parabolic contact structure or parabolic contact geometry. Moreover, define Qo '■= G/P+, which is a principal Go-bundle over M. This is the reduction of the natural frame bundle of gr(TM) with respect to Ad : Go —> GI(q—i) and, in this way, we get an additional geometric structure on T_1M. Let us recall some more facts on parabolic contact structures that will be needed: The P-bundle G —>• G/P together with the (left) Maurer-Cartan form ujg <= 511(G,0) forms a geometry that is called homogeneous model. A morphism between geometries of type (G, P) from (Q —>• M,oS) to (Q' —>• M',lj') is a P-bundle symmetries of parabolic contact structures 3 morphism ip : Q —> Q' such that ip*uj' = ui. We will suppose that the maximal normal subgroup of G which is contained in P is trivial. With this assumption, there is one-to-one correspondence between morphisms of parabolic geometries and their base morphisms. Let us recall that such geometries are called effective. The curvature is described by P-equivariant mapping k : q —>• A2(g/p)* • ker(<9*)/im(<9*). There is the following general statement, see [6]: Theorem. On a regular normal parabolic geometry, the curvature n vanishes if and only if the harmonic curvature vanishes. It can be proved that ker(<9*)/im(<9*) is a Go-submodule of A2g^ g_i, • k^2) valued in g^ A g^ Qo and • k^4) valued in q*_2 A q*_x ® q±. See the Appendix for summary of all contact gradings and corresponding geometries with their components of harmonic curvature. Detailed description and computation of components of harmonic curvature for parabolic contact structures can be found in Section 4.2. of [6]. 1.2. Adjoint tractor bundles and Weyl structures. Here, let us briefly introduce the concept of adjoint tractor bundles, which allows us to write formulas and make computations in a more convenient form. The adjoint tractor bundle is the natural bundle AM := Q XpQ corresponding to the restriction of Ad-action of G on g. For each parabolic contact geometry, the filtration of g induces a filtration AM = A~2M D A~XM D A°M D AXM D A2M such that AiM = Q xP Ql, and there is the associated graded bundle gr(AM) = A-2M © A-iM © A0M ffi A\M ffi A2M, where AtM = AlM/Al+1M equals to £0 xGo Qt. Clearly, TM ~ AM/A°M and T*M ~ AXM. On the graded bundle gr(AM), there is the algebraic bracket {—, —} : AiM x AjM —>• Ai+jM defined by means of the Lie bracket on g. Clearly, its part A-iM x A-\M -)• A-2M on gr(TM) = A-2M ffi A-\M coincides with the Levi bracket thanks to the regularity. Since each fiber of gr(AM) is isomorphic to g, the grading element defines a unique element E(x) G gr0(AxM) such that {E(x), —} is a multiplication by i on gx^AxM). In fact, these elements form a section E of gr0(_4M) which is called the grading section. Let us also remark that we simultaneously get an action • of gr (AM) on arbitrary tensor products of gr(AM) which is given using the tensoriality of the algebraic bracket. In particular, the grading section acts on each homogeneous component of the tensor as multiplication by its homogeneity. Now we should recall basic facts on Weyl structures. For any parabolic contact geometry (Q —> M, ui) with the underlying Go-bundle po : Qo —> M, a Weyl structure 4 lenka zalabová is a global smooth Go-equivariant section a : Go —> Q of the canonical projection tt : G ^ Go- Weyl structures always exist, and for any two Weyl structures a and cr, there are Go-equivariant functions Ti : Go —> 0i and T2 : Go ~* 02 such that a(u0) = a(u0) expTi(M0)expT2(Mo) for all u0 g ^?o- Clearly, Tj g T(^liM) and T := (T1;T2) is a smooth section of gr(T*M). Moreover, the Campbell-Baker-Hausdorff formula implies expT = exp Ti exp T2 = exp(Ti + T2) = exp T2 exp Ti, see [9]. For each Weyl structure cr, we can form the pullback (£_2,£-i). If crexpTiexpT2 is another Weyl structure, the isomorphism changes as £ 1—> (£-2, £-1 — {Ti, £-2})- In particular, cr and cr exp T2 induce the same isomorphism for an arbitrary T2 : Go ~* 02- The part • M,lj) be a parabolic contact structure. A (local) symmetry with the center at x g M is a (locally defined) diffeomorphism sx on M such that: (1) sx(x) = x, (2) Txsx = -id on T~1M, (3) sx is a base morphism of some (locally defined) automorphism ip of the parabolic contact geometry. The geometry is called (locally) symmetric if there is a (local) symmetry at each point x g M. Clearly, each symmetry is a local symmetry. In this article, we discuss local symmetries and local properties of locally symmetric geometries and for conciseness will say 'symmetry at x' and 'symmetric' instead of 'local symmetry at x' and 'locally symmetric', respectively. Global symmetries and their systems we will symmetries of parabolic contact structures 5 discuss elsewhere. Moreover, we will also call the automorphism p of Q and its underlying automorphism po of Qo a 'symmetry at x'. 2.2. Basic properties of symmetries. Let sx be a symmetry on a parabolic contact geometry and let p be as above. Since each symmetry sx preserves x, the (uniquely given) automorphism p has to preserve the fiber over x. Then for each frame u g p^(x) we have p(u) = ugoexpZ = ugo exp Z\ exp Z2 for suitable go g Go, Z\ g 0i and Z2 g 02, where Z = Z\ + Z2. Let us describe the element go exp Z\ exp Z2 in detail: For each £(x) = g T~XM, i.e. for each X g 0_1/p, we have and simultaneously Txsx.£(x) = —£,(x) = fu,—X]. All together, the element go exp Z\ exp Z2 has to induce —id on 0_1/P by the Ad-action. Moreover, exp Z\ exp Z2 acts trivially on 0~1/p. Indeed, there is the formula for all X g 0- and Z g p+ and if X g 0-i, all brackets on the right hand side belong to p. In fact, along the fiber over x, the P+-parts of the above elements are determined by p and can be arbitrary in general, one only has to impose the compatibility of p with the right action of P. Then the element go has to cause the sign change on 0_1/P — 0-i- Since our geometries are effective, there can exist at most one element go g Go which gives —id on 0_i and it has to be the same element along the fiber. In particular, the underlying morphism p0 18 °f the form VoC^o) = uo9o f°r each uo g Pq1(x). Clearly, the element go has to induce identity on 0_2 = [0-i,0-i]. One of basic properties of classical symmetries is their involutivity and there is a natural question on involutivity of our symmetries. Thus let us focus on s2 := sx o sx. Let us first point out that pop = idg if and only if sx osx = idj^f, which follows directly from effectivity. Clearly, po is then involutive, too. Thus it suffice to study the morphism p2. In the above notation, p2(u) = ugo exp Z\ exp Z2go exp Z\ exp Z2 holds, and using the known fact expXgo = 9o exp(AdSoX), we can rewrite this as ugg exp(—Z\) exp Z2 exp Z\ exp Z2. Moreover, acts as id on 0_i and thus on 0, because 0_i generates 0_, p = 0^ and 0O C q*_ q2. In fact, we have proved the following statement: Lemma. On a parabolic contact geometry, each symmetry sx at x defines uniquely a covector Z2 <^T*M through the equation (4) f2(u) = uexp2Z2(u) along the fiber over x. The symmetry sx is involutive if and only if the covector Z2 equals to zero. Thanks to the above observations, it is easy to describe the differential of each symmetry at its center. Proposition. For each symmetry sx at x on a parabolic contact geometry, the mapping Txsx : TXM —> TXM is involutive, thus TXM decomposes into two eigenspaces (2) Txsx.£(x) = lp(u),X} = {ug0expZ1expZ2,X] (3) 3=0 6 lenka zalabová with eigenvalues — 1 and 1. The eigenspace corresponding to the eigenvalue — 1 has to coincide withTx1M, the contact distribution, and there exists a one-dimensional eigenspace corresponding to the eigenvalue 1. Proof. For each £(x) = fu,X] from TXM we have TxsU{x) = y2(u),Xj = [uexp2Z2(u),X} = [u, Adexp(_2Z2(u))Xj which follows directly from formulas (4) and (3). The rest follows immediately from the definition of the symmetry. □ 2.3. Action of symmetries on Weyl structures. Let us now discuss relations of various Weyl structures to the symmetry sx. For each Weyl structure a we can write (5) ip(a(u0)) = a(ip0(u0)) expTi(u0) expT2(w0) for each u0 g Go and for suitable functions Ti : Go —> Qi and T2 : Go ~* 02 which are generally determined by ip and a. With the notation from the last section, we have T2{uq) = Z2{a{uo)) in the fiber over x. The Lemma 2.2 then shows that T2 does not depend on the choice of a Weyl structure a and coincides for all Weyl structures at x. Clearly, T2 vanishes at x if and only if sx is involutive. The function Ti depends on the choice of a Weyl structure a at x and with the above notation, Ti(tto) = Z\ for u{uo) = u. Let us now focus on the role of Ti for the isomorphism TM ~ gr(TM) given by the Weyl structure a: For a tangent vector £(x) = \a(uo),X-2 + X-iJ where Xi g Qi we have Txsx.[cr(uo),X-2 + X-ij = la((po(uo))expT1(uo)expT2(u0),X-2 + X-x] = la(u0),X-2-X-! - [Ti(u0),X_2]], which follows from the fact that <£>o(uo) = uo9o f°r 9o giving —id on q_i and from formulas (5) and (3). In particular, the isomorphism TM ~ gr(TM) given by a Weyl structure a reflects the decomposition of TXM into ±l-eigenspaces for Txsx if and only if the Weyl structure a satisfies Ti(tto) = 0 for each uq from the fiber over x. Lemma. On a parabolic contact geometry with a symmetry sx at x, there are Weyl structures a such that ip(a(uo)) = &((fo(uo)) expTi(tto) expT2(i*o) holds for suitable Ti such that Ti(uq) = 0 for each uo from the fiber over x. Proof. Consider an arbitrary Weyl structure a and let (T1; T2) be determined by a as above. Let us verify that the Weyl structure a{u0) = o{uo) exp(-iTi(w0)) satisfies the condition: The formula (5) and the Campbell-Baker-Hausdorff formula allow us to write o(wo)))expT2(wo) = v( g2 and f± : Qq gi where i^i(uo) = 0 in the fiber over x, see 1.2. Let us finally describe the involutivity of our symmetries in the language of Weyl structures. Proposition. On a parabolic contact geometry with a symmetry sx at x, the following facts are equivalent: (a) the symmetry sx is involutive, (b) there exists a Weyl structure a such that ip(a(uo)) = cr((f>o(uo)) holds in the fiber over x, (c) there exists a Weyl structure au such that (p{ (b) Let a be an arbitrary almost s^-invariant Weyl structure. The Lemma 2.2 says that the involutivity implies vanishing of T2 in the fiber over x. Thus if sx is involutive, the almost s^-invariant Weyl structure a has to satisfy (b). (b) => (c) Let a be an arbitrary Weyl structure satisfying (p(&(uo)) = &((fo(uo)) in the fiber over x. Consider the normal Weyl structure au such that uu{uo) = o{uq) for po{uo) = x. The condition of the normality prescribes au uniquely on a normal neighborhood of x € M, see 1.2 for definition. But then, because ip(cru(uo)) = ^uifoiuo)) holds in the fiber over x, it has to hold over some normal neighborhood of x and au satisfies (c). (c) => (a) Consider an arbitrary Weyl structure a satisfying (c). This can be equiva-lently written as ip^1 (a(ipo(uo))) = ct(m0) which means that the corresponding Weyl connection is invariant with respect to sx. Since the isomorphism TXM ~ gr(TxM) reflects the decomposition of TXM into ±l-eigenspaces, we can describe sx on a neighborhood of x nicely via geodesies of the invariant connection. Indeed, each vector (£_2(x), £_i(x)) g TXM determines uniquely a geodesic at x, and the symmetry sx maps it on a geodesic at x, which is uniquely determined by a vector (£_2(x), — £_i(x)). This describes sx on a neighborhood of x and one can see directly that it has to be involutive. □ Let us call each Weyl structure satisfying the condition (b) of the Proposition sx-invariant Weyl structure at x and each Weyl structure satisfying the condition (c) of the Proposition sx-invariant Weyl structure on a neighborhood of x. 3. Symmetries of homogeneous models In this section, we focus on homogeneous models, which are the simplest examples of parabolic contact geometries. We describe explicitly their symmetries and we give some concrete examples of homogeneous symmetric geometries. 8 lenka zalabová 3.1. Description of symmetries. Let (G —>• G/P,log) be a homogeneous model of a parabolic contact geometry of type (G, P). It is well known that all automorphisms of the homogeneous model are just left multiplications by elements of G and an analog of the Liouville theorem states that any local automorphism can be uniquely extended to a global one, see [10, 6]. Thus if the homogeneous model is locally symmetric, then it is symmetric. Moreover, because G acts transitively on G/P, it suffices to find a symmetry at the origin to decide whether the homogeneous model is symmetric. Proposition. All symmetries of the homogeneous model of a parabolic contact geometry centered at the origin o = eP are given by left multiplications by elements go exp Zi exp^2 G P, where Z\ G 0i and Z^ G 02 are arbitrary and go G Go is such that AdSo = —id on 0_i. In particular, if there is one symmetry at a point, then there is an infinite amount of them. Proof. For homogeneous models, T^1 (G / P) = Gxp0~1/p. Then we can write each tangent vector £(o) G T^1(G/P) as £(o) = \e,X\ for suitable X G fl_1/P- Since automorphisms of the homogeneous model are left multiplications \g by elements g G G, all symmetries at the origin are exactly left multiplications \g satisfying Xg(o) = o and T0\g.£(o) = — £(o) for all contact vectors £(o). The first condition is equivalent to the fact that geP. Then g can be written as g = go exp Z\ exp Z^ and the second condition means that To^g0 exp Z\ exp Zi ■ \e,X\ = \go exp Zx exp Z2,X\ = [e, Ad^ exp exp Z<2 and — £(o) = [e, —X\ coincide for each X G 0~1/p. Thus we look for elements g G P such that Adcxp(_Z2Adcxp(_ ■?iAdg-1X = —X for all X G 0_1/p and the rest follows immediately from observations in Section 2.2. □ Let us finally discuss involutivity of these symmetries. The symmetry go exp Z\ exp Z^ is involutive if and only if the element (g0 exp Z\ exp Z2)2 induces identity on G/P and effectivity says that it has to be equal to e. We have go exp Zi exp Z2go exp Z\ exp Z2 = e exp 2Z2. Thus involutive symmetries at the origin are left multiplications by elements go exp Z\ where g0 and Z\ are as above and Z2 has to be equal to zero. If Z2 is non-zero, then the symmetry is not involutive. In particular, there exist non-involutive symmetries on homogeneous models. Clearly, if g induces (involutive) symmetry at the origin o = eP, then hghrx induces (involutive) symmetry at the point hP. 3.2. Examples. Let us introduce here some examples of parabolic contact structures and discuss symmetries on their homogeneous models, see [6] for detailed description. Lagrangean contact structures. Let us start with 0 = sl(n + 2,M), the split real form of sl(n + 2, C), for n > 1. This admits a contact grading which is given by the following decomposition into blocks of sizes 1, n and 1: (00 0i 02 \ 8-1 00 flf 1 . 0-2 0^i 00 / The splittings 0±i = 0^ © 0^ are 0o-invariant and qi^1 and qr:1 are isotropic for [—,—] : 0_! x 0_! —>• 0_2- Let us choose G = PGL(n + 2,R), the quotient of GL(n + 2,R) by its center. Then P consists of classes of block upper triangular matrices and Go of block diagonal matrices. In particular, Go coincides by means of the Ad-action with the group of all automorphisms of graded Lie algebra 0-2 ®0-i symmetries of parabolic contact structures 9 which in addition preserve the decomposition g_i = fl^iffifl^i- Thus for a parabolic contact geometry of type (G,P), the underlying geometry consists of a contact distribution together with a fixed decomposition of the form T_1M = L ® R into two subbundles (of rank n), each of which is isotropic with respect to £. These geometries are known as Lagrangean contact structures. The homogeneous model is the flag manifold of lines in hyperplanes in Rn+2. Let us now discuss symmetries at the origin of the homogeneous model. We look for an element g0 g Go such that AdgoX = — X for each X g Elementary matrix computation shows that there is a solution which is represented by the matrix of the form / -1 o o g0 = o e o \ o o -1, where E is the identity matrix, and thus the homogeneous model is symmetric. All symmetries at the origin are represented by matrices of the form -1 -v 7 0 e w 0 0-1 where V*, W g Rn and 7 g R are arbitrary, and the involutive ones have to satisfy 7 = -\VW. Non-degenerate partially integrable almost CR-structures oj'hyper'surface type. Consider the real form q = su(p + 1, q + 1) of sl(n + 2, C) for p + q = n > 1. For a suitable choice of Hermitian product, this admits a contact grading of the same block form as the Lagrangean case. Denoting I the diagonal n x n-matrix with the first p entries equal to 1 and the remaining q entries equal to —1, we write the elements explicitly as a Z iz X A -IZ' ix — X'l — a where i,zeR,aeC,I, ZT g C™, A g u(n) and a+trA—a = 0. The bracket g_i x 0_! —>• 0_2 is given by [X, Y] = Y*IX — X*IY, which is twice the imaginary part of the standard Hermitian product of signature (p, q). Choose G = PSU(p+ l,q+l), the quotient of SU(p + 1, q + 1) by its center. Then P consists of classes of block upper triangular matrices, and elements of Go are represented by block diagonal matrices. For parabolic contact geometries of type (G, P), the underlying geometry consists of a contact distribution T_1M together with a complex structure J such that £(J£,Jrf) = £(£,rf) for all £,77 g r(T~xM). Such geometries are known as partially integrable almost CR-structures of hypersurface type. The homogeneous model is the projectivized null cone of a Hermitian form of signature (p + 1, q + 1), which is a real hypersurface in CPn+1. Let us now discuss briefly symmetries of the homogeneous model at the origin. We look for an element go g Go such that AdgoX = —X for all X g Elementary computation shows that the solution exists and is given by the same matrix go as in the Lagrangean case. Then all symmetries at the origin are represented by matrices of the form ' -1 -Z iz 0 e -IZ' 0 0-1 where Z g C™* and z g R are arbitrary, and the involutive ones have to satisfy iz = ZIZ*. 4. Curvature restrictions In this section, we discuss restrictions on the curvature of a parabolic contact geometry, which are caused by the existence of a symmetry. We study the torsion of symmetric parabolic contact geometries in detail. We show that there are relations between the curvature of a symmetric geometry and involutivity of its symmetries. 10 lenka zalabová 4.1. Torsion restrictions. Let us work with normal parabolic contact geometries here. In fact, the normality assumption is only some technical restriction which plays no role, if we understand symmetries as morphisms of the underlying geometry, and this clearly is the most reasonable point of view. For such underlying geometry, there are various non-isomorphic parabolic geometries inducing this underlying structure and it can be proved that the normal one always exists, see [6]. Assuming normal geometry, we can discuss its components of harmonic curvature, which are easily computable and provide information on the whole curvature of the parabolic geometry, see Section 1.1 and Appendix. Let us start with the harmonic torsion which has its values in q*_-^ A fl-i. Lemma. If there is a symmetry sx at x on a normal parabolic contact geometry, then vanishes at x. Proof. Let ip be as usual. For u G p~1(x) we have fl-i. The action of g^1 is of the form •to1 • KW(X,Y) = Adg^(KW(u)(AdgoX,AdgoY)) = -k(1)(u)(-X,-Y) = -k(1)(u)(X,Y) since the element go acts as —id on and the action of exp(—Z) is trivial. Because automorphisms preserve curvature, — k^(u)(X, Y) has to be equal to k^(u)(X, Y) in the fiber over x and then it has to vanish at x. □ Proposition. Each normal symmetric parabolic contact geometry is torsion-free. Moreover, normal symmetric • lie contact structures, • parabolic contact geometries corresponding to exceptional lie algebras have to be locally isomorphic to the homogeneous models of the same type. Proof. Thanks to the regularity, the curvature satisfies n(u)(gl, gJ) C Ql+i+e for all u G q and for some £ > 1 in general. Moreover, it can be proved that the component of degree £ mapping Qi x Qj to Qi+j+e corresponds to the component of kjj(u) of degree £, see [6]. The above lemma shows that £ > 2 for symmetric geometries. Moreover, if the component of degree 2 is non-zero, then the only possibility is that it maps g_i x g_i to go. Thus it has its values in q*_ A q*_ 3 have to have their values in this subbundle, too, and the geometry is torsion-free. Finally, let us recall that vanishing of the harmonic curvature implies vanishing of the whole curvature, see Theorem 1.1. This applies if kh coincides with which has to vanish for symmetric geometries, and they are locally isomorphic to homogeneous models. Now, it suffices to discuss components of harmonic curvature for concrete geometries, see Appendix. □ 4.2. Obstructions to flatness and involutive symmetries. One can see from the discussion of the harmonic curvature that among all normal parabolic contact geometries, only • contact projective structures, • Lagrangean contact structures, • partially integrable almost CR-structures of hypersurface type symmetries of parabolic contact structures 11 can carry a symmetry at a point with non-zero harmonic curvature. For each such symmetric geometry, there is exactly one obstruction to being locally isomorphic to the homogeneous model of the same type. For three-dimensional almost CR-structures and three-dimensional Lagrangean contact structures, there is the harmonic curvature valued in g^ A q*_2 fli- For the other ones, we have the harmonic curvature valued in q*_-^ A Qo- Let us first focus on k(2\ Let sx be a symmetry at x on a normal symmetric parabolic contact geometry and let p be as usual. For u g p—1(x) and X, Y g q-± we have n^{p{u)){X,Y) = ^2\ug0expZ)(X,Y) = exp(-Z)g^ ■ n^{u){X,Y), where • is the induced Ad-action on q*_-^ A q*_-^ Qo- For the action of g^1 we can write g^1 ■ K™(u)(X,Y) = Adg-^(^2\u)(AdgoX,AdgoY)) = Adg-^\u)(-X, -Y)) = Adg-^\u)(X,Y)). Because go is a subspace of L(g_i,g_i) ~ q*_-^ jj-i, the element g^1 has to act trivially on g0 and thus on k^(u)(X, Y) for each X, Y. Because also exp(—Z) acts trivially on k^(u), we get no additional restriction. In fact, is a tensor of type A2T~1*M 4W(Txsx.i],Txsx./z){Txsx.v) = (-i)4v^.cw%m)M = V^W^mXz.) for each £ g X(M) and r/,/z, v g r(T_1M) since Txsx gives -id on TXXM. 12 lenka zalabová Now we focus on the right hand side of (6). We use the general formula for the change of Weyl connections, see formula (1) in Section 1.2. Because our Weyl structure a satisfies Ti = 0 over x, the right hand side of (6) simplifies to vaeXp(-r1)eXp(-r2)w = y^ + ^ . w in the fiber over x. If we put the above observations together, we see that Eq. (6) can be rewritten as (7) v^a^w = v%w + {t2,s-2}*w in the fiber over x. Let us discuss some concrete choices of the vector (a) Suppose £ is contained in the contact distribution T_1M. In particular, = £_i(x). Then Txsx.£(x) = —£,(x) and Eq. (7) simplifies to at x. The algebraic bracket simply vanishes because £,-2(x) = 0 in this case. This implies W^W = 0 at x for each £ from the contact distribution. (b) Let us now suppose that Txsx.£(x) = Such vectors exist and for an almost s^-invariant Weyl structure er, these are exactly the vectors satisfying £ = £_2 at x, see Section 2.3. Then the Eq. (7) simplifies to V£W = V£W + {T2,£}«W and we get the restriction {T2, £} • W = 0 in the fiber over x. □ Part (a) is not surprising. Actually, VaW defines a tensor of type T_1*M A2T~1* M ^T^1* M ®T~1M, i.e. of odd degree, which is invariant with respect to sx. The consequences of the part (b) we formulate in the following statement. Proposition. On a symmetric normal parabolic contact geometry with a symmetry sx at x, suppose that W is non-zero at x. Then each almost sx-invariant Weyl structure has to be sx-invariant. Proof. With the above notation, we will discuss the formula (b) from the lemma for an almost s^-invariant Weyl structure a and for some vector field £ such that is non-zero and satisfies Txsx.£(x) = Let us point out that such vectors exist and satisfy £(x) = £,-2{x) in the isomorphism TM ~ gr(TM) given by er, see 2.3. In some concrete frame u = ct(mq) from the fiber over x, we can write £(x) = for suitable non-zero X G q-2. Similarly, t2(x) = {u,Zj for suitable Z G 02 and the algebraic bracket {T2, £} corresponds to fa, [Z, X]j at x. Moreover, if Z A 0, we can choose X such that [Z,X] is exactly the grading element E, see Section 1.1. Then {T2, £} corresponds to the grading section E(x), see Section 1.2. In particular, it acts by the algebraic action • on W by its homogeneity. Because W has homogeneity two, we get {T2,£}*W = 2W in the fiber over x and we have a restriction of the form 2W = 0 at x. This is a contradiction with the assumption that W is non-zero at x. Thus the only possibility is that Z = 0 and thus t2 has to vanish at x. But this means that the almost Sa;-invariant Weyl structure is s^-invariant, see Section 2.3. □ Before we proceed further, let us return to valued in A q*_2 0i- The discussion of k<4> is parallel to the discussion of and we summarize it very briefly. If is non-zero, then it defines a tensor of even degree which has to be invariant with respect to the symmetry sx. Really, for each X G 0-i and V G 0-2 we have ^4\ip(u))(X, V) = KW(ug0expZ)(X, V) = exp(-Z)g^ ■ n^{u){X,V) = symmetries of parabolic contact structures 13 Adg-,(k^(u)(AdgoX, AdgoV)) = -k^(u)(-X,V) = k^(u)(X, V). Let us denote this tensor by C. Again, we can differentiate C with respect to a Weyl connections corresponding to almost invariant Weyl structures. For an almost s^-invariant Weyl structure a we have the equation yep*aq _ yO-exp(-T1)exp(-T2)^f for each vector field £ and suitable Ti and T2 corresponding to a, where Ti vanishes at x. The left hand side can be rewritten as Vf aC(V,n) = (4Vff)4C(7?,M) = TxS-\W^S!c4C(TxSx.v,TxSx.n) for each £ e X(M), 77 e r(T_1M) and /z e X(M) such that /i = n—2 via the isomorphism given by a at x. Really, Txsx gives —id on T~XM and Txsx.\x(x) = n{x). Thus we get the restriction of the form ^ks^C = V£C + {T2,£-2}«C and we have V^C = 0 and {T2,£_2} • C = 0 in the fiber over x. Because C is of homogeneity four, the same arguments as in the proof of the above proposition shows that T2 vanishes at x and then each s^-invariant Weyl structure has to be s^-invariant. All these observations together with the last proposition give us the following statement. Theorem. On a symmetric normal parabolic contact geometry with a symmetry sx at x, suppose that its harmonic curvature is non-zero at x. Then sx is involutive. Proof. In such case, almost s^-invariant Weyl structures have to be s^-invariant and the rest follows from Section 2.3. □ Corollary. On a symmetric normal parabolic contact geometry with a symmetry sx at x, suppose that its harmonic curvature is non-zero at x. Then there are admissible affine connections which are invariant with respect to the symmetry sx: We take Weyl connections corresponding to s^-invariant Weyl structures. 5. Uniqueness of symmetries We discuss here the question how many different symmetries can exist at a point with non-zero curvature. We first give one general restriction and then some consequences for concrete geometries. 5.1. Algebraic restriction. Let sx and sx be two different symmetries at x on a symmetric normal parabolic contact geometry with non-zero harmonic curvature at x and denote by ip and ip corresponding automorphisms of the parabolic geometry. Clearly, sx =/= sx if and only if ip =/= (p. Symmetries sx and sx are involutive and there exist s^-invariant and s^-invariant Weyl structures, see Sections 4.2 and 2.3. Lemma. For each two different involutive symmetries sx and sx at x on a symmetric parabolic contact geometry, sx-invariant and sx-invariant Weyl structures form two disjoint families of Weyl structures. Proof. Suppose there is a Weyl structure a which is s^-invariant and s^-invariant at x, i.e. p(u(uo)) = u(po(uo)) and simultaneously ip(u(uo)) = u(po(uo)) in the fiber over x. Then, the corresponding Weyl connection VCT is invariant with respect to both symmetries sx and sx. But similarly as in the last part of the proof of Proposition 2.3, the connection VCT determines uniquely the symmetry via behavior of its geodesies at x. Consequently, sx = sx on a neighborhood of x. □ 14 lenka zalabová Let a be an s^-invariant Weyl structure and let a be s^-invariant Weyl structure. Then a = crexpTiexpT2 holds for suitable Ti : Go —>• Qi and T2 : Go 02- The last lemma says that Ti has to be non-zero at x. Proposition. Suppose there are two different involutive symmetries at x on a symmetric normal parabolic contact geometry and let a and a are corresponding invariant Weyl structures. For all £ from the contact distribution, the bracket {Ti, £} acts trivially by the algebraic action on W or Y, respectively, at x. Proof. Let us start with W. Let £ be an arbitrary vector field from the contact distribution, thus £ = £_i for each Weyl structure. The Lemma 4.2 gives W^^W = 0 and W = 0 at x. Simultaneously, we have a = crexpTiexpT2 and the formula (1) from Section 1.2 gives at x, since £_2(x) = 0. Because both covariant derivatives vanish at x, we get the restriction of the form {£-i,Ti} • W = 0 at x for each £ from the contact distribution. One can see from Section 4.2 that the same line of arguments works for C and we get the restriction of the form {£_i, Ti} • C = 0 at x for each £ from Remark. Let us again point out that the existence of a non-involutive symmetry at x causes vanishing of the harmonic curvature at x, see Section 4--2. 5.2. Examples. Let us now discuss the above restrictions for concrete types of geometries. The key point is to find sufficiently nice £ such that the action of the above algebraic bracket is easily understandable. Lagrangean contact structures. Let us first point out that we use here the notation from Section 3.2. The decomposition of the contact distribution into two isotropic subbundles T_1M = L ® R can be interpreted as a product structure on T~XM, which an operator J : T~XM —>• T_1M satisfying J2 = id. The subbundles L and R are simply eigenspaces of J. The Levi bracket C : T~XM x T_1M —>• TMjT~xM is non-degenerate antisymmetric bilinear map, and then, L{—, J—) is a non-degenerate symmetric map which defines a conformal class of pseudometrics on T_1M of signature (n, n). We denote the class by g. Each pseudometric is then given by the choice the identification TM/T~XM ~ R. In particular, the question whether g(£,rf) equals to zero for some £,r/ € T_1M makes sense, because the answer does not depend on the choice of the metric from the class. Proposition. Suppose there are two different involutive symmetries at x on a symmetric normal Lagrangean contact structure, and denote by a and crexpTiexpT2 corresponding invariant Weyl structures. Identify Ti with its image in T~1M = L ® R via an isomorphism given by a metric from g and denote by and corresponding components in L and R. If g(T^ ,T^) ^ 0 at x, then the harmonic curvature vanishes at x. Proof. We discuss the restriction from the Proposition 5.1 for Lagrangean contact structures in detail. Let us write Ti(x) = for suitable which has to be non-zero, see Lemma 5.1. Choose £_i e T(T 1M) such that £_i(x) = for X of the form Vf^ = Vl_W + U-i, TJ . W the contact distribution. □ symmetries of parabolic contact structures 15 The bracket {£_i,Ti} then corresponds to [u, [X,Z]J at x, where [X,Z] ( ST 0 0 It is easy to verify that with this choice, [X, Z] is simply a grading element multiplied by a non-zero number —ST. Then the bracket Ti} is a non-zero multiple of the grading section E{x). Via the identification given by the metric from g, the components S and T correspond to components and Tf of Ti in subbundles R and L at x and the fact that ST ^ 0 means that g(Tf", Tf) / 0 at i. Because the grading section acts on W by its homogeneity, Ti} acts trivially on W if and only W vanishes at x. Clearly, the same arguments work for C. □ Partially integrable almost CR-structures. Let us first point out that we use here the notation from Section 3.2. Moreover, suppose that the geometry is oriented and then, we can speak about the signature of the structure. Using the complex structure J given on T_1M, we can define a non-degenerate symmetric mapping £(—, J—), which defines a conformal class of pseudometrics on T_1M. The signature is given by the signature of the structure. Let us denote the class by g. Each pseudometric from the class is given by the choice of the identification TM/T-XM ~ R. In particular, the question whether g(£,£) ^ 0 for £ e T_1M makes sense, because the answer does not depend on the choice of the pseudometric from the class. Proposition. Suppose there are two different involutive symmetries at x on a symmetric normal partially integrable almost CR-structure and denote by a and a exp Ti exp T2 corresponding invariant Weyl structures. Identify Ti with its image in T~XM via an isomorphism given by a metric from g. If g(T1; Ti) 7^ 0 at x, i.e. if the length of Ti is non-zero at x, then the harmonic curvature vanishes at x. Proof. We discuss the restriction from the Proposition 5.1 for CR-structures in detail. Let us write Ti(x) = for suitable which has to be non-zero, see Lemma 5.1. Choose £_i e T 1M such that £_i(x) = [u, X] for X of the form It is easy to verify that with this choice, [X, Z] is simply a grading element multiplied by a number —SIS*. Then the bracket {£-1, Ti} is a multiple of the grading section E(x). Using the identification T_1M ~ T_1*M given by a metric from g, —SIS* corresponds to g(Ti,Ti) and SIS* 7^ 0 means that g(Ti,Ti) 7^ 0. Because the grading section acts on W by its homogeneity, {£-1, Ti} acts trivially on W if and only W vanishes at x. Clearly, the same arguments work for Y. □ The bracket {£_i,Ti} then corresponds to [u, [X,Z]J at x, where SIS' 0 0 0 0 0 Corollary. Suppose there are two different involutive symmetries at x on a symmetric normal strictly pseudoconvex partially integrable almost CR-structure. Then the harmonic curvature vanishes at x. 16 lenka zalabová Appendix: Contact gradings and corresponding geometries Let us sketch here briefly a classification of contact gradings of real semisiniple Lie algebras. There is the well know classification of all (complex) semisimple Lie algebras in the language Dynkin diagrams and description of all their real forms in the language of Satake diagrams, see [6, 11]. It can be proved that if a Lie algebra admits a contact grading, then it has to be simple. It turns out that except sl(2, R), sl(n,H), so(n — 1, 1), Sp(p, q) and some real forms of E& and F4, any non-compact non-complex real simple Lie algebra admits a unique real contact grading, see [6]. Let us start with real classical Lie algebras, i.e. real forms of Lie algebras of type Ae, Bn, Cg and Dg. In the first column of the following table, we indicate a real simple Lie algebra which admits a contact grading. In the second column we specify the geometry, which corresponds to the unique contact grading and in the last column we write its components of harmonic curvature. real simple g contact geometry components of kh sl(3,M) Lagrangean contact structures of dimension 3 0-2 x g_i ->• fli 0-2 x g_i ->• fli si(n + 2,R) for n > 2 Lagrangean contact structures of dimension 2n + 1 0-1 X 0_i ->• 0_i 0-1 X 0_i ->• 0_i 0-i x 0_i ->• g0 su(2,1) and «ru(l,2) partially integrable almost CR structures of dimension 3 0-2 X 0_i ->• 0i su(p + 1, q + 1) for p + q > 2 partially integrable almost CR structures of dimension 2p + 2q + 1 0-1 X 0_i ->• 0_i 0-i x 0_i ->• g0 S0(p + 2, q + 2) p + q 7^ 4 Lie contact structures of dimension 2p + 2q + 1 0-1 X 0_i ->• 0_i 0-1 X 0_i ->• 0_i S0(p + 2, q + 2) for p + q = 4 Lie contact structures of dimension 9 0-1 X 0_i ->• 0_i 0-1 X 0_i ->• 0_i 0-1 X 0_i ->• 0_i Sp(n+2) for n > 1 contact projective structures 0-1 X 0_i ->• 0o Let us also give a brief overview of contact gradings corresponding to exceptional Lie algebras. For types G2 and F4, there is exactly one real algebra admitting contact grading, the split real form. For Eq, there are three real forms which admit a contact grading, the split form and two su-algebras. For £7, there are three different real forms and for E%, there are two different real forms admitting a contact grading. The description of corresponding geometries can be found in [6]. All these geometries have harmonic curvatures only of type valued in 0_i A 0^1 (8)0_i. References [1] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academe Press, New Yourk, San Francisco, London, 1978, p. 628. [2] S. Kobayaslii, K. Nomizu, Foundations of Differential Geometry, Vol II, John Wiley & Sons, New York, 1969 [3] F. Podesta, A Class of Symmetric Spaces, Bull. Soc. Math. France 117, no. 3 (1989), 343-360 [4] L. Zalabova, Remarks on Symmetries of Parabolic Geomeries, Arch. Math. (Brno) 42 (2006), suppl., 357-368 [5] L. Zalabova, Symmetries of Parabolic Geometries, Differential Geometry and its Applications 27, No. 5, 2009, 605 [6] A. Cap, J. Slovak, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, vol. 154, AMS, 2009, 628pp. symmetries of parabolic contact structures 17 [7] W. Kaup, D. Zaitsev, On symmetric Cauchy-Riemann manifolds, Adv. Math. 149 (2000), 145-181 [8] A. Čap, J. Slovák, Weyl Structures for Parabolic Geometries, Math. Scand. 93 (2003), 53-90 [9] I. Kolář, P.W. Michor, J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, 1993, 434pp. [10] R. W. Sharpe, Differential geometry: Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics 166, Springer-Verlag, 1997 -622 [11] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Mathematics 22 (1993), 413-494. Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Branišovská 31, České Budějovice, 370 05, Czech Republic Email address: lzalabova@gmail.com Chapter 2: Local Generalized Symmetries and Locally Symmetric Parabolic Geometries article by Jan Gregorovic and Lenka Zalabova, published in SIGMA Symmetry, Integrability and Geometry: Methods and Applications 13 (2017), Paper No. 032, 33 pp. arXiv: 1607.01965 LOCAL GENERALIZED SYMMETRIES AND LOCALLY SYMMETRIC PARABOLIC GEOMETRIES JAN GREGOROVIČ AND LENKA ZALABOVÁ Abstract. We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non—zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the parabolic geometry is a generalization of an affine (locally) symmetric space. 1. Introduction Symmetric spaces are extremely useful geometric objects on smooth manifolds. There are also many generalizations of symmetric spaces appearing in several areas of differential geometry and the theory of Lie groups and algebras. We are interested in generalizations of symmetric spaces in the setting of parabolic geometries, see [3, Section 3.1]. We consider regular normal parabolic geometries (Q —>• M,ui) of type (G, P) on smooth connected manifolds M. We assume that G is a Lie group with a |/j|-graded simple Lie algebra g = ©f=_fcfli and P is the parabolic subgroup of G with the Lie algebra p = ©£_0jji such that the Klein geometry (G, P) is effective. We fix the reductive Levi decomposition P = Go xexp(p+), where p+ := ©f=1fli and Go is the Lie group of grading preserving elements of P. We write g_ := • A2(g/p)* • M, uj) is a (local) P-bundle morphism if on Q such that p*uj = uj holds. We denote by p the underlying (local) diffeomorphism of p on M. We say that a (local) diffeomorphism f on M preserves the parabolic geometry (Q —>• M, uj) if / = p for some (local) automorphism p of (Q —>• M, uj) . Local automorphisms of parabolic geometries are uniquely determined by the underlying diffeomorphisms under our assumption of effectivity of the Klein geometry (G, P). We are interested in a class of (local) diffeomorphisms f on M for which we know a priori the (local) P-bundle morphisms p on Q covering (local) diffeomorphisms / 2010 Mathematics Subject Classification. 53C10; 53C22; 53C15; 53C05; 53B15; 53A55. Key words and phrases, parabolic geometries; generalized symmetries; generalizations of symmetric spaces; automorphisms with fixed points; prolongation rigidity; geometric properties of symmetric parabolic geometries. 1 2 jan gregorovič and lenka zalabová and we ask when they preserve the parabolic geometry, see the Definition 1.1. Let us explain that these diffeomorphisms are closely related to geodesic symmetries. We recall that a normal coordinate system of a linear connection V on M given H( X) by the frame u of TXM is given by projections of flows Flt of the standard horizontal vector fields B(X) for X g Rn on the first order frame bundle starting H( X) at u, see [16, Section III.8.]. Indeed, the projection of Flt (u) onto M is the geodesic of V going through x in the direction with coordinates X in the frame u. A geodesic symmetry of V at the point x is the unique diffeomorphism with coordinates —idgn in all normal coordinate system given by any frame u of TXM. The pair (M, V) is an affine locally symmetric space if each geodesic symmetry of V is an affine transformation. In [14] or [1] the authors studied the theory of symmetric spaces, where the geodesic symmetries preserve a geometric structure such as Riemannian metric or quaternionic Kahler structure. The first author classified in [8] all parabolic geometries preserved by all geodesic symmetries on semisimple symmetric spaces. Typical examples of such parabolic geometries are provided by the projective class of V of the affine (locally) symmetric space (M, V) or the conformal class of the metric on the Riemannian symmetric space or the (para)-quaternionic geometry given by the (para)-quaternionic Kahler symmetric space. A normal coordinate system on the parabolic geometry (p : Q —> M, lS) given by u g Q is given by projections p o Fl" ^ (u) of flows of the constant vector fields uj^iX) for coordinates X g 0-. If we consider (local) diffeomorphisms / on M that are linear in some normal coordinate system of (Q —> M, lS) , then we know a priori the (local) P-bundle morphisms ip on Q covering (local) diffeomorphisms / and we ask when they preserve the parabolic geometry. The action of Go on Q induces a linear change of the normal coordinates, but the change of coordinates induced by the action of exp(p+) is highly non-linear. Nevertheless, we can consider the class of (local) automorphisms of parabolic geometries with the property that their underlying (local) diffeomorphisms on M, analogously to geodesic symmetries, have the same coordinates in all normal coordinate systems in which the coordinates are linear. Definition 1.1. For s in the center Z(Gq) of Go and u g G, let su be the (local) P-bundle morphism of Q induced by the formula S„(FLf := Flf 1(X)M = Fl^1(Ad(s)(X))HS for all X in a maximal possible neighbourhood of 0 in g_ preserved by Ad(s). (1) The (local) P-bundle morphism su is a (local) s-symmetry of the parabolic geometry (Q —> M, lS) at x = p(u) if s*u; = uj. (2) We write su for the underlying (local) diffeomorphism on M of the P-bundle morphism su which has coordinates Ad(s) g GZ(g_) in the normal coordinate system given by u. (3) All (local) s-symmetries at all x g M for all s g Z(Gq) together are called (local) generalized symmetries of parabolic geometries. (4) The parabolic geometry is (locally) s-symmetric if there is a (local) s— symmetry at each point of M. Remark 1.2. We always assume that s is not the identity element e in Z(Gq), because idg is the unique e-symmetry of each parabolic geometry, and therefore, the results presented in this article are trivial for e-symmetries. Firstly, let us focus on the automorphisms ip of parabolic geometries such that ip has coordinates — id0_ in the normal coordinate system given by u g Q. If such symmetric parabolic geometries 3 an automorphism exists, then there is m g Z(Gq) such that Ad(m) = — idg_. Therefore we will always speak about (local) m-symmetries in this case. The bundle morphisms mu (and thus diffeomorphisms mu) wee generally different for different u from the fiber Qx over x and each of them can be a (local) m-symmetry. In particular, there can be infinitely many (local) m-symmetries at x. In fact, this is the case of all models G/P of AHS-structures, where the bundle maps mu are m-symmetries for all u g Q. On the other hand, the second author proved in [23, Theorem 2.5] that projective, conformal and (para)-quaternionic geometries are the only types of parabolic geometries allowing m-symmetries at a point x with a non-zero Weyl (harmonic) curvature. Moreover, there is at most one m-symmetry at the point x with a non-zero Weyl curvature. The second author showed in [23, Theorem 3.2] that if a geodesic symmetry at x for some linear connection on M is an automorphism of {Q —> M,ui), then the geodesic symmetry has coordinates — id0_ in the normal coordinate system given by some u g Qx. We prove in this article that there is the following characterization of non-flat parabolic geometries which are preserved by all geodesic symmetries on affine (locally) symmetric spaces. Theorem 1.3. Suppose there is a parabolic geometry on a smooth connected manifold M with a non-zero harmonic curvature at one point. Then the following claims are equivalent: (1) The parabolic geometry is (locally) m-symmetric, i.e., at each point x of M, there is a (local) automorphism of the parabolic geometry such that the underlying (local) diffeomorphism on M has coordinates — id0_ in the normal coordinate system for some u g Qx. (2) The parabolic geometry is preserved by each geodesic symmetry on an affine (locally) symmetric space (M, V). In particular, if one of the above claims is satisfied, then the parabolic geometry is (locally) homogeneous, the affine (locally) symmetric space (M, V) from Claim (2) is unique and V is a distinguished (Weyl) connection of the parabolic geometry. Remark 1.4. Let us emphasize that (local) m-symmetries can appear only on |l|-graded parabolic geometries and only the projective, conformal and (para)-quaternionic geometries (and their complexifications) can satisfy the assumptions and conditions of Theorem 1.3. The global version of this statement was proved in [21] for projective geometries and in [24, Corollary 4.5] for conformal and (para)-quaternionic geometries under the additional assumption of homogeneity or under the assumption that m-symmetries depend smoothly on the point x. In [13, Theorem 1], we proved the global version of Theorem 1.3 for conformal geometries. In this article, we obtain Theorem 1.3 special case of Theorem 1.8. There are many other interesting types of parabolic geometries, e.g., parabolic contact geometries, where there is no element m g P such that Ad(m) = — id0_. Thus they cannot be preserved by geodesic symmetries of any affine (locally) symmetric space. On the other hand, there are generalizations of symmetric spaces appearing in the literature that are nearly related to contact geometries. In [2] and [15] the authors study sub-Riemannian and CR geometries preserved by so-called geodesic reflexions on reflexion spaces, see [19]. A geodesic reflexion on a reflexion space is given by an endomorphism s g Gl(Rn) such that s2 = idgn in a normal coordinate system of an admissible linear connection on the reflexion space, see [19]. We studied in [9, 10] parabolic geometries on reflexion spaces preserved by geodesic reflexions. We proved that a geodesic reflexion at x preserving a parabolic 4 jan gregorovič and lenka zalabová geometry {Q —> M,uj) is given by an endomorphism Ad(s) G GZ(g_) for some s G Go such that s2 = id in a normal coordinate system of the parabolic geometry given by some u G Qx. However, if s G Go is not contained in Z(Gq), then we cannot expect the uniqueness of the automorphisms p such that p has coordinates Ad(s) G GI(q-) in the normal coordinate system given by some u G qx. Indeed, if there is an other automorphism tp such that tp has coordinates Ad(go) <= GI(q-) for some go G Go in the normal coordinate system given by u G Qx, then ippip^1 is in general a different automorphism such that ippip^1 has coordinates Ad(s) G GZ(g_) in the normal coordinate system given by ug0 G Qx. On the other hand, the second author proved in [25, Section 5] that on some parabolic contact geometries, there is at most one s-symmetry at a point x with a non-zero harmonic curvature for s G Z(Go) such that Ad(s)|01 = —id. We prove in this article that this holds for all parabolic contact geometries. We classified in [12] all elements s G Z(Gq) that can appear as coordinates of underlying diffeomorphisms of automorphisms of parabolic geometries in a normal coordinate system at a point with a non-zero harmonic curvature. For example, we have found out that for complex |l|-graded parabolic geometries with a harmonic curvature of homogeneity 3, we have to consider elements s G Z(Gq) such that s3 = id. Moreover, we constructed in [11, Proposition 6.1] and [12, Proposition 7.2] examples of such parabolic geometries on Z3-symmetric spaces, which are generalizations of symmetric spaces that are studied in [18]. In fact, there are many known examples of (locally) s-symmetric parabolic geometries. Each locally fiat parabolic geometry is locally s-symmetric for each s G Z(Gq). We classified in [12] the elements s G Z(Gq) for which all locally s-symmetric parabolic geometries are fiat. Further, we showed in [11, Proposition 6.1] that all submaximally symmetric parabolic geometries constructed in [17, Section 4.1] are locally s-symmetric parabolic geometries for elements s G Z(Go) that do not impose flatness. Let us emphasize that some of these examples carry more than one s-symmetry at each point and explicit examples can be found in [11, Section 6]. This shows that the results we obtain in this article do not hold for all types of parabolic geometries. There are also further examples of (locally) s-symmetric parabolic geometries in [2, 7, 8, 13, 21]. Let us now summarize our main results for (local) s-symmetries and (locally) s-symmetric parabolic geometries we obtain in this article. The first main result states that there is a large class of types of parabolic geometries whose algebraic structure enforces uniqueness of (local) s-symmetries at points with a non-zero harmonic curvature. We characterize these types in a way that is related to the theory of prolongations of annihilators of the harmonic curvature and the prolongation rigidity from [17, Section 3.4] as follows. Definition 1.5. Let /i be a component of the harmonic curvature (irreducible as a Go-submodule of ff2(g_, g)) of regular normal parabolic geometries of type (G, P). (1) For

) := {A G g0 : A.cp = 0} the annihilator of (p in go. We define the ith prolongation of the annihilator of

• M,lj), i.e., S_ defines a smooth system S of (local) s-symmetries such that S_(x) are the underlying (local) diffeomorphisms of S(x) for all x. Thus the notation S_ is consistent with the Definition 1.1. If there is a smooth system S of (local) s-symmetries of (Q —>• M, oS), then we need the prolongation rigidity outside of the 1-eigenspace of s to show the existence of an 5-invariant class of Weyl connections, see Section 4.2. For all |l|-graded parabolic geometries and s such that Ad(s) = —id0_, i.e., s = m, we obtain affine (locally) symmetric spaces, because the class [V] consists of a single connection. For all parabolic contact geometries and s such that Ad(s) |8_1 = —id we obtain reflexion spaces, but the 5-invariant class [V] is not the class of admissible connections from [19]. For triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s, we get the following existence result, which in particular implies Theorem 1.3. Theorem 1.8. Suppose s g Z(Gq) is such that (g, p, p) is prolongation rigid outside of the l-eigenspace of s. Suppose that the parabolic geometry {Q —> M,uS) of type (G, P) has everywhere non-zero component of the harmonic curvature in p. Then the following conditions are equivalent: (1) The parabolic geometry is (locally) s-symmetric. (2) There is a smooth system S of (local) s-symmetries. (3) There is an S_-invariant class [V] of Weyl connections. Moreover, the smooth system S is unique and S_ consists of the underlying diffeomorphisms of S on M. The equality S(x) o S(y) = S(S_(x)(y)) o S(x) holds whenever the compositions are defined. If Ad(s) g GZ(g_) has no eigenvalue I, then [V] consists of a single S_-invariant Weyl connection, which is locally affinely homogeneous. We prove the claims of Theorem 1.8 except the last one in Section 4. The last claim does not hold without additional assumptions on the 1-eigenspace. We prove the last claim in Section 5, where we study additional properties that follow from assumptions on the position and shape of the 1-eigenspace of s in g_. Outline of the article. We recall basic facts and formulas for Weyl connections symmetric parabolic geometries 7 in Section 2. In particular, we characterize automorphisms of parabolic geometries with their actions on Weyl structures. We recall the relation between normal coordinates and normal Weyl structures. In Section 3, we prove Theorem 1.6 and we provide the characterization of the triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s. In Section 4, we prove Theorem 1.8. We also obtain further properties of (locally) s-symmetric parabolic geometries of type (G, P) that have everywhere nonzero component of the harmonic curvature in /i for the triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s. In Section 5, we classify all triples (g, p, /i) that are prolongation rigid outside of the 1-eigenspace of s such that /j, is in the 1-eigenspace of s. The classification is separated in the tables according to the common properties of the triples (g,p,/i) and elements s e Z(Go) . The notation for the tables and details on the classification can be found in Section 5.1. We show in Section 5.2 that there are triples (g,p,/i) for which the 5-invariant class [V] of Weyl connections consists of a single S_-invariant Weyl connection. In particular, such an 5-invariant Weyl connection is always (locally) affinely homogeneous. In Sections 5.3 and 5.5, we show that there are triples (g,p,/i) for which the (locally) s-symmetric parabolic geometries are locally correspondence spaces over some other s-symmetric parabolic geometries. In Section 5.4, we prove that there are triples (g,p,/i) for which the condition of (local) homogeneity is satisfied for more complicated 5-invariant class [V] of Weyl connections. In the Appendix A we recall from [11] the construction of (locally) homogeneous s-symmetric parabolic geometries that we need in Section 5. 2. Automorphisms of parabolic geometries In this Section, we introduce necessary techniques and establish notation from the theory of parabolic geometries that we will use in the article, see [3, Section 5.1]. We focus here on actions of automorphisms on Weyl structures and connections. 2.1. Weyl structures and connections. Consider a parabolic geometry (Q —>• M, bS) of type (G,P). Many geometric objects on M can be identified with sections of natural bundles V associated to the P-bundle Q for representations V of P. We can equivalently view the sections of V as P-equivariant functions Q —>• V. In other words, the points of Q are (higher order) frames and the P-equivariant functions are the coordinate functions. A crucial tool that allows us to reduce the number and order of the frames are Weyl structures. A (local) Weyl structure is a (local) Go-equivariant section a : Go —> Q of the projection tt : Q —> Goi where Go '■= QI exp(p+) and Po ■ Go M is a Go-bundle over M. Definition 2.1. Assume a : Go —>• Q is a Weyl structure. Then for a section t of a natural bundle V, we denote by (t)ct the Go-equivariant function Go ^ V satisfying (t)ct :=tou, where t : G —> V is the P-equivariant function corresponding to t. In particular, vector fields £ and 1-forms T on M are sections of bundles G x pg/p and G x p p+, respectively, and there are corresponding Go-equivariant functions (Ov : So 0- and (t)a : Go -» P+- Weyl structures always exist on parabolic geometries and for each two Weyl structures a and a, there exist a 1-form T and Go-equivariant functions Tj : Go Qi for i = 1,..., k such that er = crexp(T)cr = crexp(Ti).. .exp(Tfe). 8 jan GREGOROVIČ AND LENKA ZALABOVÁ The Go-equivariant function (T)CT : C?o —^ P+ is related to the functions Tj via the Baker-Campbell-Hausdorff (BCH)-formula. We can decompose the pullback a*uj : TGo —> g into Go-equivariant 1-forms (jjf : TGo —>• Qi according to the grading Qi of g. These forms clearly depend on the choice of the Weyl structure a. For a Weyl structure a = crexp(T)0-, there is the following formula describing the change of the forms (1) c4Texp(T)-= J2 ^(ad(Tfe)Jfco...oad(T1n°^, \i\+3=l l' where we write il = i\\.. .ik\, \i\ = i\ + 2i2 + • • • + kik and (—1)* = (—Hk for the multi-index i = {i\,.. ., i^) with i\,. .. , i^ > 0. The sum of the forms ujf for i < 0 is called the soldering form given by the Weyl structure a. Suppose = + • • • + £_i holds for the vector field £ on M and for Go-equivariant functions £j : Go —> Qi. If (^)o-exp(T)^ = £-fc + ■ ■ ■ + £-i holds for Go-equivariant functions £j : Go —> Qi and the Weyl structure crexp(T)0-, then Zi= J2 ^ad(Tfe)Jfco...oad(T1)^.ej, \i\+3=l l' where . is the algebraic action of the values of functions Go —>• p+ on the values of the functions Go ^ Q-- The form ujq is a principal connection form on Go- Suppose that the finite-dimensional representation of P on V is completely reducible as a representation of G0. Then (1) the form ujq induces a linear connection VCT on the space of P-equivariant functions Q —>• V, (2) for each P-equivariant function t : Q —> V, the connection VCT preserves the decomposition of (t)ct into Go-equivariant components. The induced connections VCT on V are called Weyl connections. The Weyl connection v°"exp(T)• GZ(g/p) such that i* 9 = ujZ and t^'jcr = <^o hold for the natural soldering form 9 on T)1M and the principal connection form 7CT on P^M of the Weyl connection V°\ This allows us to describe explicitly geodesies of Weyl connections. The geodesic of the Weyl connection V°" on TM through x in the direction £(x) g TXM is the curve (3) po o Flt (w0) for arbitrary «o g Go in the fiber over x. Indeed, since (ujZ. ® ) 1((C(^))o-) is contained in the kernel of the connection form uJq = i*7CT and Tpo o (w^ ® symmetric parabolic geometries 9 ljq) 1((£i(x))a)(x) = the claimed curve is the projection of a flow of a standard horizontal vector field of 7^ and therefore a geodesic of V°\ 2.2. The characterization of automorphisms. Let p : Q —>• be a (local) automorphism of the parabolic geometry and denote by p§ : Go —> C?q the underlying (local) Go-bundle morphism. Then for each Weyl structure a, there is a 1-form Ta'v on M such that (4) Q be a (local) automorphism. Then (5) (T M, lS) • /*Vff = Vff' holds, and • Vx f maps a point of ta'(Go) into the image ta(Go)-Then f preserves the parabolic geometry. Proof. The assumptions imply that p0 := i~x o V1f o La, is a well-defined (local) Go-bundle morphism po '■ Go ^ Go satisfying p*o^o = anc^ Po^- = w- ■ The associated graded map (6-k, ■ ■ ■ ■ TGo 0-feffi- ■ -ffifl-i corresponding to ljZ. is independent of the choice of the Weyl structure according to the formula (1). In fact, the tuple (p0 : Go ~~^ M, (6-k, ■ ■ ■, $-1)) is a regular infinitesimal flag structure with a (local) automorphism po, see [3, Section 3.1.6-8]. Therefore the claim of Theorem follows from [3, Theorem 3.1.14] except for projective and contact projective geometries. In the case of projective geometries, the claim trivially follows 10 jan gregorovič and lenka zalabová from the assumption /*V°" = V°" . In the case of contact projective geometries, (fo is a (local) automorphism of the regular infinitesimal flag structure if and only if / is a contactomorphism and the claim again follows from /*V°" = V°" , see [3, Section 4.2] for details. □ 2.3. Normal Weyl structures and generalized geodesies. There is a distinguished class of local Weyl structures, so-called normal Weyl structures at x = p(u), each of which is determined by a choice of u g Q. More precisely, we consider local Weyl structures vu given by z^MFrf 1(X)H)) := Frf 1(XV) for X in some neighbourhood of 0 in g_. The Weyl structures vu for all u g Qx exhaust all normal Weyl structures at x, see [3, Section 5.1.12]. These Weyl structures are distinguished by the fact that (6) p(Flf1(X)(u)) = Flf1(X)(p(u)) holds for all (local) automorphisms ip of the parabolic geometry and all X in some neighbourhood of 0 in g. This particularly means that holds for all (local) automorphisms ip of parabolic geometries. The curves of the form poFlf1(X\u) for X g g_ and u g Q are called generalized geodesies. They always provide the normal coordinate system given by u. The crucial observation is that the set of generalized geodesies going through x coincides with the set of geodesies of normal Weyl connections V1'" for all u. Therefore there is the following description of automorphisms of parabolic geometries. Proposition 2.4. Let ip be a (local) P-bundle morphism on Q and let f = ip be its underlying (local) diffeomorphism of M. If ip is a (local) automorphism of the parabolic geometry, then the equality /*Vly" = V^-1*"' holds for all u g Q and f maps the set of generalized geodesies going through x onto the set of generalized geodesies going through f(x). Moreover, if f has coordinates Ad(go) g GI(q—) for go g Go in the normal coordinate system given by u g Q, then p is a (local) automorphism of the parabolic geometry if and only = \7Utl holds. Proof. Since /*V°" = Vv a holds for all Weyl structures a and all (local) automorphisms ip of the parabolic geometry, the first claim follows from the formula (3). If / has coordinates Ad(go) g GI(q-) in the normal coordinate system given by u g G, then the second assumption of Proposition 2.3 is satisfied. Then the second ^ —i claim is a consequence of the first claim and Proposition 2.3, because XIVu = V "so holds. □ 3. The uniqueness of s-symmetries and the prolongation rigidity In this section, we prove Theorem 1.6. We also characterize all triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s. SYMMETRIC PARABOLIC GEOMETRIES 11 3.1. Consequences of the existence of more s symmetries at one point. Let us recall that if V is an irreducible Go-module, then the element s G Z(Go) acts on V by a single eigenvalue. In particular, we can decompose each completely reducible Go-module V into Go-submodules Vs(a) :={X eV : s(X) = aX} according to the eigenvalues of the action of s G Z(Gq). In particular, we will often consider the 1-eigenspaces 01(1), flf(l) and p+(l) in g_,0i and p+, respectively. The following Proposition is a crucial technical result for the proof of Theorem 1.6. Proposition 3.1. Let su be a (local) s-symmetry at x for some u G Qx. Then for each Weyl structure a, there is a l-form Ta'Stl on M satisfying (1) <<7 = <7exp(-(T^-)(T), (2) (Tff-S")ff(7r(w)) = G(-Ad(s)-1(y),y) for some Y G p+, where C represents the BCH-formula on the nilpotent Lie algebra p+, and (3) if (Ta'Stl)tT(Tr(u)) = Zi + ■ ■ ■ + Zk holds for Zj G Qj, then the component of Zi contained in gf (1) is trivial, where i is the smallest index j such that (Ya'Stl)a(Tr(u)) has a non-zero component in g^. Moreover, if sv is a (local) s-symmetry at x for some v G Qx, then su = sv if and only if Tff-S" (x) = Tff-S- (x) holds. Proof. The normal Weyl structure vu always satisfies vu(pj(uj) = u and therefore the set of all Weyl structures a satisfying cj(ty(u)) = u is non-empty. Let su be a (local) s-symmetry at x and consider arbitrary Weyl structure a satisfying (t(tt(u)) = u. Then (Tff's")ff(7r(w)) = 0 holds and Lemma 2.2 implies that (Tff-S")ff has the claimed properties (1) and (2) for arbitrary Weyl structure. The claimed property (3) holds, because the BCH-formula implies that C(—Ad(s)_1(Y), Y)i = -Ad(s)"1(yj) +Yt = Zt holds. If sv is a (local) s-symmetry at x for some v G Qx, then su = sv if and only if us = su(u) = sv(u) holds. Thus we need to show that if T°"'s"(a;) = Ta'Sv(x) holds, then su = sv. We can assume ct(tt(u)) = u for the Weyl structure a, because the equality Ta'Su(x) = Ta'Sv(x) is preserved if we change the Weyl structure a. Suppose g0 G Go and Y G p+ are such that v = ug0exp(Y) holds. If a is a Weyl structure such that &(tv(u)) = ug0exp(Y), then T°"'s"(a;) = 0, T^(x) = 0 and (T^)a(ir(u)) = G(-Ad(s)-1(Ad(ff0)(^)),Ad(ff0)(^)) hold. Since C(-Ad(s)-1(Ad(go)(Y)),Ad(g0)(Y)) = 0 if and only if Ad(s)(y) = Y, the element s commutes with goexp(Y) and su = sv holds. □ The harmonic curvature kh is preserved by each (local) automorphism of the parabolic geometry. Since kh is a section of an associated vector bundle to Q for a representation of P which is trivial on exp(p+), the function (kh)ct does not depend on the choice of the Weyl structure a and we will write kjj(u) instead of (k_h")o-(7t(m)). Consequently, kjj(p(u)) = 0 if and only if kjj(u) = 0. If su is a (local) s-symmetry at p(u), then s^kh = kh- Thus s.kjj(u) = kh(u) trivially follows, where we denote by . the tensorial action of go on kh ■ This proves the first claim of Theorem 1.6. The second claim of Theorem 1.6 is a consequence of the following Proposition and the Definition 1.5 of the prolongation rigidity. Proposition 3.2. Assume there are (local) s-symmetries su and sv at x for some gx. Suppose that (Tff's")ff(7r(w)) = 0 and (Tct's^)(t(ty(u)) = Zi H-----VZh hold for some Weyl structure a. Then Zi G pr{nH{u))i. 12 jan gregorovič and lenka zalabová Proof. We show that ad(Xi)... ad(Xj)(Zj).Kij(u) = 0 holds for all X\,...Xi g g_i. Consider an arbitrary Weyl structure a and consider the iterated covariant derivative (V0^)^ ^ for vector fields ... , £J such that (Cb)o- = £-i ■ Go ->• 0-i, (£6)ff(7r(U))=X6 hold for some Xb g 0-i(^) for some a& for all 1 < 6 < j. We assume j < i unless we state otherwise. We compute d:vff);£1.....£)%w=i:(vff)iSo)<£1.....(Sii)>£^)t%(«) Since we assume Xb g flii(^), we get = Ad(S)-1(e)a(7t(u))=abXb. Thus (7) QlLV%^jKh(u) = ai • • • aj(V%^jKH(u). If (T°"'s")0-(7t(m)) = Zi + - ■ - + Z]„ holds for the Weyl structure er, then the formula (2) together with Proposition 3.1 imply (sy°)ibKH(u) = \7°rPiHr°'SUMKH(u) = Vf Kjy(«) + ^(Z^X^.KHiu). In particular, if z > 1, then (s.:V°)eKH(u) = V™P(-(r""WKj?(«) = VfKjy(«). If we apply the above formulas onto the first connection in (s^V0^)^ ^kh(u), then we obtain (s;(VT')«i,..,«^H = Vf^V^-1^nH{u). In the next step, the same formulas for the second connection lead to the formula Thus before we consider the next step, we need to characterize the components of (V^T0"'S")0-(X(U)) in 0i © • • • © Qj for j < i. Firstly, let us view (T°"'s")0- as a section of the adjoint tractor bundle q Xp q. Observe that the covariant derivative V|j, coincides with the fundamental derivative on the components in 0-©0o©0i©- ■ -ffi0j according to the formula from [3, Proposition 5.1.10]. We know that (T°"'s")0- has its values in p+ and the components of (V|i,T°''s")0.(,r(u)) in 0i © • • • © Qj for j < i are tensorial both in £b and ta's''. Then, using the formula from [3, Corollary 1.5.8] and the P-equivariancy of uj, we get the following equality on the restriction to 0i © • • • © Qj for j < i (V^T^)ffWu)) = w(<7(7r(U)))([W-1(e6),w-1(Zi)]) = -ad(Xb)(Z4). Therefore If we iterate the computation of (V|i,T0''s" f°r 3 < h then we obtain by the same arguments ((v%,...^Tff's"W(«)) = (-i)Mx'') ■ ■■^{x1){zl) SYMMETRIC PARABOLIC GEOMETRIES 13 for the component in gi © • • • © Qi—j- Thus for j < i, we obtain = "h(u) and for j = i, we obtain (^V)^.....£iKff(U) = (V)^.....£iKff(«) + (-l)Jad(XJ)... ad(X1)(Z,).kh(m). If we compare the formulas (7) and (8) for (s*V°')^1 ^kh(u), we obtain (9) (-l)>i • • • a, - l)(Vff)Je_vnH{u) = ad(A^)... &d{X1){Zi).KH{u) for all j < i. If the Weyl structure a satisfies (Tcr'Sv)a(ir(u)) = 0, then we simultaneously have (ai - ■ ■ a3 - l)(VaYe^KH(u) = 0 for all j < i if we follow the proof for sv instead of su. Thus if a\ ■ ■ ■ dj — 1 =/= 0, then ad(X*)... a,d(X1)(Zi).KH(u) = 0. But since Zi has a trivial component in gf(l), we know that ad(X*)... a,d(X1)(Zi) ^ 0 implies a\ ■ ■ ■ cij — 1 ^ 0 and the claim of Proposition holds due to the linearity. □ If we follow the computations from the proof of Proposition 3.2 for a Weyl structure a satisfying T°"'s"(a;) = 0, then most of the assumptions on the vector fields £b are vacuous and (s* V0")^^(u) = W^Kff(u) holds for arbitrary vector field £. Therefore we obtain the following corollary using the formula (7) for £ from particular eigenspaces of Txsu. Corollary 3.3. Let su be a (local) s-symmetry at x = p{u) on a parabolic geometry and assume Ta'Su(x) = 0. Then we get where £fix G TXM is the component of £ G TXM such that (S,fix)lT('^(u)) G gs_(l). In particular, if Qs_(l) = 0, then \7^kh(x) = 0 holds for all £ G TXM. Remark 3.4. The authors showed in [22] and [5] that there are projective and conformal geometries satisfying V^nnix) = 0 for all x G M for a suitable Weyl connection V°", but (M, Vff) are not an affine locally symmetric spaces. Therefore Theorem 1.3 implies that the condition V" kh = 0 is necessarily satisfied on (locally) TO-symmetric parabolic geometries, but is not sufficient to distinguish the (locally) TO-symmetric parabolic geometries among the geometries satisfying V"kh = 0. 3.2. The characterization of triples that are prolongation rigid outside of the 1— eigenspace of s. We can estimate the dimension of pr(nH(u))i in the following way: The result of [17, Proposition 3.1.1] states that the dimension of ann(Ku(u)) is bounded by the dimension of the annihilator do := n0oann(o) of all minus lowest weights cpo in (the complexification of) all irreducible go-modules in which kjj(u) has a non-zero component. Moreover, the dimension of pr{nH{u))i is bounded by the dimension of the prolongation Oj := (~^4>0P'r((po)i of do- The main result of [17, Theorem 3.3.3 and Recipe 7] states that there is a semisimple Lie subalgebra g of g and a parabolic subalgebra p of g such that Oj = Qi for i > 0. Let us prove that these estimates are compatible with the decomposition of Qi into go-submodules, which allows us to characterize the triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s. 14 jan GREGOROVIČ AND LENKA ZALABOVÁ Proposition 3.5. Suppose Z G pr(Kjj(u))i decomposes as Z = Za + Zb for Za, Zb in different Qg-submodules of Qi. Then Za G pr{nH{u))i and Zb G pr(Kn(u))i. Therefore the triple (g,p,/i) is prolongation rigid outside of the l-eigenspace of s if and only if Oi corresponding to p is a subspace o/gf(l) for alii. Proof. Let ..., oij) be an ordering of simple positive roots of g such that the root space gQi. satisfies gQr G gi. Then we can uniquely assign a j-tuple (ai,..., af) to each irreducible go-component of Qi, where an is the height of all root spaces in the go-component with respect to an. This defines a multigrading of g and the Lie bracket in g is multigraded. Let us decompose the element Z G pr(Kff(u))i as the sum of the elements ^(6i,...,bj) over all possible j-tuples with respect to this multigrading. Similarly, let us decompose the module M,uj) of type (G,P). Then: 16 JAN GREGOROVIČ AND LENKA ZALABOVÁ There exists an almost S-invariant Weyl structure a and the map S is smooth. If a is an almost S-invariant Weyl structure, then crexp(T)0- is an almost S-invariant Weyl structure if and only if (T)a has its values in p^_(l). For each x G M, there is a local almost S-invariant Weyl structure a, which is invariant at x, and S_(x) = sx holds. The equality S(po(uq)) = sa(Uo^ holds for each almost S-invariant Weyl structure a for all u0 G Go- The equality S(x) o S(y) o S(x)^1 = S(S_(x)(y)) holds for x,y G M, where the compositions are defined. For each eigenvalue a, the union of the a-eigenspaces TxMs(a) ofTxS_(x) in TXM over all x G M defines a distribution TMs(a) on M that is preserved by all (local) s-symmetries for each a. The equality TMs(a) = Tpo o (u;^ + u!q )~1(gs_(a)) holds for each almost S-invariant Weyl structure a. The decomposition TM = ®aTMs(a) is preserved by all almost S_-invariant Weyl connections V°\ All almost S_-invariant Weyl connections restrict to the same partial linear connection on TM corresponding to the distribution <3)a^iTMs(a). We show that Claim (1) of Theorem 1.8 implies Claim (3) of the Theorem 1.8 and simultaneously obtain all the claims of Proposition. Proof. Let us pick an arbitrary Weyl structure a and consider the Go-equivariant function (S)a : G0 {C(—Ad(s)^1(Y),Y),Y G p+} defined by S(po(u0))*v(u0) = a(u0) exp(-(S)z(u0)) for all uo G Go- We show that (S)& is smooth. We decompose = $>4(a) + • • • + $>fe(a) a a according to the grading and the eigenvalues a of Ad(s). It follows from Claim (3) of Proposition 3.1 that Tj(l) = 0. Thus the formula (9) from the proof of Proposition 3.2 that holds under our assumptions at each point of M implies that each Ti(a) is smooth. The formula (5) from Lemma 2.2 gives (S0*exp(T)» = CC-AdOO-^T)*, C((S)a, (T)*)). If we take T = rTj(a) for arbitrary r£l, then CC-AdOO-Vifa)). C((S)», rn{a))Ua) =C(--n(a), C{n{a),rn{a))Ua) a r(l — a) + a =-n{aj a holds for the component of the BCH-formula in flj(a), while the components of the BCH-formula in Qi(b) for the other eigenvalues 6 ^ a of Ad(s) remain Tj(6). Consequently, if we take a^l and consider the Weyl structure s^)CT,(vo)) holds for vq in the fiber over y for arbitrary a' such that V°" G [VCT]. Thus Ad(s)(T^))>0) = (r°'s^)a(v0) holds and thus [V°"] is an 5-invariant class of Weyl connections. □ 5. Geometric properties of parabolic geometries of distinguished types and classification In this section, we study properties of (locally) s-symmetric parabolic geometries of particular types (G, P) for triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s for \i in the 1-eigenspace of s. The properties follow from the position and shape of fli(l) inside of g_. We classify all triples (g, p,/i) where fli(l) has such a position and shape for generic s. 18 JAN GREGOROVIČ AND LENKA ZALABOVÁ 5.1. Classification results and notation. Let us use the characterization from Section 3.2 for the classification of the triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s G Z(Gq) such that fi is in the 1-eigenspace of s. We separate the classification into the series of tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. The main reason for such a separation is that parabolic geometries from different tables have different geometric properties and we divide the tables according to these properties. Theorem 5.1. Let (g,p,/i) be a triple obtained from one of the tables 1-13 in the following way: • The Lie algebra g is a simple Lie algebra of the (complex) rank n that is at least A4, B4, C4, D§ or some explicit Lie algebra of lower rank from the column g. • The parabolic subalgebra p is the parabolic algebra from [3, Section 3.2.9] for the set £ in the column S. • The component of the harmonic curvature \i is specified by an ordered pair of simple roots of g from the column \i that provides the highest weight of \i by the affine action of corresponding elements of the Weyl group, see [3, Theorem 3.3.5]. • The component \i is contained in the l-eigenspace of s for the elements s G Z(Gq) that have the eigenvalues jia from the columns jia on the irreducible Qo-components that are determined by the iath element of the set S. Then (g, p, /i) is prolongation rigid outside of the l-eigenspace of s if the eigenvalues jia °f s satisfy the condition in the column PR. The tables 1-13 contain the complete classification of triples (g,p,/i) that are prolongation rigid outside of the l-eigenspace of s for \i in the l-eigenspace of s (except the cases that are conjugated by an outer automorphism of g to one of the listed entries). The remaining notation we will use in the tables is the following: We characterize the real form of g by a number q and a field {R, C, H}. The set £ characterizes the set of crossed nodes in the Dynkin or Satake diagram that provides the parabolic subalgebra p. We use the ordering of nodes which is consistent with [3, Appendix B] and we will not add the conjugated crossed nodes to £ in the case of complex Lie algebras, su(q, n+l — q) and so(3, 5). We distinguish the complex conjugated simple roots by '. If the column for the eigenvalue jia is blank, then the value of jia is generic. If the eigenvalue jia ^ R and ln(jia) = ria + i(pia, then either ria = 0 or cpia = 0 and we specify only the non-zero one in the table. If the column PR is missing or the condition is blank, then the triple (g, p, fi) is either prolongation rigid or the condition that \i is contained in the 1-eigenspace of s is sufficient for Oj corresponding to fi to be a subspace of gf (1) for all i. The classification tables are presented in the following subsections and the triples (g,p,/i) are obtained from the tables according to Theorem 5.1. 5.2. Parabolic geometries with fli(l) = 0. The Table 1 contains all triples (g,p,/i) with the property that if s G Z(Gq) is such that (g,p,/i) is prolongation rigid outside of the 1-eigenspace of s, then fli(l) = 0. In particular, all AHS-structures that have a component of the harmonic curvature in the 1-eigenspaces of some s G Z(Gq) are prolongation rigid outside of the 1-eigenspace of s and thus are contained in this table. Example 5.2. Before we formulate the general result, let us demonstrate how the results for (locally) symmetric conformal geometries that we presented in [13] can be obtained from the Table 1 and Theorem 5.3: symmetric parabolic geometries Table 1. Theorem 5.3 19 0 M «1(3, C) {1} 01 (ax, ax>) «1(3, C) {1} vr (ax',a2') «1(4, {M,C}) {1} (ax,a2) «1(4, C) {1} 0i (ax, ax>) «1(4, C) {1} vr (ax',a2') sl(n + 1,{M,C}) {1} (ax,a2) sl(n+ 1,C) {1} 0i (ax, ax>) sl(n+l,{M,C,H}) {2} VT (a2,ax) sl(n+ 1,C) M VT (ap,,ap+1,) «o(l, 5),«o(2,4),«o(3,3),«o(6, C), «o(l,6), «o(2,5), «o(3,4), «o(7,C) , «o(l, 7), so(2,6), so(3,5), so(4,4), «o(8, C) {1} VT (ax,a2) «o(6, C), «o(7, C), «o(8, C) {1} VT (av,a2>) «o(7, C) {3} Vi (0:3,0:2) So(q, n — q), so(n, C) {1} Vi (01, a2) so(n, C) {1} VT (or, a2r) so(2n, C) W VT («»', 0„_2') «o(2n+ 1,C) W VT (ani, an-V) sp(4, C) {i} Vi (01, a2) sp(4, C) {i} VT (or, a2r) sp(4, C) {2} VT (02',0l') sp(6,C) {2} VT (o2', O3/) sp(6,C) {3} Vi (o3', a2r) sp(2n,C) {n-1} VT (an-V, ani) sp(2n,C) W VT (an', an_x'*) e6(C) {1} VT (or, 02') e7(C) {1} Vi (or, a2r) There are rows with g = So(q, n — q) and S = {1} in the Table 1 and the triples (so(q, n—q), p{i}, M(ai,a2)) are prolongation rigid outside of the 1-eigenspace of s for n > 5 and q > 0. We read of the corresponding line that the eigenvalue ji± = VT. Thus s = m and the m-symmetries in question are the symmetries of conformal geometries presented in [13]. We get immediately from Theorem 5.3 that Theorem 1.3 holds for conformal geometries. In the following theorem, we summarize geometric properties of geometries from the Table 1 and prove the last claim of Theorem 1.8. Theorem 5.3. Assume (g,p,/i) is prolongation rigid outside of the l-eigenspace of s for s G Z(Gq) such that fll(l) = 0 holds. If the harmonic curvature of the (locally) s-symmetric parabolic geometry (Q —> M, lS) of type (G, P) has a non-zero component in p at some x, then: (1) The parabolic geometry is (locally) homogeneous, kjj(x) =/= 0 at all x G M and there is a unique smooth system of (local) s-symmetries S on M. (2) There is a unique distinguished Weyl structure a which is uniquely characterized by one of the following equivalent properties: 20 JAN GREGOROVIČ AND LENKA ZALABOVÁ (a) The equalities VaTa = 0, s.(Ta)a = (Ta)a, VaRa = 0 and s.(Ra)a = {Ra)a hold for the torsion and the curvature of the Weyl connection V°\ (b) The Weyl connection VCT is S_-invariant. (c) All (local) automorphisms of the parabolic geometry cover affine transformations of VCT. (d) All (local) diffeomorphisms sx are affine transformations o/V°\ (e) All (local) P-bundle morphisms sa(Uoj are (local) s-symmetries. (3) The pseudo-group generated by all local s-symmetries is transitive on M and its connected component of identity is generated by the flows of the Lie algebra [, which is the vector subspace o/g_ ®0o? generated by g_ by the bracket (Ta + Ra)a on A2q*_ (g) I and the natural bracket on the rest of I. (4) The equalities S_(x) = so.(„0) = s% hold for the Weyl structure a from Claim (2). In particular, • the maps S_{x) can be extended to a larger neighbourhood of x as long as the corresponding geodesic transformations of VCT are defined, • S_{x) ° Sly) °j2(:r)_1(z) = £L(£L(x)(y))(z) holds for (x,y,z) in some neighbourhood of the diagonal in M x M x M, and • for each eigenvalue a, the distribution TMs(a) is preserved by all (local) automorphisms of the parabolic geometry. Proof. Let U C M be the set of points x such that kh (x) has a non-zero component in p. Then there is a unique system of (local) s-symmetries on U due to the prolongation rigidity of the triple (q,p,p) outside of the 1-eigenspace of s. It suffices to prove Theorem under the assumption U = M, because if we prove Claim (3) on U, then the equality U = M follows from the (local) homogeneity, i.e., Claim (1) follows from Claim (3). Then Claim (4) follows from Claim (2) due to Claims (5) and (7) of Proposition 4.3. Therefore, it suffices to prove Claims (2) and (3) under the assumption U = M to complete the proof. If 01(1) = 0, then p+(l) = 0 and Proposition 4.3 implies that there is a unique S'-invariant Weyl structure a. It follows from Proposition 3.1 and Proposition 4.1 that the Weyl structure a satisfies (2b) if and only if it satisfies (2e). Further, Proposition 4.1 and the Corollary 4.2 imply that the Weyl structure a satisfies (2e) if and only if it satisfies (2d). We show now that (2b) implies (2a). The torsion and the curvature of S_-invariant Weyl connection VCT are 5-invariant. In particular, S.(T°)a(u0) = (S(p0MrT>0M))ffM = (T°)a(u0) and s.(iT)ffM = (S(PoMriT(p0M))ffM = (iT)ffM hold for all u0 G Go f°r the natural action . of Go on the values of (Tcr)a and (Ra)a. Since the same arguments can be applied on V°"T°" and V°"i?°", it follows that (V^Tff)ff = s.(V^Tff)ff = a(V^Tff)ff and (V^i?ff)ff = s.(V^i?ff)ff = a(V^i?ff)ff hold for any vector field £ on M such that (£,)a(uo) "= 0S-(a 1) f°r an uo "= Go- Thus (2b) implies (2a), because fli(l) = 0. The Claim (2a) implies that V°" is a locally affinely homogeneous connection. Therefore, according to [9, Section 1.5], the affine geometry (M, V°") can be encoded as a locally homogeneous Cartan geometry of type (g_ x GI(q-),GI(q-)) on the first order frame bundle VXM. Moreover, the assumptions of [11, Lemma 2.2] are satisfied, because (Ta + Rcr)a(ia(uo)) is the bracket of the infinitesimal affine transformation at l,j{uq) G ta{Go) C VxM. Thus there is a (local) affine transformation A of V°" such that (A)a(uo) = s. Therefore Claim (2d) follows from Claim (2a) and SYMMETRIC PARABOLIC GEOMETRIES 21 Proposition 4.1 due to the uniqueness of s-symmetries. In particular, if we consider a (local) one-parameter subgroup exp(i£) for an infinitesimal affine transformation £, then exp(t^)S_(x) exp(—££) is the (local) s-symmetry at exp(t£)(x) and the map ■^\t=oexp(t^)S(x)exp(-t^)S(x)-1 maps £ into I. If ('P1£)ff(Mo) = -X", then the element X — Ad(s)(X) is contained in I. Thus g_ C I as a vector subspace. Thus the flows of the Lie algebra I generate a sub-pseudo-group, which is the connected component of identity of the pseudo-group generated by local s-symmetries. Since Ad(s) preserves I, Claim (3) follows. We can use the results from [12, Theorem 1.3] due to the local homogeneity and (2c) follows from (2a). Clearly (2c) implies (2d), which completes the proof. □ 5.3. Parabolic geometries with distinguished parabolic subalgebras gi(l)+ p. There are triples (g, p, fi) that are prolongation rigid outside of the 1-eigenspace of s which admit 1-eigenspace in g_ for some s such that q := jjf_(l) + p is a parabolic subalgebra of g such that the harmonic curvature in \i vanishes on insertions of elements of q/p at all points of M. These are listed in the Tables 2, 3 and 4 due to [12, Proposition 6.2 and Proposition A.2]. Example 5.4. To demonstrate our results, let us look in the Table 2 on the row g = sl(n+ l,R) and £ = {1,2} which corresponds to generalized path-geometries (for systems of second order ODEs), see [3, Sections 4.4.3-5]. These parabolic geometries generally have two harmonic curvatures, one torsion k(ai,a2) and one curvature h{a2,a1)- However, they fall in the Table 2 only when the torsion k(ai,a2) vanishes and the harmonic curvature consists only of the curvature K(Q2 Ql) corresponding to H{a2,a1)- There are many s G Z(Gq) that act trivially on H{a2,a1)i but the triple (sl(n + 1, R), p{i,2}; M(a2,ai)) 1S prolongation rigid outside of the 1-eigenspace of s only for s G Z(Gq) with eigenvalues ji = l,j2 = —1. In such case, q = p{2} is the parabolic subalgebra of g corresponding to E = {2}. The torsion-freeness of generalized path-geometries implies that the space of local solutions of the corresponding ODEs carries a Grassmanian structure, which is a parabolic geometry on the local leaf space of type (G, Q) from Theorem 5.5. Therefore if {Q —> M, ui) is a (locally) s-symmetric torsion-free generalized path-geometry with a non-zero harmonic curvature, then we conclude from Theorem 5.5 that the space of local solutions N is a locally symmetric space (N, S_), while M together with the system of (local) s-symmetries S is a reflexion space (M, S_) over (N, S_), see [6]. Let us emphasize that due to dimensional reasons and the formula [3, Theorem 5.2.9], the pseudo-group generated by all local s-symmetries is locally transitive at x G M if and only if the Rho-tensor Pa(n(x)) of the S'-invariant Weyl structure a on N does not vanish on Tn^N. We summarize geometric properties of geometries from the Tables 2, 3 and 4 in the following theorem. Theorem 5.5. Assume (g,p,/i) is prolongation rigid outside of the l-eigenspace of s for s G Z(Gq) such that q = fll(l) + p is a parabolic subalgebra of g and q/p inserts trivially into the harmonic curvature kh of the (locally) s-symmetric parabolic geometry {Q —> M,ui) of type (G, P). If kh has a non-zero component in H at some x, then: (1) The inequality kh 7^ 0 holds in an open dense subset of M, and there is a unique smooth system of (local) s-symmetries S on M. (2) There are • a parabolic subgroup Q of G with the Lie algebra q such that P C Q, • a neighbourhood Ux of each x G M with the local leaf space n :UX —> N for the foliation given by the integrable distribution Tp o a;_1(q), and 22 jan gregorovic and lenka zalabova Table 2. Theorem 5.5, part with |S| = 2. 0 3i\ Jl2 PR sl(3,C) {1,2} 2ri sl(3,C) {1,2} («1', «2') T2 = 0, 02 = 2-7T sl(3,C) {1,2} -|01 (a2',ai') n = O,0i = 2tt *[(4,{R,C}) {1,2} J? (a2,a1) 32 = -1 sl(4,M) {1,2} 3i (at2,a3) sl(4,C) {1,2} 2r! sl(4,C) {1,2} -102 [av,a2>) T2 = 0, 02 = 2-7T sl(4,C) {1,2} (a2',ar) ri = O,0i = 2tt sl(4,M) {1,3} 3i (ai,a2) sl(4,C) {1,3} 2ri (ai,av) sl(n+ 1,{R,C}) {1,2} (a2,ai) 32 = -1 sl(n+ 1,C) {1,2} 2ri (ai,av) sl(n+ 1,M) {1,3} j'i (ai,a2) sl(n+ 1,C) {hp} 2 < p < n 2ri (ai,av) 7-1=0 sl(n + 1,{M,C}) {hp} 3 < p < n J? (ai,a2) 31 = -1 sl(n + 1,R) {l,n} J? (ax,a2) sl(n+ 1,C) {l,n} 2ri (ai,av) sl(n+ 1,M) {2,3} (a2,ax) sl(n+l,{M,C,H}) {2,rf 3 < p < n (a2,ai) J2 = -1 sl(n + 1,{M,H}) {2,n} j'i (a2,ai) sl(n+ 1,C) {p,p+ 1} -iP rp = 0, 0P = 2-7T so(2,5),so(3,4),so(7,C), so(2,6),so(3,5), so (4,4),so(8,C) {1,2} vT (a1, a2) 32 = 1 so (4,4) {1,4} i'2 Jl (a1, a2) so(g, n — g),so(n, C) {1,2} (a1, a2) 32 = 1 so(n,n),so(2n, C) {l,n} i2 Jl (a1, a2) 31 = -1 Table 3. Theorem 5.5, part with |S| = 3. 0 .7*2 3is M PR sl(4,M) {1,2,3} Jd'i (a2,ai) 3i = l sl(n + l,R) {1,2,3} jij'i («2, &l) 3i = l sl(n+ 1,{M,C}) {l,2,rf 3 < p < n Jd'i (a2,ax) 3i = 1, J2 = -1 sl(n + {1,2,71} 3132 (a2,ai) 3i = l so(4,4) {1,2,4} 3i (ai,a2) 32 = 1 so(n, n), so(4n, C) {l,2,n} j'i (ax,a2) 3i = -1,32 = 1 • a (locally) s-symmetric parabolic geometry {Q' —> N,uj') of type (G,Q) satisfying the assumptions of Theorem 5.3 such that (Glijx => Ux,^\uS) is isomorphic to an open subset of {Q' —> Q'/P,uj') for each x. In particular, there is a unique s-symmetry S_(n(y)) on {Q' —> N,uj') at symmetric parabolic geometries Table 4. Theorem 5.5, part with |S| = 4. 23 0 eigenvalues M PR sl(n+ 1,R) {l,2,3,p} 3 < p < n ■ -2 -i 3p — j1j2j3 (a2, ai) Jl = 1, J3 = J2 sl(n+ 1,{M,C}) {l,2,p,g} 3 < p, q < n 3q = jiji.jp 1 (a2, ai) jl = 1, j2 = -1, jp = 1 sl(n+ 1,R) {l,2,p,n} 3 < p < n = jijijp 1 (a2,a1) jl = 1, jp = 1 eac/i n(y) G -/V smc/i that n o S(y) = S_(n(y)) o n holds /or aZZ y G L^. m i/ie /i&er ot)er n(y). f5j The connected component of identity of the pseudo-group generated by all local s-symmetries is generated by the flows of the Lie algebra I, which is the vector sub-space ofqop, generated by q°p by the bracket (Ta(n(x)) + Ra(n(x)))a on A2(q7)*®l and the natural bracket on the rest of I for the S-invariant Weyl structure a on (q' —> N,ui'), where qop is the opposite parabolic subalgebra of q to q. The pseudo-group generated by all local s-symmetries is locally transitive at x if and only i/q/p C 1/(1 Hp), i.e., if and only if (R°(n(x))) a spans the whole q/p. (4) There is a bijection between • the almost S-invariant Weyl structures on Ux, and • the reductions of the image in Q' of the (unique) S-invariant Weyl structure a on n (that exists due to Theorem 5.3) to exp(gs_(l)) xi Go C Qq. A reduction corresponds to an S-invariant Weyl structure on Ux if and only if it is a holonomy reduction o/VCT. (5) In particular, • the maps S_{x) can be extended to a larger neighbourhood of x as long as the corresponding geodesic transformations of VCT on N are defined, • §_{x) 0 §_{y) 0 Sjy^x)^1^) = S_(S_(x)(y))(z) holds for (x,y,z) in some neighbourhood of the diagonal in M x M x M, • the distribution TMS{\) is the vertical distribution of the local leaf space n:Ux^ N, • for each eigenvalue a, Tn{TxMs(a)) is the a-eigenspace of Tn^S_(n(x)) in Tn{x)N, and • for each eigenvalue a, the distribution TMs{a) is preserved by all (local) automorphisms of the parabolic geometry. Proof. The Claim (1) is a direct consequence of Claims (2) and (3), because kh A 0 holds for the harmonic curvature of {Q' —> N, ui') and thus = 0 can hold only in the subset of the fiber corresponding to a (Zariski) closed subset of Q. The Claim (2) follows from [4, Theorem 3.3] and the fact that (q°/)s(l) = 0. Then Claim (3) is a clear consequence of Theorem 5.3. The Claim (4) follows from the comparison of images in Q' of the S'-invariant Weyl structure on N and the almost S'-invariant Weyl structures on Ux, because they intersect precisely in a reduction to exp(gs_(l)) x Go C Qo, i.e., in a subbundle with the structure group exp(gs_(l)) x Go- The Claim (5) is a consequence of Claim (4) of Theorem 5.3 and Claim (2). □ 5.4. Parabolic geometries with gs_1(l) = 0. There are triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s which admit a 1-eigenspace in 24 jan GREGOROVIČ AND LENKA ZALABOVÁ 0_ for some s such that 0S_1(1) = 0 holds, but which do not generically satisfy 01(1) = 0. These are contained in the Tables 5 and 6. Table 5. Theorem 5.7, part with |S| = 1. 0 S M «ru(l,2) {1} ri (ati, a2) su(l,3), £u(2,2) {1} 0i (ati, at3) su(l,3), su(2,2) {1} n (ati, ot2) su(q, n + 1 — q) {1} 0i (ati, an) su(q, n + 1 — q) {1} n (ati, ot2) su(q, n + 1 — q) {2} n (at2, ati) S0(3,5) {3} (ot3, ot2) sp(4,C) {1} 0i (ai,av) sp(6,{M,C}) {1} Vi (ati, ot2) sp(6,C) {1} 01 (ati, ot'i) sp(l,2),sp(6,{M,C}) {2} y/l (at2, ati) sp(2n, {M, C}) {1} Vi (ati, ot2) sp(2n,C) {1} 01 (ati, ot'i) sp(q,n-q),sp(2n,{R,C}) {2} Vi (at2, ati) 02({2,C}) {1} n (ati, ot2) Table 6. Theorem 5.7, part with |S| = 2. 0 S 3i\ Ji2 M sl(3,{M,C}) {1,2} (ati, ot2) äí(3,C) {1,2} (ati, ot2) «1(4, {K, C}) {1,3} if1 (ati, at3) sl(n+ í, {R, C}) {í,n} if1 (ati, Otn) so(2,3),so(5,C) {1,2} vr (ati, ot2) so(5,C) {1,2} j\ = 1 or jl = 1 i3 (ati, ot2) so(3,4) {1,3} n3 J3 (at3, ot2) Example 5.6. We see that partially integrable almost CR-structures of hypersur-face type are contained in the Table 5, i.e., 0 = su(q, n + 1 — q), q > 0, n > 1 and £ = {1}. With the exception of the case n = 2, there are two possible components of the harmonic curvature such that the triple (su(q, n+l — q),p^iy,n) is prolongation rigid outside of the 1-eigenspace of s for s e Z(Gq) with the specified eigenvalue. Moreover, fli(l) = 0-2 holds in all the cases when (su(q,n + 1 — q),p{i},n) is prolongation rigid outside of the 1-eigenspace of s. Let us emphasize that the possibility s3 = id is available for both components of the harmonic curvature. Since fli(l) = 0-2, we need some additional assumptions in Theorem 5.7 to show that (M, S) is (locally, under these assumptions) either a (locally) homogeneous one-dimensional fiber bundle over (reduced) l^-space, or a Z3-space or a symmetric space (due to [12, Proposition 7.3], see also [20]) that carries some 5-invariant Weyl connection on TM. In particular, all such parabolic geometries can be classified using [9, Theorem 5.1.4] and Theorem 5.7, if one knows the classification of §1-spaces, Z3-spaces and symmetric spaces. Let us emphasize that a part of the classification is done in [7]. SYMMETRIC PARABOLIC GEOMETRIES 25 As mentioned in the example, we need an additional assumption on where the local s-symmetries are defined for parabolic geometries in question. Theorem 5.7. Let (g, p, p) be prolongation rigid outside of the l-eigenspace of s for s G Z(Gq) such that gs_1(l) = 0 holds. Assume that for the (locally) s-symmetric parabolic geometry {Q —> M,uj) of type (G,P), the open subset U of M containing the points at which kh has a non-zero component in p is non-trivial, and the maps S_(x)(y) and S_{x) °S_(y)^1(z) are defined on neighbourhoods of diagonals in U x U and U x U x U for the unique system S of (local) s-symmetries on U. Then: (1) The parabolic geometry is (locally) homogeneous and U = M, i.e., kjj(x) 7^ 0 at all x G M and there is a unique smooth system of (local) s-symmetries S on M. (2) There is a class of distinguished Weyl structures characterized by one of the following equivalent properties for each Weyl structure a in the class: (a) The equalities WaTa = 0, s.(Ta)a = (Ta)a, WaRa = 0 and s.(Ra)a = {Ra)a hold for the torsion and the curvature of the Weyl connection V°\ (b) The Weyl connection VCT is S_-invariant. (c) All (local) automorphisms of the parabolic geometry cover affine transformations of VCT. (d) All (local) diffeomorphisms sx are affine transformations o/V°\ Two Weyl structures a and crexp(T)0- from the class differ by a Go-equivariant function (T)CT : Q0 —> p+(l) which is invariant with respect to all (local) automorphisms of the parabolic geometry and is provided by an invariant element o/p^_(l). (3) The pseudo-group generated by all local s-symmetries is transitive on M and its connected component of identity is generated by the flows of the Lie algebra I, which is the vector subspace o/g_ <3)Qo, generated by g_ by the bracket (Ta + Ra)a on A2gl ® I and the natural bracket on the rest of I. (4) The equalities S(x) = sff(uo) = sax hold for any Weyl structure a from (2). In particular, • the maps S_(x) can be extended to a larger neighbourhood of x as long as the corresponding geodesic transformations of VCT are defined, • for each eigenvalue a, the distribution TMs{a) is preserved by all (local) automorphisms of the parabolic geometry. (5) The distribution TMS{\) is integrable and for each x G M, the leaf Tx of the foliation T of TMS{\) through x is a totally geodesic submanifold for arbitrary Weyl structure. Let n : Ux —> N be a sufficiently small local leaf space ofTMs{\). (a) There is a unique local diffeomorphism S_(n(y)) of the local leaf space N at each n(y) G N such that S_(n(y)) o n = n o S.(x) holds for all y G Ux, and (b) for each eigenvalue a, Tyn(TyMs(a)) is the a-eigenspace of Tn^S_(n(x)) in Tn(y)N for all y eUx. Proof. The proof is similar to the proof of Theorem 5.3. However, we need a different method to prove the local homogeneity in Claim (3), because the existence of some S'-invariant Weyl structure does not follow from Proposition 4.3 anymore. Therefore we need an additional assumption on the system S on U in order to apply the following lemma. Nevertheless, the fact from Proposition 4.3 that S(x) = So-(Uo) holds for any almost S'-invariant Weyl structure a implies that the system S is smooth on U. 26 JAN GREGOROVIČ AND LENKA ZALABOVÁ Lemma 5.8. Suppose the smooth system of (local) s-symmetries S on M satisfies that the maps S_(x)(y) and S_(x) °S_(y)^1(z) are defined on neighbourhoods of diagonals in M x M and M x M x M. • If c(t) is a curve in M such that c(0) = x and £ := -^\t=oc{i), then the vector field Lt(y) :=||t=o5(C(t))o5(x)-1(y) is defined for y in some neighbourhood of x in M. • Then L^(y) is an infinitesimal automorphism of the parabolic geometry. • If £ is contained in the a-eigenspace of TxS_{x), then L^{x) = (1 — a)£. • The map £ i—> for £ G TXM is a linear map onto the Lie algebra of local infinitesimal automorphisms of the parabolic geometry. Its kernel consists of the l-eigenspace ofTxS_{x) in TXM, and it is injective on the sum of the remaining eigenspaces in TXM. Proof of Lemma 5.8. Since S(c(0)) o S(x)^1 = idg, there is a natural lift of L^(y) onto the P-invariant vector field jj:\t=oS(c(t)) o Si(x)_1(m) for u G G in the fiber over y. Since S(c(t)) o S(x)^1 is an automorphism, the vector field is P-invariant and -^\t=o(S(c(t)) o Slx)^1)*^ = 0. Thus L^(y) is an infinitesimal automorphism. Since S(c(t))(c(t)) = c(t), we conclude that L^x) + (5(x))*(£) = £. Thus L^(x) = £ — (5(x))*(£) and the claim follows due to the linearity of TxS_(x). □ Let us continue in the proof of Theorem 5.7. Since the map £ i—> L^ from Lemma 5.8 is injective on the bracket generating distribution given by g_i due to the assumption 0S_1(1) = 0, the local homogeneity follows from the regularity of the parabolic geometry. This implies Claim (1). Then Claim (4) follows again from Claim (2). Since we are on a (locally) homogeneous (locally) s-symmetric parabolic geometry, the parabolic geometry can be described as in Theorem A.l. It follows from [12, Theorem 1.3] that there is a if-invariant Weyl connection V on the if-homogeneous parabolic geometry described Theorem A.l such that all local automorphisms of the parabolic geometry are affine transformations of V. Therefore it follows from the last claim of Theorem A.l that the pullback of V to M does not depend on the local isomorphism with the if-homogeneous parabolic geometry. Therefore we obtain a Weyl structure a that satisfies (2c), which implies the remaining parts (2a), (2b) and (2d). It is clear that the if-invariant Weyl connection V from [12, Theorem 1.3] is not unique and the difference between two such Weyl structures is the claimed T provided by a if-invariant element of p^_(l). The Proposition 4.1 implies that the Weyl structure a satisfies (2b) if and only if it satisfies (2d). Again, results in [12, Theorem 1.3] imply that (2b) implies (2c) and the same arguments as in the proof of Theorem 5.3 show that (2b) implies (2a) and (2a) implies (2d). To prove Claim (5), we use the fact that s.(Tar(x))a = (Ta(x))a holds for the torsion of the 5-invariant Weyl connection V°\ Thus TMS(1) is involutive, because each (almost) 5-invariant Weyl connection V°" preserves TMS(1). Moreover, the formula for the difference between VCT and arbitrary Weyl connection implies that the difference in the parallel transport is an element of TMS(1) at each point of Tx. Thus Tx is a totally geodesic submanifold for any Weyl connection. We know that S_(x) = sx and this implies 5(x)|jrx = sj|jrx = idjrx. If v = Fl" ^X\u) for X G fli(l), then S(x)v = vs holds and y = p o Fl" ^X\u) G Tx, because Tx is a totally geodesic submanifold. Thus S_(x) is covered by the s-symmetry at y and S(x) = S(y) holds in some neighbourhood of x due to the symmetric parabolic geometries 27 uniqueness of s-symmetries. Consequently, Claim (5a) holds on a sufficiently small local leaf space and Claim (5b) is a clear consequence of Claim (4). □ 5.5. Parabolic geometries with gi1(l) + p in a distinguished parabolic sub-algebra. There are triples (g,p,/i) that are prolongation rigid outside of the 1-eigenspace of s that admit a 1-eigenspace in g_ for some s such that gs_1(l) + p C q C fli(l) + p holds for some parabolic subalgebra q of g such that the harmonic curvature vanishes on insertions of elements of q/p at all points of M. These are listed in the Tables 7, 8 and 9, due to [12, Proposition 6.2 and Proposition A.2]. Table 7. Theorem 5.10, part with |S| = 2. 0 Jn Ji2 M PR «1(4, C) {1,2} j'i («2,0^3) «1(4, C) {1,3} ("1,0:2) «u(2,2) {1,2} 2ri (ai, 0:3) n = 0 sl(n+ 1,C) {1,3} i 2 (0:1,0:2) sl(n+ 1,C) {2,3} i2 J2 (o2, Ol) sl(n+ 1,C) {l,n} i2 Jl (01, o2) sl(n+ 1,C) {2,n} i2 J2 (o2, Ol) su(n, n) {l,n} 2ri (01, o2„-i) 7-1=0 so (7, C) {1,3} jt (03, o2) «o(8, C) {1,3} i2 Jl (01, o2) «p(4,C) {1,2} 2ri (01, av) 7-1=0 sp(6,{M,C}) {1,2} (o2,Oi) j2 = -1 sp(6,{M,C}) {1,2} i2 Jl (01, o2) sp(6,{M,C}) {1,3} i2 Jl (oi,o2) «p(6,C) {1,3} 2ri (01, Ol/) n = 0 sp(6,{M,C}) {2,3} (o2,Oi) sp(2n, {R, C}) {1,2} IS (o2,Oi) j2 = -1 sp(2n, {R, C}) {l,n} 3l (01, o2) jl = -1 sp(f,f),sp(2n,{M,C}) {2,n} fi (o2, Ol) 32 = -1 sp(2n, {R, C}) {1,2} i2 Jl (01, o2) sp(2n, C) {l,n} 2ri (01, Ol/) ri = 0 Table 8. Theorem 5.10, part with |S| = 3. 0 S Jii Jl2 Jis M PR «1(4, C) {1,2,3} nil (o2, Ol) Ji = 1 sl(n+ 1,C) {1,2,3} iiii (o2, Ol) Ji = 1 sl(n+ 1,C) {l,2,n} iiii (o2, Ol) Ji = 1 sl(n+ 1,{M,C}) {l,P,7l}, p > 2 jljn (01, o„) Jl = On1 «o(8, C) {1,2,4} i2 Ji (01, o2) 32 = 1 sp(6,{M,C}) {1,2,3} jd'i (o2, Oi) 31 = 1 sp(2n, {R, C}) {l,2,p}, p < 71 (o2, Oi) Jl = 1, 32 = -l,jp = 1 sp(2n, {R, C}) {l,2,n} iiii (o2, Ol) Jl = 1, 32 = -1 Example 5.9. Let us focus on Lagrangean complex contact geometries, i.e., g = sl(n+ 1,C) and S = {l,n}. If we consider the triple (sl(n + 1, C), p{i,n}, ^(ai,a2)) 28 jan gregorovic and lenka zalabova Table 9. Theorem 5.10, part with |S| = 4. 0 £ eigenvalues M PR sl(n+ 1,C) {1,2,3, q}, q M,ui) of type (G,P). Assume the open subset U of M containing all points at which kh has a non-zero component in [i is non-trivial, and the maps S_(x)(y) and S_(x) o S_(y)~1(z) are defined on neighbourhoods of diagonals in U x U and U x U x U for the unique system S of (local) s-symmetries on U. Then: (1) The set U is an open dense subset of M and there is a unique smooth system of (local) s-symmetries S on M. (2) There are • a parabolic subgroup Q of G with the Lie algebra q such that P C Q, • a neighbourhood Ux of each x G M with a local leaf space n : Ux —> N for the foliation given by the integrable distribution Tp o a;_1(q), and • a (locally) s-symmetric parabolic geometry {Q' —> N,ui') of type (G,Q) satisfying the assumptions of Theorem 5.7 such that (Glii-c Uxt^uS) *s isomorphic to an open subset of {Q' —> Q'/P,u)') for each x. In particular, there is a unique s-symmetry S_(n(y)) of {Q' —> N, ui') at each n(y) G N such that n o S_(y) = S_(n(y)) o n holds for all y G Ux in the fiber over n(y). (3) The connected component of identity of the pseudo-group generated by all local s-symmetries is generated by the flows of the Lie algebra I, which is the vector sub-space ofqop, generated by q+p by the bracket (Ta(n(x)) + Ra(n(x)))a on A2(q°/)* 1 and the natural bracket on the rest of I for arbitrary S-invariant Weyl structure a on {G' -> N,w'). The pseudo-group generated by all local s-symmetries is locally transitive at x if and only if q/p C 1/(1 Hp), i.e., if and only if {R°'(n(x))) a spans the whole q/p. (4) There is a class of almost S-invariant Weyl structures on Ux given by reductions of the images in Q' of the S-invariant Weyl structures on N (that exist due to Theorem 5.7) to exp(gs_(l)) x Go C Qo- A reduction corresponds to an S-invariant Weyl structure on Ux if and only if it is a holonomy reduction. (5) We get that • the maps S_(x) can be extended to a larger neighbourhood of x as long as the corresponding geodesic transformations of VCT on N are defined, SYMMETRIC PARABOLIC GEOMETRIES 29 • the space TMS{\) is integrable, it contains the vertical space of the local leaf space n : Ux —> N, and Tn{TxMs(a)) is the a-eigenspace of Tn^S_(n(x)) in Tn{x)N, • for each eigenvalue a, the distribution TMs{a) is preserved by all (local) automorphisms of the parabolic geometry, and • all almost S_-invariant Weyl connections from Claim (4) restrict to the same partial linear connection on TM corresponding to the distribution Go XG0 tf+ > which is preserved by S_{x) for all x G M. Proof. The Claim (1) is a direct consequence of Claims (2) and (3). The Claim (2) follows from [4] and the fact that (q^p)s(l) = 0 holds. Then Claim (3) is a clear consequence of Theorem 5.7. The Claim (4) follows from the comparison of images in Q' of the S-invariant Weyl structure on N and the almost S-invariant Weyl structures on Ux, because they intersect precisely in the reduction to exp(gf_(l)) xi Go C Qo- The Claim (5) is a consequence of Claim (5) of Theorem 5.7 and the properties of Weyl structures from Claim (4). □ 5.6. Parabolic geometries with gs_1(l) that inserts non trivially into the harmonic curvature. There are also some remaining parabolic geometries, which can have a part of gs_1(l) that inserts non-trivially into the harmonic curvature. These are contained in the Tables 10, 11 and 12. Example 5.11. Let us continue in the discussion of generalized path geometries from the Example 5.4. The case when the harmonic curvature K(a2]ttl) vanishes and the harmonic torsion Q2) does not vanish can be found in the Table 10. There are several possible situations depending on the eigenvalues of s G Z(Gq). If ji = 1, then we are precisely in the situation which is not covered by any of the previous theorems and we can apply only the results of Propositions 4.3 and 5.12. If ji = —1 and .72 = 1, then we can apply Theorem 5.5 and we are in the situation of a generalized path geometry on the projectivized cotangent space of an affine locally symmetric space. If ji = —1 and J2 = —1, then fli(l) = jj-2 and we need the assumptions of Theorem 5.7 to show that we are in the situation of a generalized path geometry on a (locally) homogeneous (n — l)-dimensional fiber bundle over an affine locally symmetric space. Finally, if ji = —1 and j2 A VT, then we can apply Theorem 5.3. The properties of these geometries are as follows. Proposition 5.12. Assume (g,p,/i) is prolongation rigid outside of the l-eigenspace of s for some s G Z(Gq). Assume the harmonic curvature kh of a (locally) s-symmetric parabolic geometry {Q —> M, ui) of type (G, P) has a non-zero component in \i at all x G M and S is the unique system of (local) s-symmetries on M. Then the distribution TMS{\) is integrable and for each x G M, the leaf Tx of the foliation T of TMS{\) through x is a totally geodesic submanifold for arbitrary Weyl connection. Let n : Ux —> N be a sufficiently small local leaf space of TMS{\). • There is a unique local diffeomorphism S_(n(y)) of the local leaf space N at each n(y) G N such that S_(n(y)) o n = n o S_(x) holds for all y G Ux, and • for each eigenvalue a, Tyn(TyMs(a)) is the a-eigenspace of Tn^S_(n(y)) for all y G Ux. Proof. The proof is analogous to the proof of Claim (6) of Theorem 5.7, but, instead of an S'-invariant Weyl structure a, we need to consider some almost S-invariant Weyl structure invariant at x from Claim (3) of Proposition 4.3 for each x G M. □ 30 jan gregorovic and lenka zalabova Table 10. Theorem 5.12, part with |S| = 2. 0 k2 M «1(3, C) {1,2} 202 (ai, a2') «[(4,{R,C}) {1,2} Vl (ai, a2) «1(4, C) {1,2} 202 («1, a2/) «1(4, C) {1,3} 203 (ai, a3/) su(2, 2) {1,2} ri (a2,a1) su(2, 2) {1,2} #1 (oti,a2) sl(n+ 1,{M,C}) {1,2} VT (ai,a2) sl(n+ 1,{M,C}) {l,p}, 2 < p < n l (a1, ap) sl(n+ 1,C) {l.P} 20P («1, ay) su(g, n — q + 1) {1,2} (a2,ai) so(3,4),so(7,C) {2,3} («3,a2) «o(2,5), so(3,4), so(7,C) «o(2,6), so(3,5), so(4,4), so(7,C) {1,2} 1 (a2,ai) so(3,5) {2,3} (a3,a2) So(q, n — q), so(n, C) {1,2} l (a2,ai) sp(4, {M,C}) {1,2} #1 (ax,a2) sp(4, C) {1,2} 202 (a1, a2r) sp(4, C) {1,2} -102 (ai>,a2>) sp(6,{M,C}) {1,3} 1 (ai,a3) sp(6,{M,C}) {2,3} 1 (oi2,a3) sp(6,C) {1,3} 203 (a1, a3r) sp(6,C) {2,3} -1^3 {a2',a3>) sp(2n,{M,C}) {l,n} 1 (ai,an) sp(2n,{M,C}) {n — 1, n} 1 (dn_i, CYn) sp(2n,C) {l,n} 20n (a1, anr) sp(2n,C) {n — 1, n} (^n— 1' , <^n' ) 02({2,C}) {1,2} v7! (ai,a2) Table 11. Theorem 5.12, part with |S| = 3. 0 .7*2 3l3 M PR «1(4, {K,C}) {1,2,3} 3i (ai, a2) sl(n+ 1,{M,C}) {1,2,3} j{ («1, a2) sl(n+ 1,{M,C}) {l,2,p}, 33n — 3l32 (a2,ax), (an-x,an) 5.9. Remaining parabolic geometries with /z in the l^igenspace of s. For the sake of completeness, let us remark that there are triples (g, p, /z) that are not prolongation rigid outside of the 1-eigenspace of s for any s such that /z is in the 1-eigenspace of s. These are contained in the Table 15. Table 15. Remaining parabolic geometries with /z in the 1-eigenspace of s. 0 £ Jt2 P, sl(n + 1,{M,C}) {p,p + 1}, n—1 > p > 1 1 (ap+1,ap) So(q, n — q),so(n, C) {2,3} 1 (at3,at2) 32 jan GREGOROVIČ AND LENKA ZALABOVÁ Appendix A. A construction of locally homogeneous locally s—symmetric parabolic geometries It is proved in [11, Section 2] how to algebraically construct and classify all homogeneous s-symmetric parabolic geometries. Part of the classification is done in [7, 8] using the classification of semisimple symmetric spaces. There is the result from [9, Section 1.3] and [11, Lemma 2.2] stating that for the construction and the classification of locally homogeneous locally s-symmetric parabolic geometries, it is sufficient to find the following data: • an extension (a, i) of the Klein geometry (K, H) to (G, P) such that the action of s preserves a(£) C Q, and s acts trivially on the tensor [•, •] — a^a-1 (■), a-1 (■)]) in A2fl/p* (8)0, and • the subset A of P consisting of elements g G P, which act as local automorphisms on the parabolic geometry (K x j P —>• K/H, uja) of type (G, P) given by the extension (a,i). If U and V are open subsets of K/H such that there are k G K, g G A and a maximal open subset W of U such that kg(W) C V, then we can glue KXiP\jj —>• U with K Xj P\v —>• V by identifying w G C £/ with kg(w) G V, and glue the Cartan connection uja\u with the pullback connection (kg)*uja\y = uja\(kgY(v)-Of course, we can without loss of generality assume that U, V and W are simply connected, because we can always choose coverings of our manifolds by open sets satisfying this condition. Therefore, we can also assume that the automorphism k is given by the flow of a local infinitesimal automorphism of (K x j P —>• K/H, uja). Then we obtain the following result as a consequence of the construction in [12, Section 3] and [9, Section 1.3]. Theorem A.l. Let (Q —> M,uS) be a locally homogeneous locally s-symmetric parabolic geometry, let t be the Lie algebra of the local infinitesimal automorphisms and denote by a the inclusion oft into q given by uj(u) at some u G Q. Then: (1) Ad(s)(£) C t is an automorphism of the Lie algebra t, (2) there exist (see [12, Section 3] for the explicit construction) • a Klein geometry (K, H) such that t is the Lie algebra of K, • an extension (a,i) of (K,H) to (G,P), • an open covering Ua of M, and • isomorphisms (pa : Ui —> K/H of parabolic geometries (G\ua Ui,uj\jjb) and (K xt P|0Q(;7a) ->• 4>a{Ua), ^a|a(!7a)) of type (G,P) such that a ° ^i,1 *s the restriction of the left action of some element of K for each a, b. Acknowledgments. First author supported by the Grant agency of the Czech Republic under the grant GBP201/12/G028. The authors would like to thank the anonymous referees for their valuable comments which helped to improve the manuscript. References [1] A.L. Besse, Einstein manifolds, Classics in Mathematics, Springer—Verlag, Berlin, 1987. [2] P. Bieliavsky, E. Falbel, C. Gorodski, The classification of simply-connected contact sub- Riemannian symmetric spaces, Pac. J. Math. 188 (1999), 65—82. [3] A. Cap, J. Slovak, Parabolic geometries I: Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc, 2009. [4] A. Cap, Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172. [5] A. Derdzinski, W. Roter, Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds, TMJ 59 (2007), 565-602. [6] J. Gregorovic, Local reflexion spaces, Arch. Math. 48 (2012), 323-332. SYMMETRIC PARABOLIC GEOMETRIES 33 [7] J. Gregorovič, General construction of symmetric parabolic geometries, Differential Geom. Appl. 30 (2012), 450-476. [8] J. Gregorovič, Classification of invariant AHS-structures on semisimple locally symmetric spaces, Cent. Eur. J. Math. 11 (2013), 2062-2075. [9] J. Gregorovič, Geometric structures invariant to symmetries, FOLIA Mathematica 18, Masaryk University, 2012. [10] J. Gregorovič, L. Zalabová, Symmetric parabolic contact geometries and symmetric spaces, Transform. Groups 18 (2013), 711-737. [11] J. Gregorovič, L. Zalabová, On automorphisms with natural tangent action on homogeneous parabolic geometries, J. Lie Theory 25 (2015), 677—715. [12] J. Gregorovič, L. Zalabová, Geometric properties of homogeneous parabolic geometries with generalized symmetries, Differential Geom. Appl. 49 (2016), 388—422. [13] J. Gregorovič, L. Zalabová, Notes on symmetric conformal geometries, Arch. Math. (Brno) 51 (2015), 287-296. [14] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, Vol. 34, 2001. [15] W. Kaup, D. Zaitsev, On symmetric Cauchy-Riemann manifolds, Adv. Math. 149 (2000), 145-181. [16] S. Kobyashi, K. Nomizu, Foundations of differential geometry, Vol. 2, John Wiley and Sons, 1963. [17] B. Kruglikov, D. The, The gap phenomena in parabolic geometries, J. Reine Angew. Math. 723 (2017), 153-215. [18] O. Kowalski, Generalized symmetric spaces, Lecture Notes in Mathematics, Vol. 805, Springer—Verlag, 1980. [19] O. Loos, Spiegelungsraume und homogene symmetrische R' aume, Math. Z. 99 (1967), 141— 170. [20] O. Loos, An intrinsic characterization of fibre bundles associated with homogeneous spaces defined by Lie group automorphisms, Abh. Math. Sem. Univ. Hamburg 37 (1972), 160-179. [21] F. Podesta, A class of symmetric spaces, Bulletin de la SMF 117 (1989), 343-360. [22] R.F. Reynolds, A.H. Thompson, Projective-symmetric spaces, J. Austral. Math. Soc. 7 (1967), 48-54. [23] L. Zalabová, Symmetries of parabolic geometries, Differential Geom. Appl. 27 (2009), 605— 622. [24] L. Zalabová, Parabolic symmetric spaces, Ann. Glob. Anal. Geom. 37 (2010), 125—141. [25] L. Zalabová, Symmetries of parabolic contact structures, J. Geometry Phys. 60 (2010), 1698— 1709. J.G. E. Čech Institute, Mathematical Institute of Charles University, Sokolovská 83, Praha 8 - Karlín, Czech Republic; L.Z. Institute of Mathematics and Biomathemat-ics, Faculty of Science, University of South Bohemia, Branišovská 31, České Budějovice 370 05, Czech Republic Email address: jan.gregorovic@seznam.cz, lzalabova@gmail.com Chapter 3: On Symmetric CR Geometries of Hypersurface Type article by Jan Gregorovic and Lenka Zalabova, published in The Journal of Geometric Analysis (2018), DOI:10.1007/sl2220-018-00110-l, 25 pp. arXiv: 1707.07531 ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE JAN GREGOROVIČ AND LENKA ZALABOVÁ Abstract. We study non—degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either fiat or homogeneous. We show that non—fiat non—degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo—Riemannian metric preserved by all symmetries. We construct examples of simply connected fiat non—degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo—Riemannian metric compatible with the symmetries. 1. Introduction In [10], Kaup and Zaitsev generalized Riemannian symmetric spaces to the setting of CR geometries, i.e., smooth manifolds with so-called CR distribution endowed with a formally integrable complex structure. They consider Riemannian metrics, whose restrictions to the CR distribution is Hermitian. Such manifold is symmetric in the sense of [10] if, at each point, there is an isometric CR transformation that preserves the point and which, at that point, acts as —id on the CR distribution [10, Definition 3.5.]. They show that such isometric CR transformations are uniquely determined by the tangent action on the CR distribution [10, Theorem 3.3]. They also show that such CR geometries are homogeneous [10, Proposition 3.6]. In fact, these CR geometries may be considered as reflexion spaces in the sense of [12]. In [1], the authors study these CR geometries in the setting of so-called CR algebras. We studied in [9] filtered geometric structures that carry an automorphism at each point that fixes the point and acts as —id on a distinguished part of the filtration at the point. Let us point out that the non-degenerate CR geometries of hypersurface type, i.e., those with CR distribution of codimension 1, are among these geometries. We answered the question whether these filtered geometries are homogeneous and can be considered as reflexion spaces. However, our result [9, Theorem 5.7.] holds under weaker conditions than the result of [10] for non-degenerate CR geometries of hypersurface type. In particular, the sufficient condition for such non-degenerate CR geometry of hypersurface type to be homogeneous is that it is non-flat at one point. In this paper, we study the case of non-degenerate CR geometries of hypersurface type in more detailed way. We consider point preserving CR transformations which, at that point, induce —id on the CR distribution. We say that a non-degenerate CR geometry of hypersurface type is symmetric (in our sense) if there exists a symmetry at each point, see Definition 1. In particular, our definition does not require the existence of a metric compatible with the CR geometry. We adapt and significantly 2010 Mathematics Subject Classification. 32V05, 32V30, 53C30. Key words and phrases. CR geometry, homogeneous manifold, Webster metric. First author supported by the project P29468 of the Austrian Science Fund (FWF). Second author supported by the grant 17-01171S of the Czech Science Foundation (GACR). 1 2 JAN GREGOROVIČ AND LENKA ZALABOVÁ improve general results of [5, 8, 9] for our particular class of CR geometries. Let us emphasize that every non-degenerate CR geometry of hypersurface type that is symmetric in the sense of [10] is symmetric (in our sense). Let us say that [9, Theorem 5.7.] is formulated in the general setting of parabolic geometries. We provide here the particular results of this theorem for CR geometries. We also provide new direct proofs, because we will need the presented ideas to explain new results, see Lemmas 2, 3 and Propositions 1, 2. This allows us to compare our results with results of [10] and [1]. We prove in Theorem 2 that non-flat non-degenerate CR geometries of hypersurface type that are symmetric (in our sense) are covered by symmetric non-degenerate CR geometries of hypersurface type that carry a pseudo-Riemannian metric compatible with the CR geometry that is preserved by all our symmetries. In the Riemannian signature, these coverings are symmetric in the sense of [10], see Theorem 3. Moreover, we show in Theorem 4 that it is always possible to embed the CR geometry on these coverings into a complex manifold. In the Riemannian signature, this embedding is provided by a different construction than the one in [10, Proposition 7.3]. Finally, we construct examples of non-homogeneous symmetric (in our sense) flat non-degenerate CR geometries of hypersurface type. These examples do not admit a pseudo-Riemannian metric that would be preserved by some symmetry at each point and in particular, they are not symmetric in the sense of [10]. We also discuss examples of homogeneous CR geometries on orbits of real forms in complex flag manifolds. In particular, we show that there are homogeneous CR geometries which are locally symmetric but not globally symmetric. In fact, Proposition 8 provides complete description of all possible cases. 2.1. CR geometries. Let M be a smooth manifold of dimension 2n+ 1 for n > 1 together with a distribution % C TM of dimension 2n and a complex structure J on %, i.e., J : % —> % is an endomorphism with the property that J2 = —id. The triple (M, %, J) is called a CR geometry of hypersurface type if the z-eigenspace n1'0 of J in the complexification of U is integrable, i.e., [U1'0, "H1-0] C H1'0. A CR geometry (M, %, J) is called non-degenerate if % is completely non-integrable. On % there exists a symmetric bilinear form h with values in the line bundle TM/H given by /i(£,t?) = §tt([£, Jr]]) for all £,77 e T(M,H), where tt : TM ->• TM/H is a natural projection. Let us recall that h is the real part of the Levi form h of (M,7i,J) whose imaginary part is the map given by §7r([£,?/])■ We assume that M is orientable and denote by (p, q) the signature of the Levi form, where our convention is p < q, p + q = n. Then the signature of h is (2p, 2q). The homogeneous space PSU(p+l, q+l)/P is usually called the standard model of a non-degenerate CR geometry of hypersurface type of signature (p,q), where P in PSU(p + 1, q + 1) is the stabilizer of the complex line generated by the first basis vector in the standard basis of C™+2. We use the convention that elements of PSU(p + 1, q + 1) are represented (up to a multiple) by matrices preserving the Hermitian form given for u,v E C™+2 by where I is a square matrix of order n defining a Hermitian inner product of signature (p, q). In [2], the matrix I is chosen to be diagonal with the first p entries equal to 1 and the remaining q entries equal to —1. 2. CR GEOMETRIES OF HYPERSURFACE TYPE u T 0 l\ 1 0 ]v, 0 0/ on symmetric cr geometries of hypersurface type 3 The standard model PSU(p+l, q + l)/P is a smooth real hypersurface in CPn+1 that can be also viewed as the projectivization of the null cone of m in C™+2. In the rest of the paper, by a CR geometry we mean a non-degenerate CR geometry of hypersurface type of signature (p, q) for p < q. Such CR geometries can be equivalently described as parabolic geometries modeled on standard models PSU(p + 1, q + 1)/P. This description can be found in [2, Section 4.2.4]. We only use several consequences of this description later in the text. 2.2. Distinguished connections. There exist many admissible connections, i.e., connections preserving % and J, on CR geometries. In particular, there are several distinguished classes of admissible connections given by a particular normalization condition on the torsion of admissible connections in the class. The most commonly used are Tanaka-Webster [2, Section 5.2.12] and Weyl [2, Sections 5.1.2 and 5.2.13] connections. In this paper we will use the latter, since in some arguments we need to relate CR and geodesic transformations for the so-called normal Weyl connection [2, Section 5.1.12]. In fact, Tanaka-Webster connections and Weyl connections induce the same class of distinguished partial connections V on %. Two distinguished partial connections V and V are related by (1) Vc(r?) = Vc(r?) + F(Ov + F(r,)£ - h(£, tfh^iF) for all £, r, e T(M, H), where F is a one-form on % and h is the duality map associated to the real part of the Levi form. Note that while the duality map depends on the choice of a local trivialization of TM/H, the composition h(£, rf)h~1(F) does not depend on this choice. We will write for short "V = V + F" for the partial connection described by (1). Each Weyl connection D is associated to the choice of decompositions TM ~ H © £, T*M ~ H* © I*, for a one dimensional complementary distribution £ of H and with %*,£* being the annihilator bundles of £,%, respectively. The one-form F in H* from formula (1) then describes the difference of the two decompositions of TM (and T*M) associated with D and D. The precise formula for the change of decompositions can be easily computed using [2, Section 5.1.5]. The explicit formula for the relation between two arbitrary Weyl connections D and D is complicated and can be computed using [2, Section 5.1.6]. This relation is given by an action of a one-form T = Ti + T2 in T*M = H* © £* on the space of Weyl connections, which we write as D = D + Ti + T2, because this action is linear in T. Let us point out that the decomposition Ti + T2 of T is the decomposition associated with the Weyl connection D and that Ti coincides with F from formula (1) for the corresponding partial connections V, V determined by D, D. Let us finally point out that admissible connections provide the fundamental invariant W of CR geometries which is known as Chern-Moser or Weyl tensor and coincides with the totally trace-free part of the curvature of arbitrary Weyl or Tanaka-Webster connections. Vanishing of this invariant implies that a CR geometry is flat, meaning that is locally equivalent to the standard model PSU(p + \,q+\)/P. 3. Symmetries of CR geometries 3.1. Definition of symmetries. A CR transformation of a CR geometry (M, H, J) is a diffeomorphism of M whose tangent map preserves the CR distribution % and whose restriction to H is complex linear. We study the following CR transformations. 4 JAN GREGOROVIČ AND LENKA ZALABOVÁ Definition 1. A symmetry at x G M on a CR manifold (M, H, J) is a CR transformation Sx of M such that: (1) Sx(x) = x, (2) = -id on U. We say that a CR geometry is symmetric if there is a symmetry at each point x G M. A system of symmetries on M is a choice of a symmetry Sx at each x G M. We call the system smooth, if the map S : M x M —>• M given by S^x, y) = Sx(y) is smooth. Let us show that the standard model PSU(p + 1, g + 1)/P is symmetric. The Lie group PSU(p+ l,q+l) is the group of all CR transformations of the standard model PSU(p + l,q + 1)/P, where we consider left action. Direct computation shows that all symmetries of the standard model PSU(p+ l,q+l)/P at the origin eP are represented by (l,n, l)-block matrices of the form I-I -Z iz+\ZIZ* (2) sz,z =0 E -IZ* \ 0 0 -1 where Z G C™*, z G M* are arbitrary and E is the identity matrix of the rank n = p + q. Lemma 1. There are infinitely many symmetries at each point kP of PSU(p + l,q+ 1)/P given by matrices of the form ksz^kX for all Z G C™* and z£l*. In particular: (1) There are infinitely many involutive symmetries at each point characterized by the condition z = 0. For each such symmetry, there is a different metric preserved by this symmetry compatible with the CR geometry. (2) There are infinitely many non-involutive symmetries at each point characterized by the condition z =/= 0. They do not preserve any metric compatible with the CR geometry. Proof. An element sz,z belongs to a compact subgroup of PSU(p +l,q+l) if and only if z = 0. □ The standard model PSU(p+ l,q+ 1)/-P is endowed with a pseudo-Riemannian metric compatible with the CR geometry given by a choice of a maximally compact subgroup of PSU(p+l, q+1), which always acts transitively on the standard model. Moreover, there is exactly one involutive symmetry at each point of this model that is contained in the chosen maximal compact subgroup. These symmetries preserve the corresponding pseudo-Riemannian metric and form a smooth system. This means that in the Riemannian signature, the standard model PSU(l,n + 1)/P is symmetric in the sense of [10]. On fiat CR geometries, local symmetries are defined as in the standard model, but may not extend globally. This means that on fiat CR geometries, there is always locally a pseudo-Riemannian metric compatible with some local CR symmetry at each point. We show in Example 1 that such pseudo-Riemannian metrics, compatible with a system of CR symmetries, may not exist globally. 3.2. Involutive and non—involutive symmetries. Suppose that there is a symmetry Sx at x on a CR geometry (M,H, J). If D is a Weyl connection, then S*D is a Weyl connection, too. Therefore, there is a one-form Ti + T2 G H* ® t* such that (3) SID = D + T1+T2. ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE 5 Lemma 2. Suppose Sx is a symmetry at x G M. Let D be an arbitrary Weyl connection and let Ti + T2 G W © £* be the one-form from formula (3). Then the following claims are equivalent: (1) the symmetry Sx is involutive, (2) T2(x) = 0, and (3) the diffeomorphism Sx is linear in the normal coordinates given by the normal Weyl connection D at x that is uniquely determined by the property that D coincides with the Weyl connection D + ^Ti at x. Moreover, the partial connection V5^ induced by the Weyl connection DSj: := D + ^Ti does not depend on the choice of D at x and satisfies • ^(V5*) = V5a= at x, and • Vs*W(x) = 0. Proof. Iterating the formula (3) we compute S*XS*XD = D + T1 + ^(Tfi) + T2 + S£(T2). The component of the (dual) action of TXSX on T*M preserving the decomposition T*M = H*(x) © t(x) is -id © id, and the component that maps H*(x) into t(x) depends linearly on Ti and is antisymmetric as a map H*(x) • £*(x). Therefore, S,*(T1)(x) = -Tx(x) and S*(T2)(x) = T2(x). If the symmetry Sx is involutive, i.e., Sx = id, then 0 = T2(x) + S*x(T2)(x) =2T2(x) and thus T2(x) = 0. If T2(x) = 0, then the normal Weyl connection D that coincides with the Weyl connection D + |Ti at x satisfies s:(D + ±r1) = D + r1 + s:(±r1). At the point x, we get Ti(x) + S^TCiX*) = T^x) - It^o;) = ^Ti(x) and thus S*D = D follows from the normality [2, Section 5.1.12]. Thus Sx is an affine map, which is linear in the normal coordinates. If the symmetry Sx at x is linear in the normal coordinates of a Weyl connection, then its (dual) tangent action preserves the decomposition T*M = H*(x) © i*(x) and therefore (TXSX)2 = id. Then it follows from the linearity that Sx is involutive. Finally, the last claim follows, because (V5^ W) (x) is a tensor of type ®4'HX®'HX invariant with respect to Sx. □ Lemma 3. Suppose that there is a symmetry Sx at x G M. Let D be an arbitrary Weyl connection and let Ti + T2 G %* © £* be the one-form from formula (3). If W(x) =/= 0, then T2(x) = 0 and the symmetry Sx is involutive. Proof. Consider the covariant derivative of W with respect to D + |Ti in the direction £ and compute S*(D + ^Ti)rW(x) for r G £(x). We know that W(x) is S'aj-invariant and thus S*X(D + ±?l)rW(x) = (D+ \?l)sz(r)W(x) = (D+ ^TJrWix). On the other hand, it generally holds that S*(D + ^Ti) = D + |Ti + T2 and TT2(a;) = a6(x) for a covector 9 G £*{x) such that 6{r) = 1. Then SX(D + ^Ti)rW(x) = (D+ ^TJrWix) + 2aW(x) and thus a = 0 which implies T2(x) = 0. □ 6 JAN GREGOROVIČ AND LENKA ZALABOVÁ 3.3. Smooth systems of involutive symmetries. Let us show that the assumption that W does not vanish at any point not only implies that all symmetries are involutive, but also that at each point of M there is exactly one symmetry Sx and the Sx's make a smooth system on M: Proposition 1. Suppose that {M,%,J) is a symmetric CR geometry and that W(x) ^ 0 for all x G M. Then (1) there is a unique symmetry Sx at each x G M, (2) the map S : x i—> Sx is smooth, and (3) Sx o Sy o Sx = Ssx(y) holds for all x,y G M. In particular, (M, S) is a reflexion space, i.e., S : M x M —> M is a smooth map that for all x,y, z G M satisfies that • S(x, x) = x, • S(x, S(x,y)) = y, and • S(x,S(y,z)) = S(S(x,y),S(x,z)). Proof. We show that if there are two different symmetries at x on a CR geometry (M,H,J), then W vanishes at x. Consider two different symmetries Sx and S'x at x (both must be involutive). We know from Lemma 2 that VS:cW{x) = 0 and Vs*W{x) = 0 hold for partial connections V5^, Vs*. These partial connections are different (at x) due to the claim (3) of Lemma 2, i.e., V5* = V5^ +F holds according to the formula (1) for F(x) ^ 0. This means that the linear map 1~LX —^ given by (4) v ^ (F(0v + F(r,)£ - h(£,i])h^1(F))(x) defines a non-zero element £(F)(x) of a Lie algebra csu(p, q) for each £ G Hx, where we identify csu(p, q) with {X G Ql(Hx) : [X, Jx] = 0, hx{X{i),v) + hx{i, X{y)) = a ■ hx{i, v),a G R}. Moreover, the element £(F)(x) of csu(p, q) has to act trivially on W(x) for all vectors £. Let us denote by ann(Wa,.) the set of all A G csu(p, q) such that A acts trivially on W(x). Then we get F(x) G ann(Wa;)(1) := {F : £(F)(x) G ann(Wx) for all £ G Hx}. The result of [11] states that if W{x) is non-trivial, then ann(Wa,.)^1^ = 0, and thus tl(F)(x) = 0 for all £ G Hx. Since £(—)(x) : H* —>• csu(p, q) is a linear map at each x G M, this implies F(x) = 0, which is a contradiction. This proves the uniqueness of symmetries at x in the case W(x) ^ 0. Since Sx o Sy o Sx is a symmetry at Sx(y), the condition Sx o Sy o ^ = Sg^^ trivially follows from the uniqueness of symmetries. Thus it remains to prove the smoothness of S. Let us fix a partial Weyl connection V. For each y G M, there is F(y) such that (\7sy — V)(y) = by the formula (1), which is well-defined due to the uniqueness of V5» at y. Thus WW(y) is given by the algebraic action (4) of £(F(y)) on W(y) for each £ G 7^. Since WW(y) is smooth, the image of £(F(y)) in csu(p, q) depends smoothly on y for each £ G "H^. Since the kernel of the action coincides with ann(W2/)^1^, we conclude that F(y) depends smoothly on y. Let D be an arbitrary Weyl connection inducing the partial Weyl connection V. Then Sy is linear in the normal coordinates of the normal Weyl connection D constructed for D + \F(y) due to the claim (3) of Lemma 2. Since D depends smoothly on y, we get that S is smooth. It clearly holds that Sx(x) = x and Sx = id for all x G M. We have proved that S is smooth and satisfies Sx o Sy = Ss^^y) ° S^1 = Ss^^y) ° Sx for all x, y G M. Thus it follows that (M, S) satisfies the conditions of the reflexion space. □ on symmetric cr geometries of hypersurface type 7 Proposition 1 has the following consequence. Proposition 2. Suppose that (M, %, J) is a symmetric CR geometry. Then either (1) W = 0 and the CR geometry is locally equivalent to the standard model, or (2) W =/= 0 and the CR symmetries generate a transitive Lie group of CR transformations of M. In particular, the CR manifold (M,7i,J) and the reflection space (M, S) from Proposition 1 are homogeneous. Proof. Suppose that U C M consists of all points with non-trivial W. It suffices to prove that the group generated by symmetries at points in U acts transitively on U to obtain the claim of the Theorem, because then W is constant on U due to the homogeneity. The fact that the group generated by symmetries on a reflexion space is a Lie group can be found in [12]. Let c(i) be a curve in U such that c(0) = x and -^\t=oc(t) = X G TA.X. Then ■^\t=oSc(t)(x) is tangent to the orbit of the action of the group generated by symmetries at points in U. Differentiation of the equality c(t) = Sc(t)c(t) gives X = ^\t=oSc(t)(c(i)) = ^\t=oSc(t)(x) +TXSX.X, and we get jt\t=0Sc{t){x) = X- TXSX.X = 2X. Thus at all x G U, the CR distribution H is tangent to the orbit of the group generated by symmetries at points in U. Therefore the group generated by symmetries at points in U acts transitively on U. □ Flat symmetric CR geometries do not have to be homogeneous. We construct an explicit example in Section 6. 4. Non-flat symmetric CR geometries 4.1. Homogeneous CR geometries and their symmetries. There are several possible ways of describing a homogeneous CR geometry. We will use the description from [2, Section 1.5.15] that is closely related to the setting of Cartan geometries, but as we show in this section, it can be treated independently of the general theory. We need only to recall that the Lie algebra su(p + l,q + 1) of PSU(p + 1, q + 1) consists of the (1, n, l)-block matrices a Z iz \ X A -IZ* , ix —X*I —a / where csu(p, q) = {(a, A) : a G C, A G u(p,q), a + tr(A) - a = 0}, X G C™, Z G C™* ,i£l and z G R*. This means that we have the following decomposition SU(P + 1, q + 1) = R © Cn © CSu(p, q) © C™* © R*. The Lie algebra p of P corresponds to (l,n, l)-block upper triangular part and decomposes as p = csu(p,q) © Cn* © R*. In fact, P = CSU(p, q) exp(C™* © R*), where CSU(p, q) consists of all elements of P whose adjoint action preserves the above decomposition. Lemma 4. Let K be an arbitrary transitive Lie group of CR transformations of a homogeneous CR geometry (M, %, J) and let L C K be the stabilizer of a point. Then there is a pair of maps (a,i) such that i is an injective Lie group homomor-phism i : L —> P and a is a linear map a : I —> su(p + l,q + 1) satisfying the following conditions: (1) a : I —> su(p + 1, q + 1) is a linear map extending Tei : I —> p, 8 JAN GREGOROVIČ AND LENKA ZALABOVÁ (2) a induces an isomorphism a : t/l —> su(p + 1, q + l)/p of vector spaces, (3) Ad(i(l)) oa = ao Ad(l) holds for all I G L, (4) the linear map A2t —> su(p + 1, q + 1) given on X AY by [a(X), ct(Y)] — a([X,Y]) for all X,Y G t takes values in p and defines a K-invariant two-form k on M with values in K XAdoi p, (5) the component of n in K XAdoi csu(p, q) is a tensor that coincides with W, where Ad is the induced action of P on csu(p, q) = p/(C™* ® M*). Conversely, suppose that (a,i) is such pair of maps from (K,L) to (PSU(p + 1,<7+ 1),P). Then there is a K-homogeneous CR geometry (K/L,7i,J) satisfying %eL = o:_1(C™ ® p)/l and JeL = a*(J), where J is the complex structure on C™. A pair (a, i) satisfying conditions (l)-(3) of Lemma 4 is usually called an extension of (K,L) to (PSU(p + l,q+ 1),P). The two-form n from condition (4) is the curvature of the Cartan connection given by the extension (a,i). Finally, the normalization condition (5) on the curvature n can also be expressed by the vanishing (d*k = 0) of Kostant's codifferential [2, Section 3.1.11]. Proof. It is shown in [2, Section 1.5.15] that each homogeneous Cartan (and thus parabolic) geometry can be described by a particular extension and that each extension determines a homogeneous Cartan geometry. The formula for n in condition (4) is obtained from [2, Section 1.5.16]. Therefore, it follows from the description of CR geometries in [2, Section 4.2.4] that conditions (4) and (5) on the curvature k have to be satisfied. □ Definition 2. The pair (a, i) from Lemma 4 is called a normal extension of (K, L) to (PSU(p + 1, q + 1), P) describing the homogeneous CR geometry (M, H, J). Examples of normal extensions describing homogeneous CR geometries and the explicit formula from condition (5) of Lemma 4 can be found in [4]. It is clear from the second part of Lemma 4 that the maps i and a wee sufficient to determine a CR geometry. This means that there are many normal extensions (a, i) of (K, L) to (PSU(p + 1, q + 1),P) describing the same CR geometry. The other parts of a are completely determined by condition (5) from Lemma 4 and carry the information about Weyl connections. The remaining freedom (for fixed i) is in the choice of a complex basis of a_1(C"). In general, if h G P, then the pair (Ad(h) oa, conj(h)oi) is also a normal extension of (K, L) to (PSU(p + l, q+ 1), P) describing the same CR geometry. Let us summarize the results characterizing symmetric non-flat homogeneous CR geometries following from [7, 8]. Proposition 3. Let K be the Lie group of all CR transformations of a non-fat homogeneous CR geometry (M, %, J). Then the following is equivalent: (1) There is a (unique) symmetry at each point. (2) There is s G L such that the triple (K, L, s) is a (non-prime) homogeneous reflexion space, i.e., • s commutes with all elements of L, • s2 = e, where e is the identity element of L, and • all symmetries are of the form SkL = ksk^1 for k G K. (3) There is a normal extension (a,i) of (K,L) to (PSU(p + l,q+ 1),-P) describing (M,7i,J) such that i(L) C CSU(p,q) and so,o G i(L) (see the formula (2)). (4) For each normal extension (a,i) of (K,L) to (PSU(p + l,q+ 1),-P) describing (M,7i, J), there is a (unique) Z G C™* such that Ad(exp(Z))a(V) is preserved by Ad(so,o); and the Lie algebra automorphism oft given by Ad(sp,o) defines an automorphism of the Lie group K. ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE 9 The condition (3) of Proposition 3 immediately implies that there are .ft-invariant Weyl connections on a symmetric non-flat CR geometry (M,H, J). According to [2, Proposition 1.4.8], a ^-invariant connection on T(K/L) can be described by a map 7 : £ —>• fll(t/0 such that • 7|t = ad, and • 7(Ad(/i)(X)) = Ad(X> ° 7p0 ° AdfXT1 hold for all AT G £ and h E L, where Ad : L —>• Gl(t/i) is induced by the adjoint representation. Proposition 4. Let K be the Lie group of all CR transformations of a non-flat symmetric CR geometry (M,7i, J). Let (a,i) be a normal extension of (K,L) to (PSU(p + l,q + 1),P) describing (M,H,J) such that i(L) C CSU(p,q) and so,o G i(L). Then 7 := a* (adorn) describes a K-invariant Weyl connection, where r0 : su(p + 1, q + 1) —>• csu(p, q) is the projection along R © C™ © C™* © M*. In particular, there is a bijection between the set of K-invariant Weyl connections on M and the set of z G R* such that conj(exp(z)) o i(L) C CSU(p, q) holds for the extension (a,i). Proof. We proved the existence of If-invariant Weyl connections on non-flat symmetric CR geometries in [8]. Therefore it is enough to check that they can be described by the functions 7. Since i(L) C CSU(p,q), the projection rn is i(L)-equivariant and 7|[ = ad holds. Therefore each 7 describes a if-invariant connection. The fact that this is a Weyl connection follows directly from condition (5) in Lemma 4. It is clear that the one-form T1 + T2 measuring the "difference" between two K-invariant Weyl connections is given by an z(L)-invariant element of C™* ffiM*. Since sn.n G i(L), it has to be an element of R*. It is clear that z G R* is z(L)-invariant element if and only if conj(exp(z)) o i(L) C CSU(p, q) holds. □ 4.2. Groups generated by symmetries. The following Theorem significantly improves the characterization of non-flat symmetric homogeneous CR geometries given by Propositions 2 and 3. Theorem 1. Let K be the Lie group generated by all symmetries of a non-flat symmetric CR geometry (M,7i,J). Let (a,i) be a normal extension of(K,L) to (PSU(p + 1,<7+ 1),-P) describing the CR geometry that satisfies i(s) = s0q and i(L) C CSU(p,q). Denote by f) the l-eigenspace of s in £ and by m the — 1-eigenspace of s int. Then: (1) The following conditions hold • a(\) C u(p,q), • a(m) C C"©C™*, and • a(\)) Clffl csu(p, q) Q)R* is a Lie subalgebra. (2) There is a basis o/f)/l©m such that for a vector in f)/l©m with coordinates (x,X) holds laix Pi(X) P2(2^ \ a((x, X) + I) = Ad(exp(z)) o I X Ax -IP^X)* + a(l), \xi —X* I aix J where z eR*, Px:Cn^Cn*, P2 : R ->■ R* and (a, A) G u(p, q) normalizes (3) The maps Pi,P2 and the matrix (a, A) are completely determined by condition (5) from Lemma 4- Proof. We know from Proposition 3 that there exists a normal extension (a,i) of (K, L) to (PSU(p + 1, q + 1), P) satisfying our assumptions. 10 JAN GREGOROVIČ AND LENKA ZALABOVÁ Consider the canonical decomposition t = F)®m, where F) is 1-eigenspace of s and m is -1-eigenspace of s. Then a(m) C C"®C™* and a (I)) C M®csu(p, q)<3)R* follow from the assumption i(s) = s0]0 and is a Lie subalgebra, because dim{\)/\) = 1. We can identify m with C™ via a, because the restriction of a to the map m —>• C™ is injective. Indeed, if the restriction is not injective, then the elements in its kernel would be another symmetries at eL, but we know that there is only one symmetry. This identification uniquely determines the map i : L —> CSU(p, q). Further, [m, m] C F) holds and we have the corresponding symmetric space K/H°, where H° is the connected component of identity of the fixed point set of the conjugation by s. Therefore exp([AT,Y]) G H° for each X,Y G m. The map Ad : H° —>• GL(xn) can be restricted to the connected component of identity L° of L and the restriction coincides with i. Therefore, it suffices to show that the element ad([X, Y\) G fll(m) belongs to sl(m) for all X,Y G m. But we have ad([AT, Y]) = ad(AT) o ad(Y) — ad(Y) o ad(AT) and the trace equals to tr(&d([X, Y])) = tr(&d(X) o ad(Y) - ad(Y) o ad(AT)) = B(X, Y) - B(Y, X), where B denotes the Killing form, which is symmetric. Therefore i(L°) C U(p, q) and Tei(l) C u(p,q). In particular, the claim (1) holds. The map a can be expressed as in the claim (2), because there is always zel* such that the extension (Ad(exp(—z)) o a, conj(exp(—z)) o i) satisfies (aix 0 P2(a;)A 0 Ax 0 +Ad(exp(-z))o o;([). xi 0 aix J Since the CR geometry (M, H, J) does not depend on parts Pi, P2 and (a, A) of a, these parts are completely determined by condition (5) from Lemma 4. □ Let us remark that although the Lie algebra homomorphism i is uniquely determined by the isomorphism m = C™ given by a, the converse is not true. See [4] for examples of non-equivalent CR geometries described by extensions with the same Lie group homomorphism i. Let us further remark that we are not aware of any example of an extension (a, i) of {K, L) to (PSU(p +l,q+l),P) where zel* from the claim (2) of Theorem 1 does not correspond to an invariant Weyl connection. The main reason for this is the following result. Proposition 5. Suppose that Ad(L°)|f,/[ = Ad(L)|f,/[. Then i(L) C U(p,q) and there is a bijection between R* and the set of K-invariant Weyl connections. In particular, there is a unique K-invariant Weyl connection corresponding to the normal extension (a,i) satisfying i(L) C U(p,q), i(s) = so,o and I aix Pi(X) P2(x)i \ (5) a((x, X) +1) = ix Ax -JPi(X)*+£*([). \ xi —X*I aix J This particularly holds when the transitive group K is semisimple. Proof. If Ad(L°)|[j/[ = Ad(L)|[j/[, then i(L) C U(p, q) holds and the claim follows. It follows from the classification of semisimple symmetric spaces that H is reductive and there is a complement to I in the center of F). Consequently Ad(L°)|f,/[ = Ad(L)|f,/i. □ 4.3. Relations to CR algebras. We explain here relations between our concept and the concept of CR algebras introduced in [1]. We denote here by tic the complexification of a Lie algebra ti. ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE 11 Let (a,i) be an extension of (K,L) to (PSU(p+ l,q+ 1),P). We complexify the linear map a to obtain a map ac:tc^sl(n + 2,C). The Lie algebra sl(n + 2, C) decomposes as sl(n + 2, C) = C © (Cn © Cn*) © (fll(n, C) © C) © (Cn* © Cn) © C, where pc = (&l(n, C) © C) © (Cn* © C") © C. The subspace Cn* © pc is a parabolic subalgebra of sl(n + 2, C) that satisfies Therefore, the preimage q of H°'l in tc is a Lie subalgebra of the form q = ac1(C"*©Pc)- The pair (£, q) satisfies conditions of a Ci? algebra from [1, Section 1.2.]. It is proved in [1] that this is the minimal set of data describing a CR geometry on the homogeneous space K/L. However, CR algebras do not provide as much information as the extension (a,i). In particular, we cannot obtain directly the curvature k of the corresponding Cartan connection from the CR algebra. Therefore, it is not easy to distinguish whether two CR algebras correspond to equivalent CR geometries. There are conditions in [1, Section 1.4] that characterize CR algebras of CR geometries that are symmetric in the sense of [10]. One of these conditions ensures that there is a Riemannian metric compatible with a CR geometry. Other conditions are analogous to condition (4) of Proposition 3 which says that the Lie algebra automorphism of t given by Ad(so,o) defines an automorphism of the Lie group K. There is the following method to check whether CR geometries corresponding to CR algebras (£, q) are symmetric (in our sense) and to construct the normal extensions (a, i) that describe them. (1) We consider I = t n q n q and HeL = (t/V) n ((q + q)/(q n q)), where q is the subalgebra conjugated to q in fie- (2) We choose a complex basis of HeL ■ This choice defines a Lie algebra homomor-phism I —>• csu(p, q) and the following facts hold: (2a) If this homomorphism is not injective, then the CR geometry is fiat (we will discuss this situation later). (26) If this homomorphism is injective and the CR geometry is symmetric, then it coincides with the restriction of a to I for some normal extension (a,i) describing the CR geometry. (2c) If this homomorphism is injective and the CR geometry is not symmetric, then the homomorphism corresponds only to associated graded map corresponding to the restriction of a to I for some normal extension (a,i) describing the CR geometry. (3) Each choice of representatives (in €) of the complex basis of HeL from (2) together with a choice of an element of I complementary to HeL allows us to define (3a) a linear map a of the form (5) from Proposition 5 for (at this point) unknown linear maps a, A, Pi, P2, (36) a linear map t : A2£ —>• su(p + 1, q + 1) given for all X, Y e £ by the formula t(X,Y) := [a(X),a(Y)]-a([X,Y]), (3c) a linear map v : £ —> £ such that v equals to • —id on the representatives (in t) of complex basis of HeL, and • id on the element of £ complementary to HeL and on I. 12 JAN GREGOROVIČ AND LENKA ZALABOVÁ Moreover, we consider only the choices that satisfy the equivalent conditions of the following statement. Proposition 6. The map v is a Lie algebra automorphism of I if and only if the components (R © <8> Cn -> R © csu(p, g) © R*, Cn Cn -> Cn © C™* o/t vanish for all linear maps a, A, Pi, P2. Proof. A consequence of the formula for t is that 1/ is a Lie algebra automorphism of £ if and only if Ad(s0,o)r(iy(X),iy(Y)) = t(X,Y), Ad(s0t0)[a(v(X)),a(v(X))] = [a(X),a(Y)] hold for all X,Y E £. If a is of the form (5), then Ad(s0,0)[a(v(X)),a(v(X))] = [a(X),a(Y)] holds for all X,Y E £ and all linear maps a, A, Pi, P2, and Ad(s0t0)T(v(X),v(Y)) = T(X,Y) holds for all X, Y G £ if and only if the claimed components vanish. □ (4) There are the following possibilities for the choice in the step (3). (4a) If there is no choice such that v is a Lie algebra automorphism of £, then the CR geometry corresponding to the CR algebra (£, q) is not symmetric. (46) If there is a choice such that v is a Lie algebra automorphism of £, then the CR geometry corresponding to the CR algebra (t, q) is symmetric if and only if v induces a Lie group automorphism of K and L is contained in fixed point set of v. (5) We require from now that the CR geometry corresponding to the CR algebra (t, q) is symmetric. The remaining step is to determine the choice of an element of t complementary to %eL and i : L —> P such that (a,i) is a normal extension describing the CR geometry. We know that there is a choice such that (Ad(exp(z))o a,i') is an extension for some z G R*, where the Lie group homomorphisms i' : L —>• P is induced by (adjoint) action of L on HeL and a(l). Thus it suffices to check the vanishing of components a(l)®R^-su(p+l,q+l), £n®£n^R of t for all linear maps a, A, Pi, P2. The condition (5) of Lemma 4 provides linear equations that determine uniquely the linear maps a, A, Pi,P2 for which the extension (a,i) is normal. 4.4. Example of non fiat symmetric CR geometries. Consider the Lie group £'(2) = R2 x 50(2) of isometries of Euclidean plane. There is the following normal extension (a, i) of (E(2), {id}) to (PSU(1, 2), P) of the form ix __5_ "v^ _ 3i_ v _15 ix 16 16 1 16 2 256 (6) a I f 0 -X1\ = \X1+iX2 -f ^Xi-ffX l2 ix —X\ + iX[ 2 16 where the choice of the basis of the Lie algebra of R2 © so(2) = (x, X2) © {Xf) reflects the convention from Section 4.2, i.e., (x,(Xi,X2)) are the distinguished coordinates from Theorem 1. Indeed, since i is trivial and T((x,(X1,X2)),(y,(Y1,Y2))) = 0 |Ä - %yX2 - f xY1 + A-xY2 32» 1 32» ^ 32 1 1 32" t2 0 0 %yXx + ±yX2 %xY, ±xY2 .0 0 0 on symmetric cr geometries of hypersurface type 13 holds for the linear map t determining the curvature k, it follows that (a,i) is a normal extension describing a non-flat symmetric CR geometry. In fact, any invertible linear map B : R2 ffi so(2) —>• R ® C defines a CR algebra (t, q) for q = Sc1(C*®Pc) and we ask the following question: Which maps B correspond to non-equivalent non-degenerate symmetric CR geometries of hypersurface type on the Lie group £'(2) of isometries of Euclidean plane? We give the answer to this question (using the algorithm from previous section and [4, Lemma 3.5]) in the following statement. Proposition 7. The normal extension (a,i) of the form (6) describes the unique (up to equivalence) non-degenerate symmetric CR geometry of hypersurface type on the Lie group E(2). Proof. Consider an invertible linear map B : M2ffiso(2) —>• M®C2. The construction of the objects from the algorithm is clear in this case. We need to find for which maps B the components M®C^M®csu(l)®M*, C®C^C®C*, ££^R of t vanish for all linear maps a, A, Pi, P2. In fact, this provides three equations on the entries of the matrix B that can be solved explicitly. In the standard basis of M2®so(2) and M®C, the inverses of matrices B that satisfy these equations define the following subvariety: (P1P2-P3P4 p5 P5P3-2p6 ' Pe Pi P2 0 pi p3 Thus it remains to check the action of morphisms from [4, Lemma 3.5] that determine which extensions define equivalent CR geometries. In particular, there are • four-dimensional Lie group of derivations of R2 ® so(2) that in addition contains the homothethies, and • two-dimensional Lie subgroup that forms center of CSU(p,q). We compute that the induced action of these morphisms on the six-dimensional variety (7) is transitive and the matrix 0 0 1 corresponds to the extension (6). □ 5. Metrizability and CR embeddings In this section, we always consider the AT-invariant Weyl connection D corresponding to a normal extension (a,i) describing a homogeneous CR geometry (M, H, J) that satisfies i(L) C U(p, q), i(s) = so,o and faix PipO P2(x) \ a((x,X) + l)= X Ax -JPi(X)* +a(l). \ x —X*I aix J Moreover, we always assume Ad(L°)|[,/[ = Ad(L)|fj/[, where L° is the identity component of L. This gives almost no restriction, because this condition is always satisfied on the symmetric CR geometry on the covering K°/L° —>• K/L, where K° is the identity component of K. 14 JAN GREGOROVIČ AND LENKA ZALABOVÁ 5.1. Distinguished metrics compatible with the CR geometry. The symmetric bilinear form h generally does not define a pseudo-Riemannian metric on %, because there is no natural way, how to measure the length of elements of TM/H. The situation is different, if there is a Weyl connection preserving not only the decomposition but also a non-zero vector field r in £. Such Weyl connection is called exact and the vector field r is called the Reed field. Equivalently, each exact Weyl connection corresponds to the contact form 9 that annihilates % and satisfies 6{r) = 1 for the Reeb field r. If there is an exact Weyl connection, then 9 o h is a pseudo-Riemannian metric on %. This metric is compatible with the CR-structure, because the form h satisfies h(J£, Jv) = h(£, v) for all sections £, v of H. The exact Weyl connection preserves this metric and the Reeb field can be used to construct a pseudo-Riemannian metric on TM, for which the connection is a metric connection. This metric is usually called a Webster metric. However, the Webster metric neither has to exist nor has to be compatible with the symmetries. Therefore, if we want to find a metric compatible with the CR geometry that is preserved by all symmetries, we need to show that the distinguished Weyl connection D is exact. Theorem 2. Let K be the Lie group generated by all symmetries of a non-flat symmetric CR geometry (M,Ti, J). Suppose that Ad(L°)|f,/[ = Ad(L)|f,/[. The distinguished Weyl connection D is exact and furthermore, there exists • a K-invariant contact form 9, • a K-invariant pseudo-Riemannian metric g := 9 o h on%, and • a K-invariant Webster metric g:=9oh + 99 on TM such that (1) Dg = 0,Dg = 0, (2) g\-n = g and the Reeb field of D is orthogonal to % and has length \, (3) choosing the Reeb field of D as a trivialization of (TM/'H)®(C, the pseudo-Riemannian metric g on% coincides with the real part of the Levi form up to a constant multiple, (4) the symmetry at x is linear in geodesic coordinates of D at x, reverses the directions of ' Hx and preserves the direction of the Reeb field of D at x. Proof. The image of a is contained in M©C"©u(p, (?)©C™*©IR* and thus 7 describing the corresponding if-invariant Weyl connection has values in ad(u(p, q)). Furthermore, the assumption Ad(L°)|f,/[ = Ad(L)|f,/[ implies that i(L) C U(p,q) and therefore the maps ad-1 o 7 and i satisfy all conditions of [2, Theorem 1.4.5]. This means that the Weyl connection D is an associated connection to a if-invariant principal connection on the bundle K ^ K/L. Therefore it is an exact Weyl connection, because its holonomy is contained in U(p,q). The remaining claims then follow from general theory. □ In the Riemannian signature, Theorem 2 particularly allows to compare symmetric CR geometries (in our sense) with the symmetric CR geometries in the sense of [10], because we have found a metric compatible with the CR geometry that is preserved by all symmetries. Theorem 3. Suppose that p = 0. Then each non-flat symmetric CR geometry is covered by a symmetric CR geometry in the sense of [10], where the covering is a CR map that intertwines the symmetries. 5.2. CR embeddings. Consider the fiber bundle KXiCSU(p, q)/U(p, q) —>• K/L. If Ad(L°)|f,/[ = Ad(L)|f,/[ holds, then this bundle is trivial, i.e., K x, CSU(p,q)/U(p,q) = K/L x R. ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE 15 Let us prove the following statement: Theorem 4. Let K be the Lie group generated by all symmetries of a non-flat symmetric CR geometry (M,Ti, J). Suppose that Ad(L°)|f,/[ = Ad(L)\ij/i. Then: (1) the manifold K/L xl is a complex manifold, and (2) the inclusion K/L —> K/L x R given as a zero section is a CR embedding. Proof. We need some more details from the theory of Cartan geometries from [2, Sections 1.5.13 and 3.1.2] to proceed with the proof. First, there is a natural complement of u(p, q) in csu(p, q) given by so-called grading element, which is the unique element Z G csu(p,q) acting by —2 on R, —1 on C™, 0 on csu(p,q), 1 on C™* and 2 on R*. Furthermore, there is a Cartan connection on K/L x R induced by the CR geometry, where we identify R (via exp) with the multiples of the grading element Z. Then the Weyl connection D provides a reduction of this Cartan connection to U(p, q), which allows us to identify the tangent space of K/L xR with the fiber bundle (K xR) x4 (MffiC"fficsu(p, q)/u(p, q)). We can extend the complex structure on C™ to Mffi C™ ffi csu(p, q)/u(p, q) by declaring R to be the imaginary part of C and the multiples of the grading element in csu(p, q)/u(p, q) to form the real part of C. This definition is clearly U(p, (j)-invariant (and thus AT-invariant) and defines an almost complex structure J on KXiCSU(p, q)/U(p, q). Let us compute the Nijenhuis tensor [£,77] — [J£, Jrj] + J([J£,rj] + [£, Jrf\) of J for £,77 G T(K/L x R). For each x G K/L x R, there are vector fields £,fj such that £(x) = S,{x),f){x) = rj{x) and that the element [£,?7](:r) is identified with the element [X, Y] - [a(x + t)),a(Y + Jj)] + a([X + t),y+ Jj]) mod u(p, q) ffi Cn* ffi R*, where £(x),rj(x) are identified with x, y G R ffi C™ ffi csu(p, q)/u(p, q). This identification can be obtained using the technique analogous to [2, Proposition 3.1.8] for T(K/L x R) instead of T(K/L). Indeed, the Cartan connection in the background remains the same and we only need to restrict ourselves to normal Weyl connections that coincide with D at x and project the results given by the Cartan connection to T(K/L x R) instead of T(K/L). However, [x, y] - [a(x + Jj), a(Y + ())] + a([X + f), Y + f)]) = [X, Y] mod u(p, q) ffi Cn* ffiR* due to condition (5) from Lemma 4. Therefore we have ([Lv}-[JLJv}+J([JLv}+[LJv}))(^ = [x,y}-[jx,jy}+j([jx,y}+[x,jy}). Let us now discuss possible values of this expression for all possible incomes: • For X, Y G C" we have [X, y] - [JX, jy] + J([JX, y] + [X, jy\) = 0. • For X G C" and Y = JZ G R we have [X, Y] - [JX, JY] + J([JX, Y] + [X, JY]) = [JX, Z]-J([X, Z]) = 0. • For X G C" and Y = Z we have [X, Y] - [JX, JY] + J([JX, Y] + [X, JY}) = [X,Z} + J([JX,Z}) = 0. • For X = JZ G R and Y = Z we have [X, Y] - [JX, JY] + J([JX, Y] + [X, JY}) = [JZ, Z] + [Z, JZ} = 0. The remaining possibilities vanish trivially. Thus the complex structure is inte-grable. Then the zero section is a CR embedding, because it is a closed orbit. □ In holomorphic coordinates on U C K/L x R, the hypersurface K/LDU C C™+1 may be described as a zero set of a function F : U —> R. Theorem 4 and Lemma 2 provide distinguished holomorphic coordinates in which the function F has a specific form. 16 jan GREGOROVIČ AND LENKA ZALABOVÁ Corollary 1. Let K be the Lie group generated by all symmetries of a non-fat symmetric CR geometry (M,H, J). Suppose that Ad(L°)|f,/[ = Ad(L)|f,/[. Then for every point x G M, there is a holomorphic coordinate system on U C K/L x M. centred at x such that the function F(z, w) defining M satisfies F(z, w) = F(—z, w). 6. Locally flat CR symmetric spaces Locally fiat CR geometries are always locally symmetric (in our sense). Therefore, the following question appears: Which local symmetries are globally defined? The answer depends on the topology of the manifold. We show on series of examples that various situations are possible. There are two sources of examples that we study here that are related to flag manifolds. The first series of examples follows the construction from [15, 6] that we apply to CR geometries. The second series of examples involves CR geometries on orbits of real forms in flag manifolds from [1]. In the case p, q > 1, it is convenient for the presentation of examples to consider the Hermitian form m given for u, v G C™+2 by m(u, v) = u1 0 0 0 0 0 /' 0 0 1\ 0 0 0 °7 where the matrix I' is a diagonal square matrix of order n — 2 with the first p — 1 entries equal to 1 and the remaining q — 1 entries equal to — 1. 6.1. Non homogeneous symmetric CR geometries. Let us apply the construction from [15, 6] to CR geometries. We start with the standard model PSU(p+ l,q+ I)/P. Consider the CR manifold M := PSU(p + l,q + l)/P - {(u), (v)}, where u,v G C™+2 are arbitrary non-zero null vectors of m. The group K(u,v) of CR transformations of the fiat CR geometry on M has two connected components. The identity component of K(u, v) is the intersection of stabilizers of (u) and (v). Let us remark that if m(u, v) / 0, then the identity component of K(u, v) is isomorphic to the group CSU(p, q). If m(u,v) = 0, then p, q > 1 and there is g G PSU(p +l,q+l) such that gu and gv are the first two basis elements of C™+2 and thus elements of Lie algebra of K(u,v) are ^-conjugated to 0 Z ÍZ \ 0 a2 B ib 0 0 A -I'B* -I'Z* 0 0 0 -02 0 \o 0 0 0 -ôľ. / for some a1; a2, zn G C, b, z G R*, B, Z G Cp+q-2* and A G u(p - 1, q - 1). The other connected component of K(u, v) contains the elements that swap (u) and (v). We check whether there is a symmetry at each K(u, i>)-orbit on M. Let us emphasize that if all symmetries at one point of a K(u, v)-orbit preserve or swap the points (u) and (v), then all symmetries at all points of the whole orbit have the same property. The orbits of the action of K(u, v) on M are characterized by the fact that the action preserves • the subspace (u,v), and • the (non)-isotropy with respect to the Hermitian form m. Moreover, the action of K(u,v) on (u,v) depends on whether (u,v) is isotropic subspace or not. ON SYMMETRIC CR GEOMETRIES OF HYPERSURFACE TYPE 17 Example 1. Assume that p,q > 1, i.e., not the Riemannian signature. Consider the CR manifold M = PSU(p + l,q + 1)/P — {(u), (v)} for arbitrary non-zero null vectors u,v G C™+2 isotropic with respect to to, i.e., m{u,v) = 0. Then (u, v) — {(u), (v)} consists of a single orbit of K(u, v). Furthermore, K(u, i>)-orbits of points (x) such that x ^ (u,v) — {(u), (i>)} depend only on the (non)-isotropy of x with respect to u, v. We show that there exist symmetries at all points of each orbit of K(u,v). Instead of fixing (u), (v) and discussing symmetries at various points (x), we fix the point (x) as the point (eo) given by the first vector of the standard basis eo, • • •, e„+i of C™+2 and we choose admissible (u) and (v) such that (eo) lies in the correct orbit. Then we find all symmetries at (eo). Let us recall that all symmetries of the standard model at the origin (e0) are of the form where I = ^ o /' o J and Z = {z\,.. ., zn) G C™* and z£l* are arbitrary. Involutive symmetries are those satisfying z = 0. (1) Let us start with the orbit corresponding to the case m(eo, u) =/= 0 and m(eo, v) =/= 0. We choose u = ieg + e\ + en — ien+\ and v = ieg — e\ + en + ien+\■ Direct computation gives that there is exactly one symmetry sz,z, where Z = (—2i, 0,..., 0) and z = 0. This symmetry is involutive and swaps (u) and (v). There is no symmetry preserving them. (2) Let us now consider the orbit for the case m(eo, u) = 0 and m(eo, v) 7^ 0 (which is the same orbit as the orbit for the case m(eo,u) 7^ 0 and m(eo,v) = 0). We choose u = \/2ei and v = ien+\. Direct computation gives that there is exactly one symmetry sz,z> where Z = (0, .. ., 0) and z = 0. This symmetry is involutive and preserves (u) and (v). There is no symmetry swapping them. (3) The next possibility is the orbit for the case m(eo,u) = m(eo,v) = 0 and eo G (u, v). We choose u = \[2e\ and v = eo + y/2ei. Computation gives that there are (many) symmetries sz,z, where Z = (z\, ... , zn) with components zk = ftfe +ibk for k = 1,. .. , n satisfies "\/2&i + 1 = 0 and bi = 0, and a^, b]~ for k = 2, .. ., n — 1 and z are arbitrary. All these symmetries swap (u) and (v), and there are no symmetries preserving them. In particular, there are also non-involutive symmetries for z^O. In fact, this covers all possible orbits for the case p = 1 or q = 1, i.e., the Lorentzian signature. In the other cases, there is one more orbit. (4) Consider the orbit for the case m(u,eo) = m(v,eo) = 0 and eo ^ (u,v). We choose u = V^ei and v = e2 + en—i- Computation gives that there are (many) symmetries sz,z, where Z = (z\,..., zn) satisfies a\ = 0, b\ = 0, a2 + an-i = 0 and b2 + = 0 and a^, b^ for k = 3,... ,n — 2 and z are arbitrary. All these symmetries preserve (u) and (v) and there are no symmetries swapping them. In particular, there are also non-involutive symmetries for z^O. Altogether, symmetries at different orbits behave differently. Therefore, there is no smooth system of symmetries. In particular, there is no pseudo-Riemannian metric compatible with the CR geometry that would be preserved by some symmetry at every point. 0 1 -Z iz+ \ZIZ ) E -IZ* ) 0 -1 Let us show that this principle does not work if we remove two points corresponding to non-isotropic vectors. 18 JAN GREGOROVIČ AND LENKA ZALABOVÁ Example 2. Consider the manifold M = PSU(p + l,q + 1)/P — {(u), (v)} for arbitrary non-zero null vectors u, v G C™+2 that are non-isotropic for to, i.e. m{u,v) A 0. We choose u = en+1 and v = e0 + (2 + i)e\ — ien. Computation gives that there is no symmetry at (eo) preserving or swapping (u) and (v). Let us remark that the identity component of K{u, v) is isomorphic to the group CSU(p, q) and K{u,v) does not act transitively on (u,v) — {(u), (v)}. 0 6.2. Flat homogeneous symmetric CR geometries and orbits of real forms in complex flag manifolds. It follows from Lemma 4 that an extension (a,i) of (K, L) to (PSU(p+ 1, q+ 1), P) corresponds to a fiat CR geometry if and only if a is a Lie algebra homomorphisms. Therefore, we can present examples of extensions describing flat homogeneous symmetric CR geometries just by specifying the Lie subalgebra of su(p + l,q+ 1) that coincides with the image of a. In general, the group K does not have to contain symmetries. Moreover, symmetries do not have to preserve a(£). This is satisfied if K is the group generated by symmetries or the full group of CR automorphisms. Example 3. Consider the orbits of PSp(l, 1) on CP4 given by inclusion PSp(l, 1) C PSp(4,C) C PGZ(4,C). Due to the isomorphisms PSp(l,l) = PO(l,4), these orbits can also be interpreted as orbits in the flag manifold of 2-planes in quadric in CP5. There is a normal extension given by identifying the following Lie subalgebra of su(2, 2) with the image of a(sp(l, 1)) : I h + il2 -Xx + iX2 -X3 + l4 + iX4 + il5 i (l3 + x) \ Xi + iX2 li — il2 —i(x + l3) X3 — l4 + iX4 + il5 X3 + iX4 —ix —l\ — il2 X\ + iX2 \ ix -X3 + iX4 -Xx + iX2 -h + il2 ) where /^-entries generate the Lie algebra of the stabilizer L = CSO(2) xi S2R2 of a point in the minimal orbit. Precisely, (li, l2) = cso(2) and (l3, l4,15) = S2R2. Example 4. Consider the orbits of PSp(4, R) on CP4 given by inclusion PSp(4, R) C PSp(4,C) C PGl(4,C)- Due to the isomorphisms PSp(4,R) ^ PO(2,3), these orbits can again be interpreted as orbits in the flag manifold of 2-planes in quadric in CP5. There is a normal extension given by identifying the following Lie subalgebra of su(2, 2) with the image of a(sp(n + 2, R)) : / h + il2 Xx - iX2 X3 + l4- iX4 + il5 i (l3 + x) \ Xx + iX2 Ix — il2 i (l3 + x) —X3 — l4 — iX4 + il5 X3 + iX4 ix —Ix — il2 —Xx — iX2 \ ix -X3 + iX4 -Xx + iX2 -h + il2 ) where /^-entries generate the Lie algebra of the stabilizer L = CSO(2) xi S2R2 of a point in 5-dimensional orbit (which is not minimal). Precisely, {lx,l2) = cso(2) and (l3,l4,l5) = S2R2. In both examples, £ is simple and q is a parabolic subgalgebra of t^. In [1], the authors discuss which CR algebras (£, q) for simple Lie algebras £ and parabolic subalgebras q of the complexification of £ are symmetric. In fact, they correspond to orbits of real forms in complex flag varieties. Therefore, symmetric CR algebras of these types generalize bounded symmetric domains. We show that if a CR algebra (£, q) for a simple Lie algebra £ and a parabolic subalgebra q of the complexification £c of £ corresponds to a non-degenerate symmetric CR geometry of hypersurface type, then the geometry is necessarily flat. Therefore, we can use the results of [13] to classify all possible cases. on symmetric cr geometries of hypersurface type 19 Proposition 8. Let (t, q) be a CR algebra such that Í is simple and q is a parabolic subalgebra of be and the corresponding CR geometry is non-degenerate and of hypersurface type. Then the following statements hold: (1) If the CR geometry is symmetric, then the CR geometry is flat. (2) If the CR geometry is flat, then it corresponds to one of the following possibilities: (a) Í = su(p + 1, q + 1) and I = p, (b) t = fip(E±l,a±l) and i=co(2)(Ssp(^,^)(S(R2®Rn-2*)(SS2R2, (c) í = Sp(n+2, R) and I = co(2)ffisp(n-2, M)®^2®^™-2*)®^2^2, where (R2 Rn~2*) ffi S2R2 is the positive part of the parabolic subalgebra corresponding to the stabilizer of a Lagrangian 2-plane in Rn+2. (3) If the CR geometry is flat and be = Sp(n + 2,C) is the full Lie algebra of complete infinitesimal automorphism and n > 2, then the corresponding CR geometry is not symmetric. (4) If the CR geometry is flat and corresponds to a (2n+l) -dimensional orbit of the real form ofsp(n+2, C) in 2. Since sp(n + 2, C) is maximal subalgebra of si(n + 2, C), the only possibility for the orbit to be symmetric is to be equivalent to standard model which is compact. Since the orbit is compact if only if the orbit is minimal, the claim (4) follows. It follows from [1] that the orbit is minimal if and only if Í A Sp(n + 2, R). □ References [1] A. Altomani, C. Medori, M. Nacinovich, On homogeneous and symmetric CR manifolds, Boll. Unione Mat. Ital. (9) 3 (2010), no. 2, 221-265 [2] Cap A., Slovák J., Parabolic Geometries: Background and general theory, Math. Surveys and Monogr. 154, Amer. Math. Soc, 2009 [3] S.S. Chern, J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974) 219-271; Erratum Acta Math. 150 no. 3-4 (1983) 297. 20 JAN GREGOROVIČ AND LENKA ZALABOVÁ [4] J. Gregorovič, General construction of symmetric parabolic structures, Differential Geometry and its Applications, Volume 30, Issue 5, October 2012, Pages 450—476 [5] J. Gregorovič, L. Zalabová, Symmetric parabolic contact geometries and symmetric spaces, Transformation Groups, Volume 18 (2013), Issue 3 (September), 711—737 [6] J. Gregorovič, L. Zalabová, Notes on symmetric conformal geometries, Archivům Mathe- maticum TOMUS 51/5, (2015), 287 - 296 [7] J. Gregorovič, L. Zalabová, On automorphisms with natural tangent action on homogeneous parabolic geometries, Journal of Lie Theory, Volume 25 (2015), 677-715 [8] J. Gregorovič, L. Zalabová, Geometric properties of homogeneous parabolic geometries with generalized symmetries, Differential Geometry and its Applications, Volume 49, December 2016, 388-422 [9] J. Gregorovič, L. Zalabová, Local generalized symmetries and locally symmetric parabolic geometries, SIGMA 13 (2017), 032, 33 pages [10] W. Kaup, D. Zaitsev, On Symmetric Cauchy—Riemann Manifolds, Advances in Mathematics, Volume 149, Issue 2, 10 February 2000, Pages 145-181 [11] B. Kruglikov, D. The, The gap phenomenon in parabolic geometries, J. Reine Angew. Math. (Crelle's Journal) 2017 (723), 153-215 [12] O. Loos, Spiegelungsräume und homogene symmetrische Räume, Math. Zeitschr. 99 (1967), 141-170 [13] A. Onishchik, On compact Lie groups transitive on certain manifolds, Sov. Math., Dokl. 1 (1961), 1288-1291; translation from Dokl. Akad. Nauk SSSR, 135 (1961), 531-534 [14] N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of OPRAVIT complex variables, J. Math. Soc. Japan 14 (1962) 397-429. [15] L. Zalabová, A non—homogeneous, symmetric contact projective structure, Central European Journal of Mathematics, 2014, 12(6), 879-886 J.G. Faculty of Mathematics, University of Vienna, Oskar Morgenstern Platz 1, 1090 Wien, Austria; L.Z. Institute of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, České Budějovice, 370 05, Czech Republic and Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic Email address: jan.gregorovic@seznam.cz, lzalabova@gmail.com Chapter 4: Submaximally Symmetric Almost Quaternionic Structures article by Boris Kruglikov, Henrik Winther and Lenka Zalabova, published in Transformation Groups 23 (2018), no. 3, 723-741. arXiv: 1607.02025 SUBMAXIMALLY SYMMETRIC ALMOST QUATERNION!C STRUCTURES BORIS KRUGLIKOV, HENRIK WINTHER AND LENKA ZALABOVÁ Abstract. The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension n. The maximal possible symmetry is realized by the quaternionic projective space MPn, which is flat and has the symmetry algebra sl(n + 1, H) of dimension 4n2 + 8n + 3. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to 4n2 — 4n+9 for n > 1 (it is equal to 8 for n = 1). This is realized both by a quaternionic structure (torsion—free) and by an almost quaternionic structure with vanishing quaternionic Weyl curvature. 1. Introduction An almost quaternionic structure on a manifold M is a rank three subbundle Q C End(TM) such that locally (in a neighbourhood of each point) we can find a basis I, J, K of Q with I2 = J2 = K2 = -1 and IJ = K. A manifold M with a fixed almost quaternionic structure Q is called an almost quaternionic manifold. A (local) automorphism of (M, Q) is a (local) diffeomorphism of M that preserves Q. There exists a class of the so-called Oproiu connections [V°p] on (M, Q) that preserve Q and share the same minimal torsion Tv, which equals to the structure torsion of Q [1]. If V°p is torsion-free, then (M, Q) is a quaternionic manifold. An almost quaternionic manifold (M, Q) can be equivalently described as a normal parabolic geometry (Q —> M, ui) of type PGL(n+l,M)/P, where P is the stabilizer of a quaternionic line in H™+1 [4]. The fundamental invariant of each parabolic geometry is its harmonic curvature kjj, which has two components in the almost quaternionic case: the torsion k\ of homogeneity 1 and the quaternionic Weyl curvature n2 of homogeneity 2. In particular, k\ coincides with the torsion Tv of arbitrary V°p and vanishes for quaternionic geometries. The quaternionic projective space HP™ is the set of quaternionic lines in H™+1, and the group PGL(n+ 1,H) acts transitively on HP™ as automorphisms of the natural quaternionic structure. The subgroup P consists of transformations preserving the first basis line in H™+1. Then, HP™ = PGL(n + 1,H)/P is the flat model of (almost) quaternionic geometry. The flat model has vanishing harmonic curvature and conversely, each almost quaternionic geometry such that = 0 is locally equivalent to the flat model. In particular, every local automorphism of HP™ uniquely extends to a global one, and it is exactly the left multiplication by an element of PGL(n+ 1,H). The space HP™ has maximal possible dimension of the symmetry algebra among all (almost) quaternionic manifolds with fixed quaternionic dimension n, that is dimsl(n+ 1, H) = 4(n+ l)2 — 1 for dimM = An. For curved almost quaternionic structures, local automorphisms generally do not extend to global ones. We consider infinitesimal symmetries, which correspond to local automorphisms. We focus on the problem of establishing the submaximal symmetry dimension, i.e. the maximal dimension of the symmetry algebra of an almost 2 boris kruglikov, henrik winther and lenka zalabová quaternionic structure with kh ^ 0 and fixed quaternionic dimension. Specifically, we answer the following question: When an almost quaternionic manifold (M4n,Q) is not everywhere flat, what is the maximal dimension & of its Lie algebra of infinitesimal symmetries? Remark 1. The submaximal dimension of the automorphism groups (without the requirement kh ^ 0) is dimP = 4n2 + 4n + 3. This is achieved on the fiat manifold M = HP™ \ {p} for some p E HP™. However the symmetry algebra of this (M, Q) is of maximal dimension 4n2 + 8n + 3. From the point of view of parabolic geometry, a model with the symmetry algebra of submaximal dimension typically has exactly one non-zero component of its harmonic curvature [12]. Sometimes, the same submaximal bound is achieved for different non-zero components of kh ■ We will show that this is the case with almost quaternionic structures. Our main result is the following. Theorem 1. The maximal dimension of the symmetry algebra of almost quaternionic structures (M, Q) with dimM = 4n > 4 and kh = («i, «2) ^ 0 is © = 4n2 - An + 9. This is realized in both cases, when k\ = 0 and when k2 = 0. We exclude the case n = 1 due to the exceptional isomorphism sl(2, H) ~ so(l, 5). In this case the geometry PGL(2,H)/P can be interpreted as a four-dimensional Riemannian conformal geometry and has two components of homogeneity 2, which are the self-dual and anti-self-dual parts of the Weyl curvature. The sub-maximal symmetry dimension is 8 and is achieved by M = CP2 [7, 12]. Thus for n > 1 there is a gap 6(2n — 1) = dimsl(n + 1, H) — & in the set of all possible symmetry dimensions. If K\ 7^ 0 and n2 7^ 0, the symmetry dimension is strictly smaller than &, so the submaximal symmetry dimension is achieved only when K\ ® n2 = 0. This is explained in the beginning of Section 5. Acknowledgements: The authors thank Jan Gregorovic for suggesting that the submaximal symmetry dimensions for n = 2 should be 17. Lenka Zalabova thanks Norway Grants NF-CZ07-INP-4-2382015 and NF-CZ07-INP-5-4362015 for financial support and the University of Troms0 for hospitality. 2. Background on almost quaternionic and related geometries Almost quaternionic geometries are closely related to projective and c-projective geometries, so we recall basic concepts common to these. Two (real) affine connections on a manifold M of dimension n are projectively equivalent if their unparameterized geodesies, i.e. curves satisfying V-y7 E (7), coincide. Here (—) denotes the linear span over C°°(M). Projectively equivalent connections do not necessarily have the same torsion, but any connection V is projectively equivalent to a torsion-free connection V — \Txj. Two connections V and V with the same torsion are projectively equivalent if and only if there is a one-form T E 511(M) such that V - V = 1 ® (T o 1) + (T o 1) ® 1. A fixed class of torsion-free projectively equivalent connections [V] on a manifold M is a projective structure on M. It is proven in [6] that the submaximal symmetry dimension in the class of projective structures of dimension n > 2 is equal to (n - l)2 + 4 (for n = 2 it is 3), see also [12]. SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 3 A generalization of this concept to almost complex manifolds leads to almost c-projective structures. A connection V on M of dimension 2n > 2 with almost complex structure J is called complex if V J = 0. Each almost complex manifold (M, J) admits complex connections, because for arbitrary V the connection ^(V — JV J) is complex. A complex connection V can be chosen minimal meaning Tv = jNj, where Nj(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y] is the Nijenhuis tensor. A curve 7 on M is J-planar if V-y7 G (7, J7) for a complex connection V. Two complex connections on (M, J) are c-projectively equivalent if they share the same J-planar curves. Two complex connections V and V with the same torsion are c-projectively equivalent if and only if there is a one-form T G il1 (M) such that V-V= J2 A2(A®(ToA) + (ToA)®A). Ae{i,j} An almost c-projective structure on (M, J) is a class of c-projectively equivalent complex connections [V] sharing the same fixed torsion. It is proven in [11] that the submaximal dimension in the class of almost c-projective structures of (complex) dimension n is equal to 2n2 — 2n + 4 for n 7^ 3 and 18 for n = 3. Let us return to almost quaternionic structures. Consider an almost quaternionic manifold (M, Q) of dimension An. Analogously to the almost complex case, this admits a quaternionic connection. Indeed, for any local basis b = (I, J, K) of Q and a linear connection V, the connection \7b := 4(V — /V/ — JVJ — KVK) is quaternionic. Any other choice b = (J, J, K) is related to b via a transformation from SO(Q), so Vg = ±(V - IVI - JVJ - KX7K) coincides with Vb. Denote B := \(Ni + Nj + Nk). The canonical structure tensor of Q is given by TQ :=B + 5{TI 5{tj ® J) + 5{tk ® K), where ta(X) = j±^Tr(AB(X)) for A = I, J, K and 5 : T*M ® T*M ® TM -» A2T*M ® TM denotes the Spencer operator of alternation [1]. A quaternionic connection can be chosen minimal meaning its torsion coincides with Tq . An almost quaternionic structure Q is a quaternionic structure if Tq vanishes. A curve 7 is called Q-planar if V-y7 G (7, J7, J7, Kj) for a quaternionic connection V. Two quaternionic connections V and V on (M, Q) with the same torsion share the same Q-planar curves if and only if there is a one-form T G 511(M) such that V - V = A2(A® (T o A) + (T o A) ® A^. Ae{t,i,J,K} Analogously to the c-projective case, we fix the class of connections [V] sharing the same Q-planar curves and with the minimal torsion Tv = Tq. These are called Oproiu connections. The Q-planar curves are the (unparameterized) geodesies of all Oproiu connections [8]. Given an arbitrary quaternionic connection, one can construct an Oproiu connection by an explicit formula [1, §3.11]. An almost quaternionic structure is quaternionic if and only if some (and thus any) Oproiu connection V is torsion-free. In that case, the curvature i?v of an Oproiu connection V decomposes as i?v = Wv + Pv, where the totally trace-free part Wv is the (quaternionic) Weyl tensor of i?v and Pv is the Ricci part o/i?v[l]-Let us remark that almost quaternionic structures can be viewed as first-order 4 BORIS KRUGLIKOV, HENRIK WINTHER AND LENKA ZALABOVÁ Go = Sp(V)GL(n, H)-structures. Then the above decomposition of i?v is just the decomposition with respect to the action of this structure group. It turns out that the Weyl part Ws/ of i?v does not depend on the choice of Oproiu connection and is a complete obstruction to the flatness of a quaternionic structure. Remark 2. The equivalence class of Oproiu connections is already determined by the quaternionic structure on the quaternionic manifold. But the complex structure alone does not determine a c-projective structure. The choice of the class of c-projectively equivalent connections on a complex manifold is an additional choice. Remark 3. All three geometries discussed in this section can be described as parabolic geometries of type PGL(n+l, K)/P, where K = R, C, EI and P is the stabilizer of a K-line in K™+1 [4]. This explains many similarities between them. 3. Parabolic geometry setup and the gap problem In this section, we summarize basic facts about almost quaternionic structures from the parabolic viewpoint. We will need the notations describing parabolic subalgebras p of a real semi-simple Lie algebra g = Lie(G). The conjugacy classes of such are in bijection with some subsets of the Satake diagram corresponding to a fixed choice of (maximally non-compact) Cartan subalgebra. These will be denoted by crossing out certain white nodes on the Satake diagram, cf. [4]. Let £ be the set of crossed out nodes. We denote by ps the standard parabolic subalgebra corresponding to S. The semi-simple Levi factor pff is given by the Satake diagram with £ removed. A Lie subgroup P% C G is parabolic if its Lie algebra ps is such. We will use the standard Bourbaki enumeration of the nodes of the Dynkin diagram, and encode parabolic subgroups Ps respectively. Consider the Lie algebra g = sl(n + 1, H), which is a real form of A2n+1 = s\(2n + 2, C). The parabolic subalgebra p = Lie(P), for P = P2, corresponding to a |l|-grading g = g_i ® g0 ® gi, p = go ffi gi, is encoded by the following Satake diagram: •-x-•-o-•--•--o-• This grading can be viewed via the matrix (1, n)-block decomposition which is given by the standard representation of g on H™+1 = ix HI™. Thus g has the matrix form (u a)' whereS-i = {"e H™}' Si = iP e M*n} and0o = i(a,A) e H©gI(n,H) : Re (a) + Re(trA) = 0}. In particular, the real part of a E H is determined by tr A and the imaginary part belongs to sp(l). Thus the reductive Lie algebra go can be equivalently viewed as sp(l) + gl(n, H), and this further decomposes as g0 = sp(l) + RZ + sl(n,H), where the semisimple part is ggs = sp(l) +sl(n,H) and the grading element Z = diag(^-j-, ■ ■ ■, ^jpx) generates the center j(go)-The Lie algebra Qff is encoded by the Satake diagram produced by removing the crossed node and adjacent edges: • •-o-•--•--o-• A fundamental invariant of a regular normal parabolic geometry is the harmonic curvature kjj, taking values in the Go-module _ff2(g_,g) (that is the Lie algebra cohomology of g_ with values in g; in the quaternionic case the regularity requirement is vacuous, i.e. H2 = H\ has positive homogeneity because the geometry is Ill-graded). This is a completely reducible module, and its two irreducible components SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 5 Hi and H2 (the subscript denotes homogeneity of the cohomology with respect to Z) yield the corresponding decomposition of kh into two summands: • the torsion Ki of homogeneity 1 valued in _ff2(g_i,g), and the quaternionic Weyl curvature k2 of homogeneity 2 valued in H2(q-i, g) The harmonic curvature Ki coincides with the torsion TV = Tq of an arbitrary Oproiu connection, and if the torsion vanishes, then the harmonic curvature n2 coincides with Weyl tensor Wy of an arbitrary Oproiu connection. For an almost quaternionic structure that is not quaternionic kh = ki + k2 and both components are non-vanishing in general. To compute the structure of these modules, where ki and n2 have their values, we invoke the complexification: the corresponding parabolic subalgebra pc C gc induces a |l|-grading of gc and iř2(g_,g) ® C ~ H2(qc:,qc). Explicit algorithmic description of the Gq-module structure of the latter follows from Kostant's version of the Bott-Borel-Weil theorem [4]. The real curvature module is then a real completely reducible p-submodule of its complexification, which is a pc-module. In the case of almost quaternionic structures, the submodules corresponding to the quaternionic torsion ki and curvature n2 are encoded by minus lowest weights of the complexified modules (adapting the convention of [2]) as follows, where the number over the z'th node is the coefficient of the fundamental weight uii\ „.3-301 01 „.0 -4 30 01 KjI . K2 . •-x-•-o--•--o-• •-x-•-o--•--o-• Remark 4. Let us point out that H+(q-,q) is a real go-module that we identify with a real g0-submodule of H^_(q_, g) ® C. Note also that minus the lowest weight is equal to the highest weight of the dual module. Let us recall how to get a universal upper bound it on the submaximal symmetry dimension & (the gap problem), and explain the role of the Go-module ff2(g_,g). Each element £ of the infinitesimal symmetry algebra inf has to preserve (both components of) kh, and thus the isotropy subalgebra of inf embeds into the annihilator of kh in p. Moreover, for arbitrary fixed u G Q there is the inclusion inf "—t g of filtered vector spaces, given by £ i—> ui(£(u)). Its image f ~ inf is a filtered Lie algebra with the bracket obtained by a deformation of the Lie bracket [•, •] of g. The associated graded algebra s = gr(f) is a graded subalgebra of g [3, 12]. The grading s = satisfies s0 C a0 = {

■ nH = 0}. Furthermore it is proven in [12] that & < it for it = maxldim^) : 0 ^ tp G H^(g_,g)}, where the graded subalgebra C g is the Tanaka prolongation of the pair (g_, üq ), and üq is the annihilator of ip in go. Moreover, & < it < itc, where itc is the universal upper bound for the complexified geometry, and the universal upper bound is realized by the stabilizer of minus the lowest weight vector in the complex case. By [12, Corollary 3.4.8] the parabolic structures of type A2n+i/P2 are prolongation-rigid, i.e. the Tanaka prolongation = 0 for any ip ^ 0. This implies the corresponding statement for real geometries [10, Proposition 3]. Thus almost quaternionic structures are prolongation-rigid af = 0, and so = g_i © for each non-zero element %p G H+(q-i,q). 6 boris kruglikov, henrik winther and lenka zalabová Let &i be the maximal symmetry dimension of the almost quaternionic geometry of dime = n in the case kj 0. We are going to bound this &i < ili = max{dim(a^) : 0 7^ ip G -ff2(g_i, g)} and prove that the submaximal symmetry dimension is it1=61 = 6 = 62=it2. We, however, cannot directly apply the methods from complex parabolic geometry. It turns out that the corresponding upper bounds are strictly less than the upper bounds for the complexification: ilj < itf and thus it < itc. A similar phenomenon was noticed for Lorentzian conformal geometries in [5], where the submaximal symmetry dimension was computed by listing all subalgebras of high dimensions that stabilize a non-zero element in the harmonic curvature module. In this paper, we choose a different approach by identifying a real analogue to the lowest weight vector in the real harmonic curvature module. 4. Minimal orbits Recall that in the case of complex parabolic geometries, obtaining the upper symmetry bound is based on the Borel fixed point theorem, which states that there is a unique closed orbit, which is of minimal dimension, in the projectivization of _ff2(g_,g). Then the upper bound is given by dimension of the stabilizer of a weight vector corresponding to minus the lowest weight (generating the minimal orbit). The Borel theorem cannot be applied in the case of almost quaternionic structures, but we still consider the g0-orbits in the projectivization of i72(g_1;g) to find one of the minimal dimension. The following statement is immediate. Lemma 2. The annihilator ofO 7^ Kj G Hf (i = 1,2) is of maximal dimension in go if and only if the Go-orbit through Ki has minimal dimension in the projectivization o/ff2(g_!,g). We will need the following result on existence of closed orbits. Here we denote by [v] the projection of a non-zero vector v G V to the projective space PY. Lemma 3. Let Y be a real, finite-dimensional, irreducible L-module for a real connected Lie group L, such that the center of L acts by real scalars. Then there exists 0 7^ v G V such that for [v] G PY the orbit L ■ [v] C PY is closed and of minimal dimension. Ln fact, any orbit in PY of minimal dimension is closed. Proof. We may quotient the group by the kernel of the representation, to obtain a faithful representation, and this will not affect the (projective) orbits. Therefore we assume without loss of generality that the representation is faithful. Because the representation is irreducible, I = Lie(L) is the direct sum of a semi-simple ideal and a central subalgebra [9, Chapter 3]. Our assumption on the center action means that the module V is a tensor product of an irreducible module Vo over the semi-simple ideal, and a one-dimensional module M. over the center. Let L ■ [v] be an orbit of minimal dimension d = dim L ■ [v]. We want to prove it is closed. Consider the complexification of the group, the action and the representation. The element v G V + 0 • i C Vc determines the complex orbit Lc ■ [v] C PYC of the same complex dimension d (because the annihilator of v in lc intersects I by the annihilator of v in I). If the closure of the orbit L ■ [v] contains another orbit L ■ [v'] (necessarily of the same dimension d), then the closure of the complex orbit Lc • [v] contains the complex orbit Lc • [v'] (again of the same complex dimension d). To exclude the latter note that the action of Lc on PYC is algebraic, because SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 7 both the semi-simple action on Yq and the central action on C are algebraic [13, Chapter 3]. But then the boundary of any orbit can only contain orbits of strictly smaller dimensions, which are less than d, cf. proof of Corollary in [13, III;§1.5]. Since such are non-existent in the real case, this proves the claim. □ Let's consider the minimal orbits in the projectivizations PH2 and PH2 of both irreducible components of ff2(g_i,g). Conjugacy classes of parabolic subalgebras are in bijection with conjugacy classes of Z-gradings of semi-simple Lie algebras g = g_fc © ... © Qk- If 0 has a proper non-trivial parabolic subalgebra p, then there exists an element Z G g which acts diagonalizably on g, with an eigenspace decomposition of the form cu).z|gri = nIdgrl, such that the direct sum of the non-negative eigenspaces equals the parabolic sub-algebra p = g>o- This Z is called a grading element for p. Changing the grading element Z to Adp(Z) for any p in P C G we get another grading element and another grading of g. Moreover, given a grading of g, the subalgebra p = g>0 of non-negative gradation is parabolic (all parabolics arise in this way [4]). Recall that in the quaternionic case go = sp(l) + RZ + sl(n,H). Let V be a go-module which is irreducible under the restricted representation of sl(n,H). The group Go = PGL(n,M) acts effectively on PY, and the Lie algebra go = sl(n,H) is simple. Consider the following parabolic subgroups of Go: H = P2 in the case n = 2, and H = P2t2n-2 in the case n > 2 (tilde in P indicates that numeration of the parabolics is with respect to Go, the indices indicating the crossed nodes on the Satake diagram of go). This parabolic determines the grading on go with respect to which the parabolic (go)>o is equal to F) = F)o © f)+ as a vector space, where f)o = sp(l) © 3l(n — 2, H) © sp(l) © RZ', fi+ = fii©fi2 = fieis(8n-12,H). Here rjeis(8n — 12, H) is the quaternionification of the real (nilpotent) Heisenberg algebra rjeis(2n—3), and Z' is a grading element in go. The action of rjeis(8n—12, H) on HP is given bynxn quaternionic matrices with zeroes everywhere except for the first row and the last column, and with zeroes on the diagonal. To distinguish the summand sp(l) in go from those in F)o we will use the notations Sp(l)ieft and Sp(l)right for the latter (marking them in the appearing order). The grading element Z' G go acts on V, and V decomposes as V = ®iYgi with respect to the action of Z', where Z'\yg. = #ildye, and 9i is real for all i. Indeed, the weights A of the representation V are rational linear combinations of the roots of go, hence the eigenvalues X(Z') of Z' on V can be rationally expressed via eigenvalues of Z' on go, which are integers. Therefore, exp(tZ') = ©ieteiIdye.. For p G we have [Z' ,p] = p = Z'p — pZ'. This implies that for all v G Yg{ we have Z'-(p-v) = (p + pZ') ■ v = (1 + 6i){p ■ v), which implies p ■ v G V^+i. Lemma 4. Let Y be an irreducible Qg-module and 0 =/= v G V. Suppose that the orbit Go • [v] C PY is closed. Then there exists 0 =/= w G Yemao. = kerr]+, Omax = niaxi{#i}, such that [w] G Go • [v]. Proof. Decompose v = ^2g. vg{ into Z'-eigenvectors as above. Let vgj be the nonzero component of the greatest index. If 8j < 9max then, due to irreducibility, there exists p G f)i such that p ■ vgj =/= 0 (in the opposite case v generates a proper 8 BORIS KRUGLIKOV, HENRIK WINTHER AND LENKA ZALABOVÁ submodule). Then wq = exp(rp)v for small t > 0 has a non-zero component in Vg.+i. Repeat this procedure for wq, w\, .. ., until Wk has non-zero component in Vemax (if 6j = 9max then Wk = v). This takes a finite number of operations, because 9max is finite, and the index 6i G [—0max,8max] is incremented by a fixed amount on each iteration. Since the greatest eigenvalue dominates and the orbit of v is closed, the limit (w) = lim etz (wk) G PVemax exists. Here (wk) = R ■ Wk- t—> + 00 max Moreover, there exists an element w in Vemax which is projected to the limit (w). □ Note that the proof of Lemma 4 would work if we switched out Fj for any parabolic subalgebra in q$, but the particular choice Fj will turn out to be well adapted to describing the minimal orbits in those modules we are interested in. 4.1. Minimal orbits in the curvature module. The irreducible Go-submodule iff C if2 (fl-, fl) will be denoted in this subsection by V11 to indicate its homogeneity 2. Since the grading element Z acts on it by multiplication by 2, it cannot be in the annihilator of k2- It follows from the description of the (complexified) curvature module in the previous section that the action of sp(l) preserves n2 and is always contained in the annihilator. Therefore we can restrict our attention to the action of go = sl(n,H). With respect to it the curvature module has the highest weight uj\ + 3cj2n-i (we flip the Satake diagram), and hence can be identified with an irreducible real Go-module V11 = slu*n 01". Here 0 denotes the Cartan product (kernel of the tensor contractions), and we use the complex notations (the complex tensor products are taken with respect to an arbitrary invariant complex structure, say I G ImH C End (HI™), whose choice is inessential). For real description we refer to [4, Proposition 4.1.8], see also Remark 5, but we use the complex notations (even in describing the real objects). We would like to find a Go-orbit of minimal dimension (closed by Lemma 3) in PV11. Due to Lemma 4 we can assume it is represented by a non-zero element w G Vg1 = ker Fj+. The element w has pure grading with respect to Z' and hence max ^ ^ its annihilator in 0o is also graded: cmn(w) = ® cmn(w)s (= ® cmn(w)s for s——2 s——1 n = 2). We already know that ann(w)+ = Lemma 5. We have: ann(w) n (0o)- = 0, i.e. ann(w)- = 0. Proof. Let us consider the case n > 2 (The case n = 2 is a simple adaptation). The f)o module (jjo)-i is reducible - it is the sum of two irreps: (0o)'_i = (the first column in the matrix from sl(n, H) with the first and last entries zero) and (0o)_i = (the last row in the matrix from sl(n,H) with the first and last entries zero). This is also true when we restrict to si(n — 2, H) C f)o- Let q = q' + q" G ariri(w)_i be non-zero. Due to the highest weight of the curvature module, s\{n—2, H) C anri(w), and both (jjo)!_i and (0o)" i are s\{n—2, H)-modules, so we conclude that at least one of (0o)'_i and (0o)" i is entirely in the annihilator. But then, since F)i C cmn(w) (and computing the brackets), we conclude that at least one of sp(l)ieft and sp(l)right is entirely in ann(w), which is impossible. Thus ann(w)_i = 0. If there is 0 7^ q G ann(w)_2, then taking brackets with f)i C cmn(w) we get a non-zero element in cmn(w)_i, which is impossible by the above. This contradiction proves the claim. □ SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 9 As already noted in the above proof, due to the highest weight, sl(n — 2,H) acts trivially on v11, and so from F)qs only sp(l)2 = sp(l)ieft©Sp(l) right acts non-trivially. With respect to this algebra the module Vlg has highest weight uii + 3ui2, and as an irreducible real module it has real dimension 8. We want to maximize the annihilator of an element w. Lemma 6. Dimension of the sp(l)2-orbit through a non-zero element w E Ylg is either 5 or 6. Thus nontrivial annihilator can be only so(2) C Sp(l)2 of dim = 1. Proof. The complex sp(l)2-module Vg1 ® C of the highest weight uii + 3ui2 is the outer product of the irreducible sp(l)-modules C2 and S^C2. The algebra sp(l)2 is a compact real form of the rank 2 algebra sl(2,C) ©sl(2,C), therefore the subalgebra of dimension 2 is a Cartan subalgebra t2, which is unique up to conjugation. Any subalgebra of dimension > 2 contains a Cartan subalgebra, but t2 does not annihilate any vector in the module. Therefore the maximal possible annihilator dimension is 1, and any subalgebra of dimension 1 is isomorphic to so(2). This is realized by annihilator of the highest weight vector, and its real part has the same annihilator. This annihilator is generated by 3eieft — eright, where eieft and eright are generators of Cartan subalgebras in the two ideals of sp(l)2. □ Corollary 7. The largest annihilator of a non-zero w E Vg1 with respect to the action of go is -Sp(l) © (S0(2) © R © Ql(n — 2,H)) K rj+, where R is generated by a suitable linear combination of the grading elements Z and Z' of q and Qq. We realize this annihilator in complex notations as follows. Let H™ = Hi ©... © H„ and vm be the standard basis of Mm (v = l,i,j,k), be the real dual basis, m = 1, ... ,n. Denote by qrs E sl(n,H) the matrix that contains q on the r-th row and s-th column, and that contains zeros elsewhere. The action on H™ is qr,s • Vt = (qv)r 5sj and the action on H*™ is minus the transpose. Let w = l*3 M*n). We refer to [4, Proposition 4.1.8] for the description of V1 as the real module. One could expect that a generator of a minimal orbit can be realized as the tensor product of such generators in each factor, and this is indeed the case. A minimal non-zero sp(l)-orbit in S^M. can have dimension no less than 2, because the maximal proper subalgebra of sp(l) is of dimension 1. This means that any element of the torsion module which achieves maximal stabilizer in sl(n, H) and a stabilizer of dimension 1 in sp(l) generates a minimal orbit. Thus, we analyze the torsion module under the action of go = sl(n, H) alone, which yields V1 = C4 0C ACH*™ 0 HP, where the first factor C4 is a trivial module. This decomposes as a direct sum of modules isomorphic to A2H*™ 0H". One can always find a minimal orbit in a completely reducible module which is contained in an irreducible summand. Moreover this orbit is closed by Lemma 3 and we again utilize Lemma 4 to ensure that the minimal orbit has an element in kerrj+. Using the grading element Z' of go we identify Yle = ker Fj+. Since the module A2H*™ 0 H™ has highest weight uii + ui2n-2 and is quaternionic, so diiriRVg = 4. Taking 0 7^ w € (°f pure grade), its annihilator is a graded algebra (containing Fj+). Lemma 8. We have: cmn(w) n (go)- = 0, i.e. cmn(w)_ = 0. Proof. Let us consider the case n > 2 (The case n = 2 is a simple adaptation). First, we show that the evaluation map (jjo)-i ® Vg —> V1 is injective. Take any element q = q' + q" E (flo)-i = (flo)-i © (flo)-i, where the latter splitting into irreps is the same as in the proof of Lemma 5. Using the same argument as in this proof, given a non-zero annihilator element in one of these submodules, we conclude (because sl(n — 2, H) is in the annihilator) that the whole submodule is in the annihilator. So it is enough to check injectivity of the action on the two elements only, which are q's ± and q'^ s for 1 < s < n. Notice that Vg = {1* A j* 0 v± : v e H}. If q = q'sl, then the action is 1 ' (1£ A Jn ® ^1) = K A in ® (lv)s 7^ 0, and if q = q^s, then the action is q ■ (1* A j* 0 Vl) = -(q'J* A j* + 1* A (q"j)*s) ®Vl^0. Thus ann(w) n (g0)-i = 0. submax. symmetric almost quaternionic structures 11 The rest mimics the proof of Lemma 5: if a non-zero annihilator element exists in (flo)-2, then bracketing with F)i we obtain a non-zero annihilator element in (flo)-i, which is a contradiction. □ Thus it remains to consider the action of F)o on Vg . Since in the semi-simple part F)os = sp(l)ieft®sl(n—2, H)®sp(l)right the last two summands are in the annihilator (because of the weight of the representation), this reduces to considering Ylg as sp(l) = sp(l)ieft-module (of the highest weight uii). This is the standard left action of sp(l) on H, any element of sp(l) acts as a complex structure, and so this part gives no contribution to the annihilator of any 0 7^ w E Vg . Also, similar to the curvature module, a combination of the grading elements acts trivially. Corollary 9. The largest annihilator of a non-zero icefj with respect to the action of go So(2) ffi (Mffigl(n — 2,H) ffisp(l)right) x f)+, where R is generated by a suitable linear combination of the grading elements Z and Z' of q and Qq. Let us give the generators of this annihilator in the complex tensor notations. Fixing w = 1* A j* ® li (again this element is contained in the Cartan product) we conclude that cmn(w) in go is generated by the elements vn^n (v = i,j, k) and the elements qr^s for 1 < r < n, 1 < s < n (q = l,i,j,k); if r = s and q is real, then qrjS is compensated by 1„ „ + 2 • li 1 to belong to g0. To get the annihilator in g0 we add one element from sp(l) and the element Z + i11 — ln,n. Taking the semi-direct product of this annihilator and the Abelian algebra HP, we get the graded algebra a^1 of maximal dimension provided 0 ^ tpi E V1: a* = (so(2) ffi (R ffi gl(n - 2, H) ffisp(l)right) * fjeie(8ra - 12, H)) k H™. This will be shown to be associated to the filtration on the symmetry algebra s of a geometry with k\ 7^ 0 in the next section. Remark 6. The annihilator algebras from Corollaries 7 and 9 are very similar but not isomorphic. The following is an explanation of this phenomenon. We reduce the curvature- and torsion-modules via the parabolic subalgebra p2,2n-2 (or p2 for n = 2) of sl(n,H). This yields the diagrams 3 -3 0100 0001 •-x-•—X—•-o--•--o—•—x—• 0 -4 3000 0001 •-X-•-X-•-o--•--o-•-X-• We note that the numbers above connected pieces correspond to the action of a semi-simple subalgebra, and the numbers above crosses only affects the scaling factors of the center of go. One can express this by the diagram with crosses removed 0 1 3 00 00 • • • •—o--•--o—• that is the same for the two modules, after a permutation. Hence the contribution from Qqs to the annihilator must be abstractly isomorphic in the two cases. The difference then comes from the action of ggs on f)+. 5. Realizations of sub-maximal models In the previous section we found the annihilator algebras a^1 and a^11 of maximal dimension that is it = 4n2 — An + 9 in both cases. To prove this is realizable, we follow the idea of [12, §4.2] and deform the graded bracket structure on a to obtain 12 BORIS KRUGLIKOV, HENRIK WINTHER AND LENKA ZALABOVÁ a new filtered Lie algebra f. We use the real highest weight vector in our modules (that correspond to the minus lowest weight vectors of the duals - note that we used flip of the Satake diagram in our construction). This is expected to correspond to the symmetry algebra of a submaximally symmetric model, which is (non-flat) homogeneous with the isotropy being f>o = cio (because of the prolongation-rigidity), and we show it is the case. To do this we follow the approach in [12, §4.1] that allows to establish an abstract model, basing on the extension functor construction. However we also provide explicit matrix models of the corresponding almost quaternionic manifolds (M, Q), for which the direct computation confirms the amount of symmetry is submaximal & = it. We consider the curvature and torsion cases separatetly. The corresponding theorems imply the main result of this paper. We consider the cases when non-zero k = (ki, k2) is either (tpi, 0) or (0, V'n)- One could also question if the submaximal symmetry dimension can be achieved when both torsion and curvature are non-zero, but even though abstractly the maximal annihilator algebras a^1 ~ a^11 the discussion in Remark 6 shows that the answer to the above question is negative. 5.1. Realization of the curvature model. Consider the case of non-vanishing curvature and vanishing torsion first. In order to realize the symmetry algebra, the minimal orbit in the abstract curvature module V11 needs to be reinterpreted as a deformation of the graded algebra a^11. This can be done by finding a Gq-equivariant map b : V11 = SlU*n 01MB = A2H*™ 0O, and using the Lie bracket deformation given by the image b(v) of a generator O^nG V11 of the minimal orbit Go • [v]. Lemma 10. The equivariant map b exists and is unique up to scale. Proof. The real module B is completely reducible, and we compute its decomposition into irreducible submodules by finding an sl(n, H)-invariant real subspace in the complexification after applying standard methods from the complex representation theory of A2n-\ = sl(n, H) ® C. We have (in complexification HP = C2™): (A2H*")C = A2C(2 • B*n) = 3 • A2H*™ © Sge*n. Here 2 • M*n means the direct sum of 2 copies of M*n, and similarly for 3 • A2H*™. The Cartan product S(M*n 0 adjt(n H) has the same highest weight as V11, and so is isomorphic to it as a complex ^n-i-module, with the isomorphism mapping the sl(n, H)-invariant real submodules into each other. This is the unique submodule in B of the required isomorphism type, so the map b is defined and is unique up to scalar multiplication (since End0o(Vn) = R, this scalar is a real number). □ We construct b in the complex tensor notations as in the previous section. Proposition 11. The bracket deformation on an extremal generator w E Ylg , corresponding to a minimal orbit Go • [w] in PY11, is given by the formula: b(w) =(i* A j* - 1* A k*) ® (i* ® li - 1* ® ]i + ]*n ®k1-k*n® n) -(In A fn +CA A*) ® (1* ® ll + C ® il + J* ® 31 +K® h). SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 13 Define the deformed Lie bracket on the space of a^11 via b(w): [> ]fn = [> ]o*n +&H(> )■ Similarly to [12, Lemma 4.1.1] one can check that this is a Lie bracket (the Jacobi identity holds), and the space a^11 equipped with it is a new (now filtered) Lie algebra fn- This deformation changes the Lie brackets of the previously Abelian subalgebra H™, and this subspace H„ C H™ becomes non-Abelian. The new bracket component takes values in the center of the subalgebra rjeis(8n — 12, H): [H„,H„]flI ca(fjeie(8n-12,e)). Notice that the semi-simple Levi factor of the symmetry algebra is unchanged by this deformation: fff = (a^n)ss = sp(l) ©sl(n — 2,H). Due to the presence of the subalgebra sl(n — 2, in fn, a sub-maximal model can be realized as a direct product (although this notion is coordinate dependent) of the flat structure H™~2 and a sub-maximal structure of dimension 2. Although the symmetry algebra is not solvable, its solvable radical acts locally transitively, which allows us to integrate the algebra and produce a coordinate description of the model. For n = 2 the operator I on H2 = M8(/ii,..., is given by the matrix I = 0 Ci Bi Ar = 1 0 -1 0 0 0 0 0 0 -1 0 1 Bt=^ ( 0 a2-2 h2 h4 \-2h2 h3 2h2 2h\ - a2 0 -2h2 h3 —2 h2 h4 —2 h2 hi 2h2 h3 0 a2 - 2h2 2h2h3 \ 2 h2 hi 2 2h2 0 / c\ 1 = 2^ 0 h2(4a2 -hi(Zhl - ,2 /h2h5 h2h6 h2h7 \h2hs (note the transpose). Here a2 -3h2) -a2) 3h2) -h3h7+ -h3hs + h3h5 --h3h6 - J = h2(2a2 - Zhl) h3(a2 - 2,h22) h4(a2 - 3^)\ n _UJh2<_n,2\ u_lu2_i_n2\ h3(3h2 h4(3hj h4hs h4h7 h4h^ h4h§ 0 + a2) + a2) -h2h6 -h2h5 --h2hs -h2h7 -+ -h4(h_ h2(3h2 - h3h8 h3h7 h h3h6 h3h5 a2) '3 1" hl) 2h2h3h4 - h4h7 h4h% - h4h§ h4h^ h3(h2 + a2) 2h2h3h4 h2(h2 + 3h2)J °\ 0 0 0/ hi -hi -h\. The operator J is given by B.j = Aj Cj 0 Bj 0 2h2h3 —2h3h4 \2h / 0 0 1 , Aj = 0 0 0 1 -1 0 0 0 \0 -1 0 V -2h2h3 2h3h4 a2- -2hf\ 0 a2 - 2h23 —2h3h4 2h23 - a2 0 -2h2h3 2h3h4 2h2h3 14 BORIS KRUGLIKOV, HENRIK WINTHER AND LENKA ZALABOVÁ c\ 1 4a2 h3(6h2-a2) —3h4a2 3h2(a2 -2hf, —6/12/13/14 / 3h4a2 h3(6h2-5a2) —6/12/13/14 \h2(6h2-a2) I h2h7 — h3h5 + h4h& 0 —h2h% — h3he — /14/15 0 —h2h5 — h3h7 + h4h% 0 \ h2liQ — h3h% — h4h7 0 (note the transpose). Here (52 = h\ — h\ — h\ To get the quaternionic structure for general quaternionic dimension n, re-denote the above operators for n = 2 by /(2) and J(2). Now let / be given as the block matrix with /(2) in the top 8x8 block, Aj on the following diagonal 4x4 blocks and zeroes elsewhere. Similarly let J be given as the block matrix with J(2) in the top 8x8 block, Aj on the following diagonal 4x4 blocks and zeroes elsewhere. Define K = I J. Denote the obtained quaternionic structure (/, J, K) by Qu. h2{<6h23-a2) 2/12/13/14 h3(ß2-4h2) -h4(3a2 + 4h2) h2h& + h3h8 — h4h7 —h2h5 +h3h7+h4h8 h2hs - h3h6 + h4h5 -h2h7 - h3h5 - h4h6 6h2h3h4 \ h2(3a2 -2h23) h4(3a2 -Ahf, h3(ß2-4h2)J °\ 0 0 Then we let K = IJ. Theorem 12. The quaternionic structure (M, submaximal dimension &2 = An2 — 4n + 9. )n) has symmetry algebra Su of Proof. The proof of [12, Lemma 4.1.4] gives the abstract parabolic model via the extension functor construction. The symmetry algebra s of this model contains (by construction) the deformed algebra fn constructed above. Thus we already have at least &2 symmetries. Since this coincides with the universal upper bound ii2 = it, there can be no more symmetries: Su = fn- □ Remark 7. There is a reductive decomposition s = f) + m, where m = g_i = H™. Moreover, we have [m, m] C f). Thus (F), m) is a symmetric pair. This reflects the fact that the submaximally symmetric quaternionic geometry is a locally affine symmetric space in the sense of [14]. Direct computations (in Maple's DifferentialGeometry package, for n = 2) gives locally a unique quaternionic invariant connection and this connection has vanishing torsion and parallel curvature. Thus the connection corresponds to the canonical connection on the symmetric space for the pair (f),m). This local connection is hypercomplex for the above I, J, K, and it is the unique Obata connection [1]. Because the structure is torsion-free, the connection is also one of the Oproiu connections and determines the class of Oproiu connections on the submaximal model. Moreover, the connection is Ricci-fiat and its curvature coincides with the quaternionic Weyl curvature (and in particular it is harmonic). However, Oproiu connections in the class are not Ricci-fiat in general. Finally, direct computation shows that there is no invariant metric (of any signature) on the submaximal model. 5.2. Realization of the torsion model. The case of non-vanishing torsion and vanishing curvature can be treated similarly. In this case, we immediately interpret the element as a deformation to the graded algebra a*, so the deformed Lie bracket on the space of a^1 is [> ]fi = [> ]o*i )■ As in the curvature case, the previously Abelian subalgebra HP will become non-Abelian, but in this case we have pi,Hi]fl CH„Ci", SUBMAX. SYMMETRIC ALMOST QUATERNIONIC STRUCTURES 15 which means that HP remains a subalgebra. It is a nilpotent ideal of the algebra fi acting locally transitively on the corresponding (local) homogeneous model F/K (cf. [12, Lemma 4.1.4]). Therefore, the minimal model is locally equivalent to a left invariant structure on the nilpotent Lie group corresponding to (HP, [, ]j:). As in the curvature case, the semi-simple Levi factor of the symmetry algebra is unchanged by the deformation, and due to the presence of sl(n—2, H) the model can once again be realized as a direct product of a submaximal structure in quaternionic dimension 2 and a flat structure in quaternionic dimension n — 2. The matrices for n = 2 in the torsion case turn out to be considerably simpler than in the curvature case. Namely, the operators / and J on H2 = M8(/ii,..., /i§) are given by the following matrices and IJ = K. I = J = (° -1 0 0 0 h7 0 1 0 0 0 hr 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 V 0 1 0 0 0 -hr °\ 0 0 0 1 0 0 0 0 -1 0 0 0 -h7 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0/ To get the quaternionic structure for general quaternionic dimension n, we again extend the above I by the 4x4 block-matrices that form the block-diagonal of the 8x8 matrix, and do similarly for J; then we define K = I J. Denote the obtained quaternionic structure (/, J, K) by Q\. Theorem 13. The quaternionic structure (M,Qi) has symmetry algebra S\ of sub-maximal dimension &i = 4n2 — 4n + 9. Proof. The proof mimics that of Theorem 12, and we conclude &i = iti = it, and therefore si = fi- □ Remark 8. The submaximally symmetric almost quaternionic geometry of torsion type is locally representable as a group. Such structures always have at least a one-parameter family of invariant connections [14]. Direct computations (in Maple's DifferentialGeometry package) shows that (for n = 2) there is a six-parameter family of invariant connections, each with vanishing curvature and parallel torsion. However, only a two-parametric sub-family is quaternionic, and all the invariant quaternionic connections are hypercomplex. Among all these connections, we can find exactly one connection such that its torsion coincides with the structure torsion of the hypercomplex structure, and this is the Obata connection of the hypercomplex structure [1]. However, the structure torsion of the hypercomplex structure differs from that of the almost quaternionic structure, because the submaximal model has non-vanishing (harmonic) torsion. There is no invariant quaternionic 16 boris kruglikov, henrik winther and lenka zalabová connection such that its torsion coincides with the structure torsion of the almost quaternionic structure. Thus, no Oproiu connection is invariant. Clearly, the class of Oproiu connections is invariant, but unlike the quaternionic submaximal model (torsion-free with curvature), there is no fixed point in the class. Remark 9. Suppose the almost quaternionic structure Q is induced by an almost hypercomplex structure /, J, K. Then the hypercomplex symmetry algebra consists of the quaternionic symmetries that preserve each of /, J, K by itself. In particular, the almost hypercomplex structure inducing the sub-maximal quaternionic structure Qu has hypercomplex symmetry algebras of dimension 4n2 — 4n + 6, while that for the almost quaternionic structure Qi has that dimension 4n2 — 4n + 8. References [1] D.V. Alekseevsky, S. Marchiafava, Quaternionic structure on a manifold and subordinated structures, Annali di Matematica Pura ed Applicata 171 (1996), no. 1, 205—273. [2] R.J. Baston, M.G. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, Oxford Math. Monographs, Clarendon Press, Oxford, 1989. [3] A. Cap, K. Neusser, On automorphism groups of some types of generic distributions, Differential Geometry and its Applications 27 (2009), no. 6, 769-779. [4] A. Cap, J. Slovák, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc, 2009. [5] B. Doubrov, D. The, Maximally degenerate Weyl tensors in Riemannian and Lorentzian signatures, Differential Geometry and its Applications 34 (2014), 25-44. [6] LP. Egorov, Collineations of projectively connected spaces (Russian), Doklady Akad. Nauk SSSR 80 (1951), 709-712. [7] LP. Egorov, Maximally mobile Riemannian spaces V4 of nonconstant curvature (Russian), Doklady Akad. Nauk SSSR 103 (1955), 9-12. [8] J. Hrdina, J. Slovák, Generalized planar curves and quaternionic geometry, Annals of Global Analysis and Geometry 29 (2006), 349-360. [9] N. Jacobson, Lie Algebras, Dover Publications, Inc., New York, 1979. [10] B. Kruglikov, Submaximally symmetric CR-structures, Journal of Geometric Analysis 26 (2016), no. 4, 3090-3097. [11] B. Kruglikov, V. Matveev, D. The, Submaximally symmetric c-projective structures, International Journal of Mathematics 27 (2016), no. 3, 34pp. [12] B. Kruglikov, D. The, The gap phenomenon in parabolic geometries, Journal fur die Reine und Angewandte Mathematik (Crelle's Journal) 723 (2017), no. 723, 153-216. [13] A.L. Onishchik, E.B. Vinberg, Lie Groups and Algebraic Groups, Springer—Verlag, 1990; translated from Russian Seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988. [14] K. Nomizu, Invariant affme connections on homogeneous spaces, American Journal of Mathematics 76 (1954), no. 1, 33-65. B.K. and H.W. Institute of Mathematics and Statistics, University of Troms0 Troms0 90-37, Norway; L.Z. Institute of Mathematics and Biomathematics, University of South Bohemia in České Budějovice, Branišovská 1760, České Budějovice, 370 05, Czech Republic Email address: boris.kruglikov@uit.no, henrik.winther@uit.no, lzalabova@gmail.com Chapter 5: Local geometric control of a certain mechanism with the growth vector (4, 7) article by Jaroslav Hrdina and Lenka Zalabova, submitted to Journal of Dynamical and Control Systems, revision requested arXiv: 1802.08480 LOCAL GEOMETRIC CONTROL OF A CERTAIN MECHANISM WITH THE GROWTH VECTOR (4,7) JAROSLAV HRDINA AND LENKA ZALABOVÁ Abstract. We study local control of the mechanism with the growth vector (4,7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories. 1. Introduction Originally, the general trident snake robot has been introduced in [11]. Let us recall that the trident robot is a mechanism composed of three snake-legs, each connected to an equilateral triangle root block in its vertices [10, 11, 12, 21] for further details. Generally, the branches can be multi-link, assumed that each link has its own passive wheel, which provides footing for the robot. Active elements, which affect controllability, are placed on branches. Its simplest non-trivial version, corresponding to one-links, has been mainly discussed, see e.g. [12, 10, 21]. In this case, the control distribution is that of the growth vector (3, 6) [20]. We are interested in the modification corresponding to one or more prismatic joints such that the control distribution will be a that of the growth vector (4,7). Local controllability of such robot is given by the appropriate Pfaff system of ODEs. The solution space gives a control system q = ^UiXi where the vector fields X\, X2, X3, X4 describe the horizontal distribution and u : R ->• R4 is the control of the system. Consequently, the system is controllable by Chow-Rashevsky theorem [1, 14, 5], see Section 2. We construct a nilpotent approximation to get nilpotent Lie algebra and corresponding Lie group to study the controllability of approximated left invariant control system, see Section 3. We study geometric properties and symmetries of the nilpotent approximation in Section 4. We use the theory of Hamiltonians and Pontryagin's maximum principle to study local control and extremal trajectories, see Section 5. In particular, we provide analysis of the system and present explicit solutions. 2. Analysis of the mechanisms In this Section we describe a mechanism that is a modification of the trident snake robot (for more details see [10, 11, 12, 21]). We present our mechanisms as a new example of non-holonomic systems with multi-generators and discuss local controllability of our mechanism based on the principles of non-holonomic mechanics. 2.1. Description of the mechanism and its movement. In the sequel, we study a generalized trident mechanism which consists of a root block in the shape of an equilateral triangle with three 1-link branches with variable length. Each of 2010 Mathematics Subject Classification. 53C17, 93C15, 34H05. Key words and phrases, local control, sub—Riemannian geometry, Pontryagin's maximum principle, nilpotent Lie group. 1 2 JAROSLAV HRDINA AND LENKA ZALABOVÁ Figure 1. Description of the mechanism the branch is connected to one vertex of the root block and they form prismatic joints. Second leg is in addition connected to the root block via the revolute joint, connection of the two remaining is fixed. Each link has a passive wheel on its branches, which is assumed neither to slip, nor slide sideways. The configuration space of the planar mechanism in question corresponds to a manifold that locally coincides with W7 (but not globally). Since we study local problems, we can consider the configuration space to be W7 with the coordinates (x, y, 9, if, li, £2, ^3)- With respect to a fixed coordinate system, the first three coordinates describe completely the global position of the mechanism in the plane. The coordinates x and y give the position of the centre of mass of the root block in M2 and 9 gives the amount of the counter-clockwise rotation. Remaining four coordinates represent the input for the mechanism's active elements. Thus as active elements we consider the revolute joint of the branch £2 with the root block, namely p, and prismatic joints, which can change absolute lengths of branches £i,£2 and £3, see Figure 1. To provide the description of the robot's movement, we shall only point out that we do not get any singularities as long as the robot's configuration satisfies that 0 < £i and p is not too far from 0. Using the method of moving frame, the kinematic model is the set of equations for i—th wheel position in the form Xi = x + cos(9 + on) + £i cos(9 + on), ^ yi = y + sin(0 + on) +^sin(6» + at), for z = l,3, where ct\ = — |7T, a3 = |-7r, and x2 = x + cos(0) + £2 cos(9 + ip), ^ v2 = v + sin(0) + £2 sin(0 + p). Consequently, we differentiate the position equations (1) and (2) with respect to time t and obtain the velocity equations as follows &i = x — s'm(9 + aAO — £i s'm(9 + aAO + cos(9 + aAti, yi = y + cos(9 + cti)6 + £i cos(9 + cti)6 + s'm(9 + Ui)£i, LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 3 for z = l,3, where ol\ = — |7T, a3 = |-7r, and x2=±- s'm(9)9 - £2 s'm(9 + p)(9 + ip) + cos(9 + ip)l2, y2=y + cos(9)9 + l2 cos(6> + p){9 + ip) + sin(6> + p)£2. The conditions preventing slipping lead to the non-holonomic constraints of the form 0 = (-sin(0 + al),cos(9 + at)) ■ (xl,yl), 0 = (-sin(0 + p),cos(9 + p)) ■ (x2,y2), where i = 1, 3, ol\ = — |7T, a3 = |-7r and • is the Riemannian scalar product on the Euclidean space W2. We obtain the following differential kinematic equations, so the movement of the mechanism is described by the Pfaff system of three nonlinear homogeneous equations 2"7T 2"7T 0 = - sin(0--)dx + cos(0--)dy + (1 + ^i)d0, 3 3 (3) 0 = - sin(0 + ^)dx + cos(0 + ^)dy + (- cos(p) + £2)d9 - £2dp, 2tt 2tt 0 = - sin(0 + — )dx + cos(0 + — )dy + (1 + £3)d9. 2.2. Local controllability of the system. The space of solutions to the system (3) forms four dimensional distribution on the configuration space, so-called horizontal distribution. It follows from (3) that the solution space always contains the vector fields X2 := d^, X3 := dn2 and X4 := dn3 as generators. In the case £2 / 0 the last generating vector field X\ is a combination of dx,dy,de and dv which is generically complicated and we do not need to write it here in the biggest generality while X\ = dv in the case £2 = 0. In fact, from the mechanical point of view, zero length of each leg makes no sense, so we suppose in the following that li > 0. We can equivalently rewrite the solution space of the Pfaff system (3) in the following form (4) q = uxXx{q) + u2X2(q) + u3X3(q) + uAXA{q), where q = (x,y, 9, p, £1, £2, £3). This is a 4-input symmetric affine control system. In general, controllability of symmetric affine systems is completely characterized by the controllability Lie algebra by Chow-Rashevsky's theorem [14, 23, 19]. Control system (4) satisfies the Chow's condition at the point q if Lie(Xi, X2, X^,X^){q) = TgW7, where the controllability Lie algebra Lie{X\, X2, X3, X4) is the Lie algebra generated by Xi, X2, X3, X4. In this case point q is regular and can be connected with any point in a suitable neighbourhood of q by a horizontal trajectory. If the Chow's condition is satisfied at all points (of a connected space) then any two points can be joined by a horizontal trajectory and the system is locally controllable. In our case, mechanical description leads to an observation that local controllability depends on the shape of the mechanism only, not on its configuration in the plane. In other words, regular points have to be invariant with respect to the rigid body (Euclidean) transformations of the plane. So we choose x = y = 0 and 9 = ^ without loss of generality. In particular, for points of the form qo = (0, 0, -|, ip, £±, £2, £3) the vector fields Y , , (li-l3)V3a 1 sin(y)v^(£1 - £3) + 3cos(y)(L + 1) + 3l2 Xl = 9x +-3L-dv-J]d6 +-3£7L-9v X2 = de±, X3 = de2, X4 = de3, 4 JAROSLAV HRDINA AND LENKA ZALABOVÁ generate the solution space of our Pfaff system, where we denote L = £\ + £3 + 2. Moreover, at qo, the controllability Lie algebra is obtained by the Lie bracket operation as -2(45 + 1). 1 „ -2sm((f)(£3 + í) + VŠcos((f) + VŠ£ Y lY Y] sin(y)(£1-4) + v/3cos(y)(L + l)fl X13 . [Xl,X3]--9, 2h+2 1 2sin(v3)(4 + l) + v/3cos(v3) + v/3^: and remaining brackets are trivial. Then the matrix G consisting of coordinates of vector fields Xl, X2, X3, Xi, X12, X13, X14 spans full tangent space W7 as long as det(G((jo)) A 0 and the system is locally controllable at q0. Thus our system is locally controllable at qo (and without loss of generality everywhere because of mechanical meaning) because £2 =/= 0 and L =/= 0. We consider the filtration A1 = (X1,..., XA) C A2 = (X1,..., XA, X12, X13, Xu). We have dim A1(q) = 4 and dim A2(q) = 7 at all points and we have filtration with the growth vector (4, 7). 2.3. Remark on corresponding dynamical systems. If we restrict our considerations only to nontrivial movements of the root block, i.e. movements in the X\ direction or in the direction of iterated bracket of Xi and Xi for i = 2, 3,4, we can use methods of dynamic pairs. In general, each control affine system m x = X(x) + J2uJYj(x) i=i on a manifold M, where X, Y±,... ,Ym are smooth vector fields on M and u = (ui, .. ., um)T are controls, defines a dynamic pair (X, V), where V is a distribution spanned by Y\, ..., Ym. Then there is a sequence of distributions defined inductively, using Lie bracket, by V° := V, Vl+1 := V1 + [X, V1] and one imposes the regularity conditions [13, 8]: rk V1 = (i + l)m, for i = 0,. .. , k, Vk © (X) = TM. Then our control system (4) can be adapted to this situation as X := f(q)Xi, where / is an arbitrary non-zero function and Y\ := X2,Y2 := ^3,13 := X4, and the regularity conditions are satisfied. Indeed, V° = (Y1,Y2,Y3), V1 = (Y, ,Y2,Y3, [X, Yj], [X, Y2], [X, Y3]), such that rk V° = 3 = to, rk V1 = 6 = 2m and V1 © (X) = TM. In fact, each f(q) defines a dynamical system with specific drift and all of them are regular. 2.4. Remark on dual curvature. Following [18, 7], curvature of a distribution H on a manifold Q is a linear bundle map F : A2H —>• TQ/H defined by F(X, Y) = — [X,Y] mod H. Denote by H1- the bundle of covectors that annihilate H. Since the curvature F is a linear bundle map, the dual of curvature is a linear map F* : H^- —>• A2H*, called the dual curvature. Because our distribution is equipped with the growth vector (4,7), the space Im(F*) is three-dimensional subspace of A2H*. We can define the Pfaffian H^- ->• A4H* as \i ^ F*(n) A F*(n). One can LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 5 see that A4H*, q G Q, is one-dimensional vector space and the Pfaffian may be understood as a real valued quadratic form on H4- by choosing a volume form. Then possible signatures for the Pfaffian are (3, 0), (2,1), (2, 0), (1,1), (1,0), (0, 0). Note that the signatures (p,r) and (r,p) must be considered as identical because A4H* is not oriented. From parabolic geometry point of view, the generic distribution with growth vector (4, 7) corresponds to quaternionic contact structures and split quaternionic contact structures for signature (3,0) and (2,1), respectively. The distribution corresponding to our mechanism has signature (0, 0) and we will see that it is a parabolic geometry called generalized path geometry [6]. Let us point out that there exist different modifications of trident snake robot that lead to the growth vector (4,7). However, it turned out that all of them have non-regular signature. We have not found a mechanism with signature (3, 0) or (2,1), yet. 3. NlLPOTENT APPROXIMATION We recall a constructive method to approximate vector fields of a nonlinear control system by a similar system on the same configuration space. The method leads to an approximate distribution which has a nilpotent basis. The techniques of nilpotent approximation have been developed by various researchers, see e.g. [2, 9]. We recall the following concept of orders of functions or vector fields and distribution weights. Let Xi, i = 1,. .. , to denote a family of smooth vector fields on a manifold M and Cco(p) the set of germs of smooth functions at p G M. For / G Cco(p) we say that Lie derivatives Xif,XiXjf,... are non-holonomic derivatives of / of order 1,2,.... The non-holonomic derivative of order 0 of / at p is f(p). Then the non-holonomic order ordp(/) of / at p is the biggest integer k such that all non-holonomic derivatives of / of order smaller than k vanish at p, i.e. ordp(/) = minjs G N : 3h ... is G {1,..., to} s.t.(Xl± ■■■XlJ)(p) ^ 0 Denote by VF(p) the set of germs of smooth vector fields at p G M. Then the notion of non-holonomic order extends to vector fields as follows: For X G VF(p) the non-holonomic order ovdp{X) of X at p is a real number defined by ordp(X) = supjcr G R : oidp(X f) > a + ordp(/), V/ G C°°(p) Note that ordp(X) G Z. Moreover, the zero vector field X = 0 has infinite order, i.e. ordp(0) = oo. Furthermore, X\,. .. ,Xm are of order > —1, [Xi,Xj] of order > -2, etc. 3.1. Construction. We construct a nilpotent approximation of the distribution with respect to the given filtration at point qo = (0, 0,^,0,1,1,1). We use Bella'iche algorithm, which may be found in [4]. Let us point out that all constructions are local in the neighbourhood of qo. In our case, as the first step of Bella'iche algorithm, the adapted frame (5) Xi,X2, X3,X4, Xi2,-X13, Xi4 was chosen. Then we use four local coordinates (x, l\, £2, £3) as first four adapted coordinates. The others can be obtained from the original coordinate system by an 6 JAROSLAV HRDINA AND LENKA ZALABOVÁ affine change in the form yi = -2x - 2v/3y - 86», 4 4 8 (6) y2 = -^p- -^x+ -9, y3 = -2x + 2y/3y - 89, where we use the following conditions: dx\q0 = Xxiqo), d^lqg = X2(q0), de2\qo = X3(q0), dk\qo = X4(q0), dyi\qo = Xi2(qo), dV2\qo = Xi3(q0), dya\qo = X23(q0). Finally, note that the adapted coordinates (x, £i, £2, £3, yi, y2,2/3) are privileged coordinates in the case of adapted frame with growth vector (4, 7) [14]. Following [14], we group together the monomial vector fields in Taylor expansions of the same weighted degree and thus we express Xi as a series xi = x(r1Hx^ + x^ + -, for i = 1,2,3,4, where X^ is a homogeneous vector field of order s. By [14, Proposition 2.3.] we set Ni := x\ ^ for i = 1,2,3,4. The family of vector fields (Ni, N2, N3, N4) forms called first order approximation of (Xi, X2, X3, X4) at qo and generates a nilpotent Lie algebra of step r = 2, i.e. all brackets of length greater than 2 are zero. The family (Ni, N2, N3, N4) is called the (homogeneous) nilpotent approximation of {X\, X2, X3, X4) at qo associated with coordinates (x,e1,e2J3,y1,y2,y3). In the sequel, by the above algorithm, we obtain the following vector fields: JVi = ax - y~irx + ^ - lJ dvi - (£2 - iH2 - + £3 -1J dya, N2 = d£l, N3 = di2, N4 = de3. In particular, the family of vector fields (Ni, N2, N3, N4) is the nilpotent approximation of vector fields {X\, X2, X3, X4) at (0, 0,^,0,1,1,1) in the coordinates (x, y, 9, if, £i,£2, £3), while it is the point (0,1,1,1, — A-k, |"7t, — Air) in the coordinates (x,e1,e2,£3,y1,y2,y3). The remaining three vector fields Ni2, Ni3, N14 are generated by Lie brackets of (N^N^N^Ni) as N12 = [N1}N2] = 8yi, N13 = [iVi, N3] = dy2, N14 = [iVi, N4] = dV3. Note that due to the linearity of coefficients of (Ni, N2,N3, N4), the coefficients of (Ni2, Ni3, N14) must be constant. To show how the nilpotent approximation affects integral curves of the distribution and the resulting control, we compute the Lie brackets of relevant vector fields. In Fig. 2, there is a comparison of the Lie bracket motions in the original distribution (red line) and in the nilpotent approximation (blue line). The following figures show the trajectories of the root centre point, vertices and wheels when a particular Lie bracket motion is realized. To simulate the bracket motion of the nilpotent approximation, we choose the initial state qo and apply the periodic input on couples (Ni, N2), (Ni, N3) and (Ni, N4) to receive the displacement approximately parallel to [Ni,N2], [Ni,N3] and [A^i,^] respectively [19]. More precisely, for one cycle we apply ui(t) = — Auj sin(u;£), Ui(t) = Aujcos(ujt), Uj(t) = 0, Uk(t) = 0 local control of a mechanism with the growth vector (4,7) 7 A) C) D) Figure 2. Motions of the mechanism in directions: A) N12, X12, B) N13, X13, C) JVu, X14, D) iVi, Xlt for i e {2, 3,4} and j, k e {2, 3, 4} — {i}, respectively, all with respect to the control system (10), amplitude A = 0.4 and angular speed uj = Then we apply the same process to original vector fields Xi,X2, X3 and X4. 3.2. Control theory on nilpotent Lie groups. The above construction led to vector fields Ni, N2, N3, N4, N12, N13, N14 which determine a 7-dimensional nilpo-tent Lie algebra n. The computation gives that [N1,N2]=N12, [N1,N3] = N13, [N1}N4] = N14 and the remaining brackets are trivial. The corresponding connected simply connected nilpotent Lie group N ~ R7 with the nilpotent Lie algebra n is then endowed, in new coordinates (x, £1, £2, £3,yi,y2,y3), with the following group structure (7) fx\ (x\ £1 £1 £2 £2 £3 X £3 Vi Vi V2 V2 W x £1- £2-£3- x £1 £2 £3 \ Vi ^ xx £xx V2 £2x \V3 ^■xx and the vector fields N,, N9, N3, NA, N- 12, V2 H f y3 - yi-i- - t3A Ni3, N14 are left-invariant with respect £3xj to the left action given by the group structure. In particular, the vector fields Ni for i = 1,2,3,4 determine a left-invariant distribution T> on N, which has the growth vector (4, 7) everywhere. Altogether, T> C TN defines an invariant 4-input symmetric affine control system (8) q = U1N1 + u2N2 + u3N3 + U4N4, with q = (x,£i,£2,£3,yi,y2,y3), that approximates the original control system. It clearly satisfies the Chow's condition and is controllable. 4. Infinitesimal symmetries We focus on basic geometric properties and infinitesimal symmetries of the nilpo-tent approximation. By infinitesimal symmetries we mean vector fields such that their flows preserve the geometric structure at any time [22]. In our case, infinitesimal symmetries preserve the distribution and also the metric. 8 JAROSLAV HRDINA AND LENKA ZALABOVÁ 4.1. Generalized path geometries and their symmetries. Let us discuss the geometric structure that occurs behind the control problem (8). In the previous section, we constructed a nilpotent Lie group N with a filtered nilpotent Lie algebra n with the growth vector (4, 7), where the 4-dimensional distribution T> is generated by the left-invariant fields N\, N2, N3 and N4. Consider subbundles E = (Ni), V = (N2, N$, N4) in TN. One can see from the structure of Lie brackets that the following holds: (1) Er\V = 0, (2) Lie bracket of two sections of V is a section of E ® V, and (3) for sections £ G T(E) and v G T(V) and a point q G N, the equation [£, v\{q) G Eq © Vq implies that £(q) = 0 or v{q) = 0. Such geometric structures are usually called (generalized) path geometries (in dimension 7) [6, Section 4.4.3]. General theory [6] says that generalized path geometries have finite-dimensional Lie algebras of symmetries and in the case of 7-dimensional manifolds, the maximal possible dimension is 24. This is the case of generalized path geometries that are locally equivalent to a generalized flag manifold PSL(5, R)/Pi,2, where by PSL(5, R) we denote the projectivised special linear group with Lie algebra sl(5,M), and by P\2 the stabiliser of the flag of a line in a plane for the projectivised standard action of PSL{b, R). In particular, the Lie algebra of symmetries for such generalized path geometry is exactly the simple Lie algebra sl(5,M) and the symmetries with a fixed point form its 17-dimensional parabolic subalgebra. There is a general method to find all infinitesimal symmetries of a nilpotent filtered structure [22]. One can apply this method to our structure (N, E(&V C TN) and it turns out that the symmetry algebra is of dimension 24 and is exactly sl(5, R). In other words, symmetries of (N, E(&V G TN) are left multiplications by elements of a Lie group with the Lie algebra sl(5, R) and the structure is left-invariant with respect to this action. From this point of view, N ~ PSL(P, R)/Pit2- Altogether, nilpotent approximation forms a flat generalized path geometry. Remark 1. The concept of Cartan geometries [24] generalizes the concept of Klein geometries [15, 16] and generalized flag manifolds are special cases of Klein geometries for the case of parabolic subgroups in semisimple groups. The Cartan's generalization then leads to a wide theory of parabolic geometries [6], that are curved versions of flag manifolds. Generalized path geometries are examples of such parabolic geometries [6, Definition 4.4.3.]. 4.2. Sub—Riemannian structure and its symmetries. To study extremal trajectories in the next section, we need the sub-Riemannian structure on the nilpotent approximation. We consider a control metric g in T> = (Ni, N2, N3, N4) such that the fields Ni for i = 1,2,3,4 are orthogonal and have the length one with respect to g. This clearly determines a left-invariant sub-Riemannian structure g of T> (with respect to the action given by group structure (7) on N). Let us now focus on the symmetries of the nilpotent control problem (M, T> = E + V, g). Thus we are interested in symmetries that preserve not only the flat generalized path geometry, but also the control metric. The symmetry algebra I of (M, E + V, g) clearly is a subalgebra of sl(5, R). In fact, both £ and sl(5, R) contain the same nilpotent subalgebra that reflects the nilpotent group structure and that LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 9 acts effectively and transitively onJV. It is generated by vector fields Wi ■■= -ax V3 2 w2 ■= dgl - xdVl, w3 ■= % - xdy2, W4 := dt3 - X3y3 , W12 — 3 ■— uyi' Wl3 — 3 Wl4 ■= dy3- X3y3 , In general, I also contains symmetries preserving arbitrary fixed point. Since all points are equivalent, we can fix the origin o = (0, 0, 0, 0, 0, 0, 0). If we study symmetries preserving the origin, intuition suggests that the sub-Riemannian metric g shall be preserved by an orthogonal algebra so(4). However, each such symmetry shall also preserve the control distribution T> and its decomposition into E and V. Thus it acts trivially on the 1-dimensional subspace EQ and restricts to the action of so(3) on the 3-dimensional subspace V0. Direct computation gives that there really is the symmetry algebra so(3) C £ preserving the origin generated by fields V3x2 vx := -4% + £2de3 - ( —--x + y3)dV2 - (x - y2)dV3, ,V3x2 ^ ,V3: 2 (9) v2 := £33ei - hdn3 + (—--x + y3)dyi + (—---h x - y{)dy Jy ,V3_x2 4 We can write it also in a 'vector-matrix-like' notation as v3 := -£2de1 + hde2 + (x - y2)dVl - (—--h x - yi)dV2. /0 -h l2 v2 = h 0 -h v3J 0 y ( 0 (V3x2 \ 4 2/3 \ * -2/2 j VSx2 y 4 - x + y3) 0 + x-yi) One can verify directly that LVV C V and CViN\ = 0 and CVig = 0 for i = 1, 2, 3, where £ denotes the Lie derivative, and that [fi,f2] = — v3, [vi,v3] = v2 and [^2, ^3] = -vi- 4.3. Properties of the so(3)—action. The action of so(3) is clear if the action of the generators vi,v2,v3 is explained: the action decomposes into two independent actions first of which is on W3 given by d^, di2 and di3 and the second one on W4 given by dx, dVl, dV2 and dV3. Moreover, the algebra so(3) acts trivially on dx, which then defines a one-dimensional invariant subspace. The action of the algebra so(3) on (d^, dn2,de3) G M3 corresponds to the rotations around all axes passing through (0, 0, 0) and this does not depend on the remaining variables x, y\, y2 and y3. In particular, for i = 1,2,3 corresponds to the rotation around the axis generated by d^. Analogously, for arbitrary fixed x, the algebra so(3) acts on (dVl, dV2, dya) via rotations around axes going through (x + ^3^ , x, x — ). In particular, vi for i = 1,2,3 corresponds to the rotation around the axis generated by dVt. One can also see from the shape of the generators that the rotation around dii is tied to the rotation around dVi for i = 1,2,3. Thus the action on (de1, dg2, dg3) 10 JAROSLAV HRDINA AND LENKA ZALABOVÁ determines the action on (dyi, dy2, dya) and vice versa. In fact, x parametrizes (in the coordinates (x, £\, £2, £3, yi, V2, 1)3)) the curve (x, 0, 0, 0, x+ , x, x — ) C N which can be viewed as a curve of centres of the above 'double-rotations'. In particular, the nilpotent sub-Riemannian structure (N, T> = V + E, g) is invariant with respect to the action, and we can study its action on curves passing through the origin. Assume that c(t) is a (parametrized) curve such that c(0) = o. Consider the flow Fl* of the infinitesimal symmetry v := a\V\ + a2v2 + C13V3 G So(3) for some a\,a2,a3. It clearly preserves the origin o. Assume that the point c(to) 7^ o for some to 7^ 0 is preserved by the action of Fl*. Then either the curve c(t) is preserved by this action on [0, to], or the action determines a family of curves of the same length from o to c(to) on [0, to]. In particular, if such a curve c(t) that is not invariant with respect to the action of Fl* is an extremal curve for the invariant control system, then it is no more minimiser after it reaches the point. Moreover, if one finds one such point then its orbit with respect to the action of so (3) consists of such points. Indeed, a family of curves from o to c(irj) is mapped to the family of curves of the same length from o to c(t0), where c(t0) is image of c(t) with respect to the action of so (3). We can describe explicitly the set of such points that are fixed for the action of (the flow of) some infinitesimal symmetry a\V\ + a2v2 + a3v3. First, one can check that each point of the curve (x, 0,0,0,:r+ Xf-,x,x- Xf-) is preserved by each such symmetry. Then the fixed points of any symmetry are given by axes of the corresponding 'double-rotations'. Explicitly, the fixed points of the symmetry ol\V\ + a2v2 + CL3V3 form the set / , V^x2 , , VZx2 , , , ^ (x, ka\, ka2, ka3, x H-----h ka\, x + ka2, x-----h ka3) : k G R Let us finally say that for each ki7 i = 2, 3,4 it holds that [Ni, k2N2 + k3N3 + k4N4] = k2N12 + k3N13 + k4N14. So the triple (N1,k2N2 + k3N3 + k4N4, k2N12 + k3Ni3 + k4Ni4) determines a subalgebra which has the structure of the Heisenberg algebra. One can see from the above mentioned that the action of the symmetry algebra so (3) simply maps each such Heisenberg subalgebra to another Heisenberg subalgebra. 5. PONTRYAGIN'S MAXIMUM PRINCIPLE We study local control of particular mechanisms. We use Hamiltonian formalism and Pontryagin's maximum principle to find local length minimisers. We study the corresponding Hamiltonian system of ODEs in detail. Finally, we model several explicit movements of the mechanism. 5.1. Formulation of the problem. Consider two configurations qi,q2 in the nilpotent approximation N. Among all admissible curves c(t), i.e. locally Lips-chitz curves such that c(0) = q± and c(T) = q2 that are tangent to T> for almost all t G [0,T], we would like to find length minimisers with respect to g. We would like to minimize the length I among all the horizontal curves c, where the length is given by 1(c) = jQ g(c, c)dt for the control metric g. Let us recall that the distance between two points qi, q2 G N is defined as d : M x M —>• [0, 00], d{li,l2) = inf{ce5qi q2j 1(c), where Sqi,q2 = {c : c(0) = qlt c(T) = q2,c admissible} [1, 5, 14]. However, since minimizing of the energy of a curve implies minimizing of its length, we will rather minimize energy of curves. We know from Chow-Rashevsky theorem that the control system of the nilpotent approximation is controllable, see Section 2.2. In particular, any two points can be joined by a horizontal curve and the distance of arbitrary two points is finite [1]. LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 11 We study the following nilpotent control problem / 1 \ ui -4 +1 V -^x-e3 + i J u2 ( 0 \ 1 0 0 0 0 V°7 "3 / 0 \ 0 1 0 0 0 V 0 y U4 ( 0 \ 0 0 1 0 0 V 0 ) for q G N and the control u = (1*1,1*2, «3, «4) G M4 with the boundary condition q(0) = (x0,4o,4o,4o, 2/10,2/20,2/30) and q(T) = (xi, ^n, ^2i,4i,yn, 2/21,2/31) arbitrary but fixed, and we minimize (11) u\) dt. Without loss of generality, we choose the origin o = (0, 0, 0, 0, 0, 0, 0) as the initial point q(0) = q\. Since we solve nilpotent control problem, we get curves starting at different points using the left action coming from the multiplication in N, see Section 3.2. 5.2. Hamiltonian formalism. Let us consider a cotangent bundle T*N —>• N and the coordinate functions hi = (A, Ni), where A G T*N. Then we consider the Hamiltonian of the maximum principle H(X, v) = uihi + u2h2 + u3h3 + u^h^ + -(w2 +u\ + u\l + u\), which is a family of smooth functions affine on fibres that is parametrized by controls (ui, U2,u3, U4) G M4 and a real number v < 0. The Pontryagin's maximum principle can be formulated as follows [1]: Assume (u(i), c(i)) to be a pair such that c(i) is a length minimizer for (10), (11) with control function u = u{i). Then there exist a Lipschitz curve A(t) G T*^M and a number v < 0 such that (X(t),u)^0, \(t) = Hm(\(t)) for Hamiltonian vector field H corresponding to H and -fffi(t) = max H(X(t), v). If \(t) satisfies the principle for v = 0, then it is called abnormal and it is called normal otherwise. The abnormal minimiser is strictly abnormal, if it is not normal. It follows from the Goh condition [17, 1] that there are no strictly abnormal minimisers in the case of 2-step distributions. Thus we will focus on the case v < 0 and we can normalize it in such a way that v = — 1. The extreme is achieved when = hi—Ui = 0 for i = 1,2,3,4 and this implies for the controls that Ui = hi for i = 1,2, 3,4. In this case, the Hamiltonian of the maximum principle is of the form H= \{h\ + h22 + hl + hl). Then the corresponding Hamiltonian system associated with H is the following (12) q = ftiiVi(g) + h2N2(q) + h3N3(q) + ^4^4(0), (13) hi = {H,hi}, where Ni wee the generators of the sub-Riemannian structure, q G N and { , } is the usual Lie-Poisson bracket [1]. Using the fact that {hi,hj} = (A, [N,Nj]) we 12 JAROSLAV HRDINA AND LENKA ZALABOVÁ conclude that the fibre system (13) is of the form -^i(t) = -h5(t)h2(t) - he(t)h3(t) - hr(t)h4(t) at ih2(t) = ft5(t)fti(t) at (14) ±h^t) = K{t)hl{t) at ^-h4{t) = ft7(t)/ii(t) at ih5(t) = ^-h6(t) = ^-h7(t) = 0. at at at The equations are clearly independent of the horizontal coordinates q. The base system has the form jtx{t) = h^t) Jtm = k2[t) Jtm = hd{t) ^-Mt) = h4{t) (15) at jty2(t) = (l-e2(t))h1(t) ^y3(t)=(l-^x(t)-e3(t^j h.it). 5.3. Analysis of the fibre system. Let us first discuss the fibre system (14). Obviously, functions h5, h§ and h7 are constant. If they are all zero, then the functions h\, h2, h3 and h4 wee constant, too. Let us denote the solution constants corresponding to h$, and h? by C$, C$ and C7, respectively, and assume that at least one of them is non-zero. Define K = ^/C2 + C2 + Cj. Then we get (16) h\ = -Cbh2 - C6h3 - C7h4 = -(C2 + C*62 + C?) hx = -K2hx. Since K2 > 0, the solution of (16) is (17) h1 = C11 cos(Kt) + C12 sm(Kt), for some constants Cu and C12. Now, if C5 A 0, we have h2 = C*5/ii = C*5(C*ii cos(Kt) + C12 sm(Kt)) and hence (18) h2 = §(Ciisin(2ft) - C12(cos(Kt)) + C13. This analogously holds for h& and h? if C§ A 0 and C7 A 0, respectively. In the same way we get h3 and h4. Indeed, only the equation for function h\ merges everything together. We know from the above mentioned, that the functions hi for i = 1, 2, 3,4 equal to the controls Ui of the system. Thus we control each vector field Ni with a function Ui which is either a constant or which oscillates. In fact, Ni, i = 2,3,4 LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 13 corresponds to dt , j = 1, 2, 3 and reflects the movement of legs which is natural from the mechanical point of view. The field Ni is the crucial field for the movement and if it is controlled by a constant, then the remaining fields Ni are controlled by constant, too. Let us point out that the choice of h$, and h-[ to be zero or non-zero, in fact, corresponds to the choice of constants in the solution. Since that corresponds to the choice of initial conditions, we can interpret the choice of zero or non-zero solution h5, h§ and h? as the choice of initial conditions, equivalently. Moreover, we are interested in the solution of the whole system (14,15) and we are basically interested in the curves, that are images in N with respect to the canonical projection T*N —> N. There can be curves of that type in N that differ only by the parametrization. It is reasonable to consider the curves parametrized by the arc-length, only. These are exactly the images of restrictions of the canonical projection to the solutions of (14, 15) with initial conditions satisfying h2 + h2 + h3 + h2 = 1 at the initial point (and thus everywhere). Finally, let us remark that the choice he = h-[ = 0 implies that h3,h^ are constants and the remaining equations result in the system h\ = —h^h2,h2 = h5hi,h5 = 0, which is the fibre system of the control problem on the Heisenberg group. 5.4. Analysis of the base system. Let us first say that it is enough to study solutions with the initial condition q(0) = o. Then one can use the action given by the multiplication on N to get a solution starting at an arbitrary point in N. The first four equations of the system depend on hi, i = 1, 2, 3, 4 only and can be computed by direct integration. Thus the equation x = h\ gives that either x = C\t for some constant C\ in the case that h\ is constant, or X=~K am(Kt) ~ ~K~ cos(Kt) + -fern the case that hi is of the form (17), where K,Cn,Ci2 are from the previous section. Then we compute directly from (18) that £i = fl(Cl1 - Cl1 co,9. We do not mention all the solutions explicitly. We rather provide several examples to demonstrate the explicit paths and movements of the mechanism in the following section. Let us finally remark that in the case that h5 = h§ = hj = 0, we can find for each solution an infinitesimal symmetry such that the solution is contained in the fixed points set of the symmetry. The situation is more complicated in the case when some of h^,h^, h-[ is non-zero. 14 JAROSLAV HRDINA AND LENKA ZALABOVÁ 5.5. Examples of solutions. Let us present several particular solutions (satisfying all initial conditions of the system and also constraints coming from the mechanical setting). To get an information about the movement of our mechanism, we transform the solutions into the original coordinates which express the kinematics of the mechanism. So we use the transformation between coordinates (x, £i,£2, h, Vi, 2/2,2/3) and (x,y,d,ip,£i,£2,£3) in the form 1 1 V = "2/2 ■ 7 2/1 ;2/3, 16 2/i —y-?--x, 16 4 ' 2/ = ~j^V3(yi - 2/3), which is the inverse transformation to the transformations (6). We also add illustrative graphs of behaviour of the control parameters which are prismatic joints £i and the revolute joint ip. Example 1. In the case that /15 = h§ = hi = 0, we can choose hi = j^, h2 = h3 = \, /14 = jq and, with suitable choice of constants, we get the solution in the form 7t To' 2/=( 49ť ~ 200 7VŠ 49 600 2A '1 10 ' 800 = t2 1600 2' 3 10' 21* "80"' Figure 3. The graphs of control parameters: Example 1 Example 2. In the case that /15 =/= 0 and h^ = /17 = 0, let us choose the constants in such a way that /15 = 1 and hi = — ^ sin(t) + \ cos(t), h2 = | sin(t) + \ cos(t) and /13 = /14 = ^. Then we get the solution in the form x(t) = ^sin(ŕ) + ^cos(i) - y(t) = -— (y/Ž{sm(ť) - l)(cos(ŕ) - 1) + cos(í)2 + t cos(ŕ) + (t - 2) sin (ŕ) + t - í) 48 V / 0® = ~hcos^2 + t~7urcos^ ■ RA —- RA ' t\A' 64 64 64 64 64 1 . ,9 36 - 11í . . 58 - 11í . . . í 37 M) = Y2 -(t)2 + —-- cos(í) + —^- am(t) + - - -. 4 (í) = 1 sin(í) - i cos(í) + i i2 = i3 = Í í — 12 í 11 ■ sin(í)---h local control of a mechanism with the growth vector (4,7) 15 0 1 2 3 0 1 2 3 Figure 4. The graphs of control parameters: Example 2 Example 3. Let us finally discuss the case that all h5, /i6, h% are non-zero. Let us choose the constants in such a way that hi = — -^pcos(t), h2 = —^^-sm(t) + \, h3 = /i4 = sin(t) + \, h§ = -^r, hs = h-? = — -^r. We get the solution in the form Figure 5. The graphs of control parameters: Example 3 6. Acknowledgment The first author is supported by the grant of the Czech Science Foundation no. 17-21360S, 'Advances in Snake-like Robot Control' and by a Grant No. FSI-S-17-4464. The second author is partially supported by the grant of the Czech Science Foundation no. 17-01171S, 'Invariant differential operators and their applications 16 jaroslav hrdina and lenka zalabová in geometric modelling and control theory' and by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme "Projects of Large Research, Development, and Innovations Infrastructures" (CESNET LM2015042), is also greatly appreciated. The algebraic computations are partially calculated in CAS Maple package Dif-ferentialGeometry [3]. We would like to thank to Sebastiano Nicolussi Golo and Wojciech Kryňski for helpful discussions and Pawel Nurowski for initial motivation. We thank the anonymous reviewers whose comments have greatly improved this manuscript. References [1] A. A. Agrachev, D. Barilari, U. Boscain. A Comprehensive Introduction to sub-Riemannian geometry Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2019 [2] A. A. Agrachev, A. V. Sarychev. Filtrations of a lie algebra of vector fields and the nilpotent approximation of controllable systems. Dokl. Akad. Nauk SSSR, 295(4):104--108, 1987. [3] I. M. Anderson, C. G. Torre. New symbolic tools for differential geometry, gravitation, and field theory. J. Math. Phys., 53(013511):1-12, 2012. [4] A. Bellaiche. The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry:l— 78, 1996. [5] O. Calin, D. Ch. Chang. Sub-Riemannian Geometry: General Theory and Examples, volume 126. Cambridge University Press, 2009. [6] A. Cap, J. Slovák. Parabolic geometries I, Background and general theory,, volume 154. AMS Publishing House, 2009. [7] Ch. De Zanet. Generic one—step bracket—generating distributions of rank four. Archivům Mathematicum, 51(5):257-264, 2015. [8] B. Doubrov, I. Zelenko. On local geometry of vector distributions with given jacobi symbols. arXiv:1610.09577 [math.DG], 2016. [9] H. Hermes. Nilpotent approximations of control systems and distributions. SIAM J. on Control and Optimization, 24, 1986. [10] J. Hrdina, R. Matoušek, A. Návrat, P. Vašík. Geometric control of the trident snake robot based on cga. Adv.Appl. Clifford Algebr., 27(l):633-645, 2017. [11] M. Ishikawa. Trident snake robot: locomotion analysis and control. IFAC Symposium on Nonlinear Control Systems, 6, 2004. [12] M. Ishikawa, Y. Minami, T. Sugie. Development and control experiment of the trident snake robot. IEEE/ASME Trans. Mechatronics, 15(1), 2010. [13] B. Jakubczyk, W. Krynski. Vector fields with distributions and invariants of odes. Journal of geometric mechanics, 5(1):85—129, 2013. [14] F. Jean. Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. Springer, 2014. [15] F. Klein. A comparative review of recent researches in geometry. Complete English Translation: https://arxiv.org/abs/0807.3161., 1872. [16] F. Klein. Elementary Mathematics from an Advanced Standpoint: Geometry, part 3: Systematic discussion of geometry and its foundations. Dover, 1939. [17] E. Le Donne, R. Montgomery, A. Ottazzi, P. Pansue, D. Vittone, Sard property for the endpoint map on some Carnot groups, Annales de lTnstitut Henri Poincare (C) Non Linear Analysis Volume 33, Issue 6, 2016, Pages 1639-1666 [18] R. Montgomery. A Tour of Subriemannian Geometries, Their Geodesies and Applications. AMS, 2002. [19] R. M. Murray, Z. Li., S. .S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton, FL, USA, 1994 [20] O. Myasnichenko. Nilpotent (3, 6) sub-riemannian problem. Journal of Dynamical and Control Systems, 8(4):573-597, 2002. [21] Z. Pietrowska, K. Tchoň. Dynamics and motion planning of trident snake robot. Journal of Intelligent & Robotic Systems, 75(l):17-28, 2014. [22] Y. L. Sachkov. Control theory on lie groups. J Math Sci, 156(3):381-439, 2009. [23] J. M. Selig. Geometric Fundamentals of Robotics. Springer, 2004. [24] R. W. Sharpe. Differential Geometry, Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, 1997. LOCAL CONTROL OF A MECHANISM WITH THE GROWTH VECTOR (4,7) 17 J.H. Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic; L.Z. Institute of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, České Budějovice, 370 05, Czech Republic and Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic Email address: hrdina@fme.vutbr.cz, lzalabova@gmail.com