Contents
Chapter 1. Lie groups 1
1. Lie groups 1
2. Lie algebras 2
3. Subgroups and subalgebras 6
4. Homomorphisms of Lie groups and algebras 9
5. The exponential map 11
6. Homogeneous spaces 13
7. The adjoint representation 17
8. Fundamental vector fields 19
9. Locally isomorphic Lie groups 21
10. Problems 23
Chapter 2. Bundles 26
1. Bundles 26
2. Basic operations with bundles 28
3. Jet bundles 30
4. Principal and associated bundles 33
5. Further properties of principal and associated bundles 38
6. Problems 39
Chapter 3. Connections 40
1. Connections 40
2. Principal connections 44
3. The covariant differential on associated bundles 47
4. The structure equation 50
5. The canonical form on P1
M (solder form) 54
6. The second tangent space TTM 55
7. Morphisms of connections 56
8. Problems 58
Chapter 4. Riemannian geometry 60
1. Interpretation of Riemannian geometry 60
2. The curvatures of a Riemannian space 62
3. Normal coordinates 65
4. The second fundamental form of a hypersurface 69
5. The geodesic curves of a Riemannian space 71
6. Geodesic variations 76
7. Problems 80
0
CHAPTER 1
Lie groups
1. Lie groups
Definition 1.1. A Lie group G is a group, which is at the same time a smooth manifold in
such a way that
• the multiplication µ : G × G → G is smooth,
• the inverse ν : G → G is smooth.
By a homomorphism of Lie groups we understand a smooth group homomorphisms.
Notation. We denote by e the unit and write a−1
instead of ν(a). We will be using the left
and right translations λa, ρa : G → G defined by
λa(b) = ab ρa(b) = ba
Theorem 1.2. The smoothness of the inverse follows from the smoothness of the multiplica-
tion.
Proof. The defining equation for the inverse is µ(a, ν(a)) = e. By the implicit function
theorem it is enough to verify that the derivative of µ(a, −) at a−1
is invertible. This follows from
the fact that µ(a, −) = λa has an inverse λa−1 .
Remark. Every Lie group is a topological group, i.e. a group and a topological group such
that the multiplication and the inverse are continuous. The fifth Hilbert problem states that every
topological group G that is at the same time a (topological) manifold admits a smooth structure
for which G becomes a Lie group. This was proved in 1952 (in fact the structure is even analytic).
If time permits we will get to the implication C2
⇒ C∞
.
Let M, N be smooth manifolds. Then the projections p : M × N → M and q : M × N → N
provide the canonical isomorphism
(p∗, q∗) : T(x,y)(M × N)
∼=
−−→ TxM × TyN.
The inverse isomorphism is obtained from the inclusions
iy : M → M × N jx : N → M × N
a → (a, y) b → (x, b)
Under the above identification the pair (X, Y ) ∈ TxM × TyN corresponds to (iy)∗X + (jx)∗Y ∈
T(x,y)(M × N).
Lemma 1.3. The following formulae hold for A, B ∈ TeG:
µ∗(A, B) = A + B, ν∗A = −A.
Proof. These are just simple calculations
µ∗(A, B) = µ((ie)∗A + (je)∗B) = (µie)∗A + (µje)∗B = A + B
and by differentiating e = µ(a, ν(a)) in the direction A ∈ TeG we get
0 = µ∗(A, ν∗A) = A + ν∗A
Examples 1.4. The classical gropups:
1
2. LIE ALGEBRAS 2
• The general linear group GL(n, R) - the group of invertible matrices (aij). Since GL(n, R) ⊆
Rn×n
can be described as GL(n, R) = det−1
(R − {0}) it is an open subset and hence a
manifold. Multiplication is clearly smooth (even algebraic).
• The general linear group GL(n, C) with coefficients in C. We think of GL(n, C) as a
subgroup of GL(2n, R) via the identification Cn
= Rn
⊕ iRn
. The embedding becomes
A + iB →
A −B
B A
On the other hand GL(n, C) ⊆ Cn×n
is again open and hence a manifold.
• The special linear groups
SL(n, R) = {A ∈ GL(n, R) | det A = 1}
SL(n, C) = {A ∈ GL(n, C) | det A = 1}
are certainly closed submanifolds and also subgroups. Later we will prove
Theorem. Every closed subgroup of a Lie group is a submanifold and with the
submanifold smooth structure a Lie group (i.e. a Lie subgroup).
• Let β : Rn
× Rn
→ R be a bilinear form represented by a matrix B = (bij). A linear
map α : Rn
→ Rn
is said to preserve β if
β(αx, αy) = β(x, y) ⇐⇒ AT
BA = B
Such linear automorphisms clearly form a closed subgroup of GL(n, R).
– Specifically for β = , , the scalar product, we have B = E, the identity matrix
and we obtain the orthogonal group
O(n, R) = {A ∈ GL(n, R) | AT
A = E}
and also the special orthogonal group
SO(n, R) = O(n, R) ∩ SL(n, R)
– Consider on R2n
the (nondegenerate antisymmetric) bilinear form
n
i=1
(xiyn+i − yixn+i)
with its matrix J =
0 E
−E 0
. The group of linear automorphisms preserving this
form is called the symplectic group Sp(2n, R). Analogously we obtain Sp(2n, C).
• The unitary group U(n) = {A ∈ GL(n, C) | ¯AT
A = E} and the special unitary group
SU(n) = U(n) ∩ SL(n, C). There is also a complex orthogonal group which is different
from the unitary group. One of the main qualitative differences is that O(n, C) is a complex
manifold and a complex Lie group (reason being that the defining equation AT
A = E
is holomorphic unlike that for the unitary group - it contains complex conjugation).
• The spin group Spin(n). We will say more about it later. It is related to SO(n, R) by a
short exact sequence of groups
1 → Z/2 → Spin(n) → SO(n, R) → 1.
• Sp(n) = {A ∈ GL(n, H) | ¯AT
A = E}, the group of linear automorphisms of the quaternionic
space Hn
preserving the scalar product. Also Sp(n) = Sp(2n, C) ∩ U(2n).
2. Lie algebras
Definition 2.1. A vector space L over R is called a Lie algebra if there is given a bilinear
map [ , ] : L × L → L satisfying
• the antisymmetry: [X, X] = 0,
• the Jacobi identity: [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.
2. LIE ALGEBRAS 3
From bilinearity we obtain
0 = [X + Y, X + Y ] = [X, X] + [X, Y ] + [Y, X] + [Y, Y ] = [X, Y ] + [Y, X]
implying [Y, X] = −[X, Y ].
Example 2.2. The vector fields on a smooth manifold M with the bracket [X, Y ]:
X =
i
Xi
∂
∂xi , Y =
i
Yi
∂
∂xi =⇒ [X, Y ] =
i,j
Xj
∂Yi
∂xj − Yj
∂Xi
∂xj
∂
∂xi
Let L be a finite dimensional Lie algebra and e1, . . . , en its basis. Then [ei, ej] = k ck
ijek.
The numbers ck
ij are called the structure constants of L with respect to the basis. They satisfy
the following identities:
• ck
ji = −ck
ij,
• k(ck
ijcm
kl + ck
jlcm
ki + ck
licm
kj) = 0.
Conversely, by giving the basis e1, . . . , en and the structure constants ck
ij satisfying the above
equalities we obtain a Lie algebra L. The complete classification of Lie algebras is not yet known.
Example 2.3. Let V be a vector space and denote L = hom(V, V ). On L we define a bracket
[f, g] = f ◦ g − g ◦ f.
In this way we obtatin a Lie algebra gl(V ).
For a Lie group G we define g = Lie(G) = TeG as a vector space. Now we proceed to introduce
a bracket on g.
Definition 2.4. A vector field X : G → TG is called left-invariant if (λa)∗ ◦ X = X ◦ λa for
any a ∈ G.
TG
(λa)∗
GG TG
G
X
yy
λa
GG G
X
yy
In other words X is λa-related with itself which we denote by X ∼λa X.
Remark. The f-relatedness of vector fields X and Y has the following characterization via
the flow lines, easily verified by differentiating both sides.
f(FlX
t (x)) = FlY
t (f(x))
In other words f transfers the flow lines of X into the flow lines of Y . We will use this property
quite often.
Remark. Let A ∈ TeG be an arbitrary vector. It defines a vector field λA : G → TG by the
formula λA(a) = (λa)∗A. This vector field is clearly left-invariant as
λA(ab) = (λab)∗A = (λaλb)∗A = (λa)∗((λb)∗A) = (λa)∗(λA(b))
It remains to verify its smoothness. Since (λa)∗A = µ∗(0a, A) this is achieved by the following
diagram
TG × TG
T µ
GG TG
G
(0,A)
yy
λA
ee
with (0, A) being the map with components the zero section 0 and the constant map sending
everything onto A.
Theorem 2.5. Let X, Y be left invariant vector fields. Then X +Y , kX and [X, Y ] are again
left-invariant.
2. LIE ALGEBRAS 4
Proof. Since X and Y are λa related with X and Y respectively, the same is true for their
sum, multiples and bracket.
Definition 2.6. The vector space g = Lie(G) = TeG together with the bracket [A, B] =
[λA, λB]e is called the Lie algebra of the Lie group G.
Remark. For every finite dimensional Lie algebra g there exists a Lie group G for which
Lie(G) = g.
We would like to explain now why this is a reasonable object of study. We have seen that
the first derivative at e does not see anything from the structure of the Lie group. The second
derivative does but in order to make sense of the second derivative one has to fix the coordinate
charts (which we will do later and for them the second derivative will be described exactly by the
Lie bracket). Without a fixed choice of the charts the second derivative only makes sense when
the first derivative vanishes at that point which is not the case for the product. The way out is to
“subtract from µ the sum of the two coordinates” by considering
[ , ] : G × G −→ G
(a, b) → aba−1
b−1
We will see shortly that the first derivative of the commutator vanishes at e and the essential part
of the second derivative is exactly the Lie bracket.
Notation. Let X, Y be two vector fields on a manifold M. Then we denote
(FlX
t )∗
Y (x) = (FlX
−t)∗Y (FlX
t (x)) ∈ TxM
the pullback of Y along the flow FlX
t of X. For each x ∈ M it is defined for t small.
Lemma 2.7. d
dt t=t0
(FlX
t )∗
Y (x) = (FlX
t0
)∗
[X, Y ](x).
Proof. First assume that t0 = 0 and let f : M → R be a smooth function. We differentiate
f in the direction of the left hand side:
d
dt t=0
(FlX
t )∗
Y (x) f = d
dt t=0
(FlX
t )∗
Y (x)f
= d
dt t=0
(FlX
−t)∗Y (FlX
t (x))f
= d
dt t=0
Y (FlX
t (x))(f ◦ FlX
−t)
= Y (x)(−Xf) + d
dt t=0
(Y f)(FlX
t (x))
= −(Y Xf)(x) + (XY f)(x) = [X, Y ](x) f
For a general t0 we have (FlX
t )∗
Y (x) = (FlX
t0
)∗
(FlX
t−t0
)∗
Y (x). Since (FlX
t0
)∗
is a linear map we can
interchange with d
dt .
Corollary 2.8. The following conditions are equivalent:
• [X, Y ] = 0,
• (FlX
t )∗
Y = Y , i.e. Y is FlX
t -related with itself for all t,
• FlX
t FlY
s (x) = FlY
s FlX
t (x), i.e. the flow lines commute.
In general we have FlY
−s FlX
−t FlY
s FlX
t (x) = x + st[X, Y ](x) + o(s, t)2
.
Proof. Differentiating twice we get
∂
∂t t=0
∂
∂s s=0
FlY
−s FlX
−t FlY
s FlX
t (x) = ∂
∂t t=0
−Y (x) + (FlX
t )∗
Y (x) = [X, Y ](x)
The remaining derivatives of order at most two are clearly zero.
Example 2.9. Let M = G, a Lie group. What does [A, B] for A, B ∈ g express? Let us
consider the following integral curves
• ϕ(t) the flow line of λA with ϕ(0) = e,
2. LIE ALGEBRAS 5
• ψ(t) the flow line of λB with ψ(0) = e.
A flow line of λA through a general a ∈ G is easily a · ϕ : t → aϕ(t). This follows from the
λa-relatedness of λA with itself: d
dt (aϕ(t)) = (λa)∗
d
dt ϕ(t). In other words
FlλA
t = ρϕ(t)
This implies that ϕ(t1 + t2) = ϕ(t1)ϕ(t2) and it is a homomorphism of groups. We now compute
FlλB
−s FlλA
−t FlλB
s FlλA
t (x) = ϕ(t)ψ(s)ϕ(−t)ψ(−s) = ϕ(t)ψ(s)ϕ(t)−1
ψ(s)−1
.
In other words the group theoretic commutator [ϕ(t), ψ(s)] has a Taylor polynomial
[ϕ(t), ψ(s)] = [A, B]st + o(s, t)2
This can also be rewritten as d2
[ , ](e,e)((A, 0), (0, B)) = [A, B]. The Lie bracket thus measures
the non-commutativity of the Lie group. More precisely [A, B] = 0 if and only if all the elements
ϕ(t) commute with all ψ(s). We will see later that the connection between commutativity of G
and vanishing of the bracket works perfectly for connected Lie groups.
Definition 2.10. Let L, L be two Lie algebras. A linear map ϕ : L → L is called a
homomorphism of Lie algebras if ϕ[A, B]L = [ϕA, ϕB]L .
Theorem 2.11. Let f : G → H be a (smooth) homomorphism of Lie groups. Then its
derivative f∗ : g → h at e is a homomorhpism of Lie algebras.
Proof. Let us rewrite f(ab) = f(a)f(b) using the left translations as
f ◦ λa = λf(a) ◦ f
Differentiating in the direction A ∈ g we obtain f∗(λa)∗A = (λf(a))∗f∗A or
f∗λA(a) = λf∗A(f(a))
which means that λA is f-related to λf∗A. Since the bracket respects relatedness, [λA, λB] must
be f-related to [λf∗A, λf∗B]. Evaluating at e yields the result.
Definition 2.12. A smooth map f : G → H between Lie groups is a local isomorphism if
it is both a homomorphism and a local diffeomorphism at e (i.e. the derivative f∗e : g → h is an
isomorphism).
Two Lie groups G, H are called locally isomorphic if there exist neighbourhoods U e and
V e, in G and H respectively, together with a diffeomorphism f : U → V which satisfies:
• f(ab) = f(a)f(b) whenever a, b, ab ∈ U,
• f−1
(ab) = f−1
(a)f−1
(b) whenever a, b, ab ∈ V .
Clearly if there exists a local isomorphism f : G → H then G and H are locally isomorphic.
Theorem 2.13. Locally isomorphic groups have isomorphic Lie algebras.
Example 2.14. The additive groups R and T = SU(1) (the group of complex units in C) are
locally isomorphic. We think of the first as the group of translations of the line while the second
is the group of rotations of the circle (or C for that matter). This is because there exists a local
isomorphism R → T sending t → e2πit
.
Definition 2.15. Let L, L be Lie algebras. On their product L × L we consider the bracket
[(X1, Y1), (X2, Y2)] = ([X1, X2]L, [Y1, Y2]L ).
We call L × L together with this bracket the product of Lie algebras L and L .
Theorem 2.16. Lie(G × H) ∼= Lie(G) × Lie(H).
Proof. The projections p : G × H → G and q : G × H → H are homomorphisms and hence
they induce homomorphisms of the Lie algebras in question. This means
p∗[(X1, Y1), (X2, Y2)] = [p∗(X1, Y1), p∗(X2, Y2)] = [X1, X2]
and similarly for q. The canonical isomorphism (p∗, q∗) : Lie(G×H) → g×h is then an isomorphism
of Lie algebras.
3. SUBGROUPS AND SUBALGEBRAS 6
Remark. With the above Lie algebra structure L × L forms a product in the category of Lie
algebras. The previous proof is then just a demonstration of the fact that Lie is a functor and
preserve products (which is obvious from the fact that this happens already at the level of tangent
vector spaces at e).
What happens if we change sides? Denoting ρA the right-invariant vector field with value
A at e the next theorem asserts that the Lie bracket defined via the right-invariant vector fields
agrees with the usual one up to the minus sign.
Theorem 2.17. For A, B ∈ g the following holds: [ρA, ρB]e = −[λA, λB]e.
Proof. Consider the opposite group G∗
with multiplication a ∗ b = ba. The inverse ν : G∗
→
G is a group homomorphism and
[A, B]∗
= [λ∗
A, λ∗
B]e = [ρA, ρB]e
Thus −[ρA, ρB]e = ν∗[A, B]∗
= [ν∗A, ν∗B] = [−A, −B] = [λA, λB]e.
Corollary 2.18. For a commutative group G the bracket on its Lie algebra is identically
zero.
3. Subgroups and subalgebras
Definition 3.1. A Lie subalgebra L ⊆ L is a vector subspace closed under [ , ].
Theorem 3.2. If H ⊆ G is both a submanifold and a subgroup then h ⊆ g is a Lie subalgebra.
Proof. In the diagram
H × H
µ
GG
 •

H •
ι

G × G µ
GG G
the map µ (which exists since H is a subgroup) is smooth since H is a submanifold. Hence H is
a Lie group and the inclusion ι : H → G is a homomorphism. Thus its derivative ι∗ : h → g is a
homomorphism of Lie algebras (saying that the bracket of h is a restriction of the bracket on g)
and its image is therefore a subalgebra.
Example 3.3. Consider R2
. Then every line {(x, kx) | x ∈ R} (for k ∈ R) is a subgroup (and
a submanifold). Now consider the torus T2
= R2
/Z2
. Again we get subgroups for any k ∈ R. For
k ∈ Q this subgroup is a submanifold but not for irrational k when this subgroup is dense.
Definition 3.4. A subset A ⊆ M of a smooth manifold M is called an initial submanifold
(of dimension k) if for each x ∈ A there exists a chart
ϕ : U
∼=
−−→ Rm
= Rk
× Rm−k
such that ϕ−1
(Rk
× {0}) is exactly the path component of U ∩ A containing x.
Theorem 3.5. Every initial submanifold is the image of an (essentially unique) injective
immersion i satisfying the following universal property:
A •
i

N
g
bb
f
GG M
For every smooth map f : N → M with the property f(N) ⊆ i(A) the unique map g : N → A
satisfying ig = f is also smooth.
3. SUBGROUPS AND SUBALGEBRAS 7
Proof. Let ϕ : U −→ Rm
be a chart on N from the definition of an initial submanifold.
Declare its restriction
Cx(U ∩ A)
∼=
−−→ Rk
× {0}
to the path component of U ∩ A containing x to be a chart for A. This does endow A with a
smooth structure. It differs from the subspace topology (which is inevitable) but the inclusion is
clearly an injective immersion.
We verify the universal property for inclusions of initial submanifolds. Let y ∈ N with
f(y) = x and V a path connected neighbourhood of y which maps into U. Since its image is
also path connected it must be contained in U ∩ A. Thus g in the chart provided by ψ is just a
restriction of f and hence smooth.
Suppose now that i : A → M is another injective immersion with the same image as i. Then
there exists a factorization
A
h GG p
i 33
Axn
i~~
M
with h an immerison and a bijection at the same time. Since its inverse is also an immersion by
the same argument h must be in fact a diffeomorphism.
Remark. It is also true that any injective immersion i satisfying the above universal property
is in fact an inclusion of an initial submanifold but we will not need this fact.
Remark. We have not proved that A has a countable basis for its topology. In fact A might
well have an uncountable number of components. However each of the components of A is second
countable.
Definition 3.6. A Lie subgroup H ⊆ G is an initial submanifold which is at the same time
a subgroup.
Theorem 3.7. A Lie subgroup H ⊆ G with its canonical smooth structure (and multiplication)
is a Lie group. Moreover h ⊆ g is a Lie subalgebra.
Proof. The whole proof is contained in the diagram
H × H
µ
GG
 •

H •
ι

G × G µ
GG G
Our new definition includes the wild subgroups of the torus T2
. In fact we are able to construct
a Lie subgroup for any Lie subalgebra of g. To motivate our construction observe that for a Lie
subgroup H ⊆ G and a ∈ H we have TaH = (λa)∗h and H is an integral submanifold of the left
invariant distribution determined by h.
More generally for a linear subspace P ⊆ g of dimension k the left translations (λa)∗P =:
λP (a) ⊆ TaG form a k-dimensional distribution λP on G. This distribution is smooth: if
A1, . . . , Ak is a basis of P then λA1 (a), . . . , λAk
(a) is a basis of λP (a).
A distribution S on M is called involutive if for every two vector fields X, Y ∈ S their bracket
[X, Y ] also lies in S.
Theorem 3.8 (Frobenius theorem). If S is involutive then for every x ∈ M there exists a local
coordinate system y1
, . . . , ym
in a neighbourhood U of x such that the vector fields ∂
∂y1 , . . . , ∂
∂yk
form a basis of the distribution S on U. In particular S is integrable.
3. SUBGROUPS AND SUBALGEBRAS 8
Proof. Let X1, . . . , Xk be local vector fields which, near x, span the distribution S and let
us choose a coordinate system around x in which Xi(x) = ∂
∂xi . We then define a map
ϕ : Rm
⊇ U −→ M
(t1
, . . . , tm
) −→ FlX1
t1 · · · FlXk
tk (0, . . . , 0, tk+1
, . . . , tm
)
The partial derivatives at the origin clearly consist of the vectors ∂
∂xi and thus ϕ is a local diffeo-
morphism.
Let us compute the partial derivative with respect to ti
for i ≤ k at a general point.
∂ϕ
∂ti
= FlX1
t1
∗
· · · Fl
Xi−1
ti−1
∗
Xi Fl
Xi+1
ti+1 · · · FlXm
tm (x)
To conclude the proof it is therefore enough to show that for any Y belonging to S the pullbacks
(FlY
t )∗
Xi also belong to S. Denote this pullback by
Yi(t) = (FlY
t )∗
Xi(x)
and write [Y, Xi] = aijXj. By Lemma 2.7 the paths Yi(t) satisfy the following system of
differential equations
d
dt Yi(t) = (FlY
t )∗
[Y, Xi] = aij(FlY
t (x))Yj(t)
We have Yi(0) = Xi(x) ∈ S(x) and since the system is linear we must have Yi(t) ∈ S(x) for all
t. Namely applying any linear form α to this system we see that α(Yi(t)) satisfy the very same
linear system of differential equations. Using the uniqueness and the existence of the zero solution
we see that α(Yi(0)) = 0 for all i implies α(Yi(t)) = 0 for all i and t.
By an integral submanifold we will now understand a connected initial submanifold A ⊆ M
for which TxA = Sx for all x ∈ A. A maximal integral submanifold is one that is not contained in
any bigger.
Theorem 3.9. If S is involutive then to every point x ∈ M there exists a unique maximal
integral submanifold going through that point.
Proof. We will obtain this initial submanifold as the set A of all points y ∈ M which can be
joined with x by a path γ : I → M tangent to the distribution S, i.e. with the properties
• γ(0) = x, γ(1) = y,
• ˙γ = d
dt γ ∈ S.
We need to verify that A is indeed an initial submanifold, maximality should be obvious. In a
coordinate chart ϕj : Uj → Rm
from the Frobenius theorem Uj ∩ A is clearly the disjoint union
(ck+1,...,cm)∈Cj
Rk
× {(ck+1, . . . , cm)}
It is enough to show that each Cj is at most countable since every countable subset of Rm−k
is
totally disconnected (in between any two distinct x, y in a countable set X ⊆ R there lies some
z ∈ X). First we prove an auxiliary fact:
Let B be an integral submanifold which is second countable. Then B intersects each Uj in at
most a countable number of leaves Rk
× {(ck+1, . . . , cm)}: if, by contradiction, the number was
uncountable then choosing a point from B in each leaf we would find an uncountable discrete
subset of B.
In particular every leaf of ϕj intersects at most countable number of leaves of ϕk. Now start
with A0 = {x} and at each step “leaf complete” Ai to obtain Ai+1. Then A = Ai and it is
second countable, hence intersects only a countable number of leaves of each ϕj.
Let us return to a linear subspace P ⊆ g and the distribution λP on G.
Lemma 3.10. λP is involutive if and only if P is a Lie subalgebra.
Proof. Since [X, fY + gZ] = f[X, Y ] + (Xf)Y + g[X, Z] + (Xg)Z it is enough to check the
brackets of vector fields of the form λA with A ∈ P. But [λA, λB] = λ[A,B].
4. HOMOMORPHISMS OF LIE GROUPS AND ALGEBRAS 9
Theorem 3.11. Let h ⊆ g be a Lie subalgebra. Then the maximal integral submanifold H
passing through e is a Lie subgroup.
Proof. Let a ∈ H. Since (λa−1 )∗λh = λh, the map λa−1 preserves integral submanifolds. As
λa−1 (a) = e and both a, e ∈ H we must have λa−1 (H) = H and thus a−1
b ∈ H for all a, b ∈ H.
Now we tackle the uniqueness issue. First a lemma.
Lemma 3.12. Let f : G → H be a homomorphism of Lie groups whose derivative at identity
is surjective. Then the image of f is a union of components of H.
Proof. The image is certainly a subgroup which is open. Since any open subgroup is necessarily
also closed (its complement being a union of cosets which are open) the assertion follows.
Remark. Later we will use a simple variation of this lemma: Let U be a connected neighbourhood
of e in a Lie group G. Then the subgroup generated by U is exactly the connected
component Ge of G containing e. Here Ge is a subgroup since the pointwise product of a path
from e to a and a path from e to b is a path from e to ab.
Theorem 3.13. Let H ⊆ G be a connected Lie subgroup. Then H is the maximal integral
submanifold of λh. In particular two connected Lie subgroups are equal if and only if they have
the same Lie algebra.
Proof. Let H0 be the maximal integral submanifold of λh passing through e. Since both H
is also an integral submanifold it must be contained in H0 and the inclusion H → H0 is both
injective and surjective by the previous lemma (the derivative at e is the identity on h) and thus
H = H0.
4. Homomorphisms of Lie groups and algebras
Lemma 4.1. A group homomorphism f : G → H which is smooth near e is smooth everywhere.
Proof. This is a classical homogeneity argument. Denoting by U the neighbourhood of e
where f is smooth pick any a ∈ G and consider the diagram
U
f
GG
λa

H
λf(a)

aU
f
GG H
in which aU is a neighbourhood of a and thus f is smooth everywhere.
The essential idea of this section is to construct homomorphisms through their graphs. Let us
consider ϕ : g → h, a linear map between Lie algebras. The graph of ϕ is the subset Graph(ϕ) =
{(A, ϕ(A)) | A ∈ g}.
Lemma 4.2. Graph(ϕ) is a Lie algebra if and only if φ is a homomorphism of Lie algebras.
Proof. By the definition of the bracket in the product
[(A, ϕ(A)), (B, ϕ(B))] = ([A, B], [ϕ(A), ϕ(B)])
which lies in Graph(ϕ) if and only if [ϕ(A), ϕ(B)] = ϕ[A, B].
Let ϕ : g → h be now a homomorphism of Lie algebras, Graph(ϕ) ⊆ g × h its graph, a Lie
subalgebra. There exists a unique connected Lie subgroup F ⊆ G × H with Lie(F) = Graph(ϕ).
Assuming that the composition F → G × H → G is a diffeomorphism F will be a graph of
a homomorphism f : G → H with f∗ = ϕ. In general however this projection is only a local
4. HOMOMORPHISMS OF LIE GROUPS AND ALGEBRAS 10
diffeomorphism: its derivative at e is the isomorphism Graph(ϕ) → g and at other points this
follows from the diagram
TeF
f∗
∼=
GG
(λa)∗ ∼=

TeG
(λf(a))∗∼=

TaF
f∗
GG Tf(a)G
Definition 4.3. A continuous map f : X → Y is a covering if for each y ∈ Y there exists its
neighbourhood U such that
f−1
(U)
∼= GG
f
33
c∈C
U
c∈C
id
~~
U
Lemma 4.4. Every local isomorphism of Lie groups is a covering.
Proof. Let f : G → H be the local isomorphism, U a, V b open neighbourhoods for
which f|U : U
∼=
−−→ V with inverse g. Then we will show that
f−1
(V ) =
k∈ker f
k · U
Therefore let x ∈ f−1
(V ). Then x = (x·g(f(x))−1
)·g(f(x)) is the decomposition. Also kx = k x
implies that x(x )−1
= k−1
k ∈ ker f and thus f(x) = f(x ). Since f in injective on U, x = x and
necessarily k = k .
The proof is finished by recalling that the image of f is a union of components (so that for
any b the a above exists).
Theorem 4.5. Let X be a path connected and locally simply connected topological space. Then
X is simply connected if and only if every connected covering of X is a global homeomorphism.
Before going into the proof we draw a corollary:
Theorem 4.6. Let G be a simply connected Lie group, H any Lie group. Then for every
homomorphism ϕ : g → h of Lie algebras there exists a unique homomorphism of Lie groups
f : G → H with the property f∗ = ϕ. For connected G the uniqueness part is still valid.
Proof. The above constructed homomorphism F → G is a covering and according to the
previous theorem a diffeomorphism. Thus F is the graph of f.
Corollary 4.7. Two simply connected Lie groups G and H are isomorphic if and only if
their Lie algebras are isomorphic.
The assumption of simple connectivity is essential: the canonical projection map R → R/Z =
T is a homomorphism but there is no non-trivial homomorphism in the opposite direction despite
the fact Lie R = Lie T.
Proof of Theorem 4.5. Let us construct the universal covering of X. Set
˜X = {[γ] | γ : (I, 0) → (X, x)}
where [γ] denotes the class with respect to homotopies preserving both endpoints. The projection
p : ˜X → X sends [γ] → γ(1). Then clearly
• p−1
(x) ∼= π1(X, x).
• p is a covering: Let U be a simply connected neighbourhood of x . Then
p−1
(U) ∼=
[γ]
γ(0)=x
γ(1)=x
[γ] ∗ {[δ] | δ : (I, 0) → (U, y)}
in bijection with U by
simple connectivity
5. THE EXPONENTIAL MAP 11
This bijection defines a topology on ˜X for which p is a covering. Therefore ˜X is a smooth
manifold if X was to start with (again we leave out the proof that ˜X is second countable).
Remark. We have shown that π1(X, x) is at most countable since p−1
(x) is discrete
and X second countable.
• p is universal: let q : Y → X be a covering with connected Y and let y ∈ q−1
(x). Then
there exists a unique f : ˜X → Y satisfying qf = p and f(˜x) = y where ˜x = [x] ∈ ˜X is
the class of the constant path
( ˜X, ˜x)
f
∃!
GG
p
55
(Y, y)
q
{{
(X, x)
This is about the path lifting property: the path γ : (I, 0) → (X, x) has a unique
continuous lift to ( ˜X, ˜x), namely t → [γ|[0,t]]. Denote the unique lift to (Y, y) by ˜γ. Since
the lifts must be preserved f must send [γ] → ˜γ(1).
• If π1(X, x) = {e} then ˜X → X is a homeomorphism: it is a local homeomorphism from
the definition of a covering and surjective from the path connectedness of X. We will
prove injectivity. Let p[γ] = p[δ], i.e.
γ, δ : (I, 0, 1) → (X, x, x )
The concatenation γ ∗ δ−1
is a loop in X, hence contractible to a point which gives
[γ] = [δ].
5. The exponential map
Definition 5.1. A one-parameter subgroup in G is a homomorphism γ : R → G.
Theorem 5.2. For every A ∈ g there exists a unique one-parameter subgroup γA : R → G
such that ˙γA(0) = A.
Proof. R is simply connected and Lie R = R with the trivial bracket and thus a homomorphism
R → g of Lie algebras is the same thing as a linear map.
The one-parameter subgroup γA is an integral curve of λA and more generally for every a ∈ G
the curve t → a · γA(t) is:
d
dt t=t0
aγA(t) = d
dt t=t0
aγA(t0)γA(t − t0) = (λaγA(t0))∗A = λA(a · γA(t0))
Theorem 5.3. The flow of the left-invariant vector field λA is
FlλA
t (a) = aγA(t) = ργA(t)(a)
Moreover λA is complete (the integral curves are defined for all t ∈ R).
Definition 5.4. The map exp : g → G sending A → γA(1) is called the exponential map of
the Lie group G.
Example 5.5. For G = (R+
, ·) the associated Lie algebra is Lie G = R, the left-invariant
vector field λA(a) = (λa)∗A = aA. The equation for the flow is
d
dt γA = γAA
and its solution is clearly γA(t) = etA
. Hence exp(A) = eA
.
Example 5.6. More generally for G = GL(n, R) the exponential map is
exp : gl(n, R) −→ GL(n, R)
A −→ eA
=
∞
k=0
1
k!
Ak
5. THE EXPONENTIAL MAP 12
Theorem 5.7. It holds exp(tA) = γA(t).
Proof. γA(t) = FlλA
t·1 (e) = Flt·λA
1 (e) = FlλtA
1 (e) = exp(tA).
Theorem 5.8. The map exp : g → G is smooth and a diffeomorphism on a neighbourhood of
0.
Proof. The vector field λA depends smoothly on A and thus also exp. We compute the
derivative of exp by considering a curve t → tA in g. Its image under exp is t → exp(tA) = γA(t)
whose derivative at 0 is ˙γA(0) = A. We conclude that exp∗ = id : g → g.
Theorem 5.9. For every homomorphism of Lie groups the following diagram commutes.
G
f
GG H
g
exp
yy
f∗
GG h
exp
yy
Proof. f(γA(t)) is a one-parameter subgroup with initial speed f∗A and thus equal to
γf∗A(t). Evaluating at t = 1 yields the result.
Lemma 5.10. Let f : G → H be a homomorphism of Lie groups with G connected and let
K ⊆ H be a Lie subgroup. Then f(G) ⊆ K if and only if f∗(g) ⊆ k.
Proof. Suppose that f∗(g) ⊆ k. Then f(exp(g)) = exp(f∗(g)) ⊆ exp(k) ⊆ K. Since exp(g)
is a neighbourhood of e in G, f−1
(K) is an open subgroup of G. As G is connected f−1
(K) must
equal G.
Theorem 5.11. Let ϕ : R → G be a continuous group homomorphism. Then ϕ is smooth.
Proof. In a neighbourhood of 0 ∈ R we can write uniquely ϕ(t) = exp(A(t)) with X(t) a
continuous path in g starting at 0. We would like to show that X(t) is linear. Let ϕ[−t0, t0] ⊆ exp U
where U is a ball centered at 0 and such that exp is a diffeomorphism on 2U. Let n ∈ N. We will
show that kX t0
n = X k t0
n for 0 ≤ k ≤ n by induction on k. For k = 0 or k = 1 this is clear.
Assuming the statement true for k write
(k + 1)X t0
n = kX t0
n + X t0
n ∈ 2U
Since
exp (k + 1)X t0
n = exp X t0
n
k+1
= ϕ t0
n
k+1
= ϕ (k + 1)t0
n = exp X (k + 1)t0
n
and exp is injective on 2U this finishes the induction step. As a particular case nX t0
n = X(t0)
and thus X k
n t0 = k
n X(t0) which easily holds also for all integers k with |k| ≤ n. From continuity
X(rt0) = rX(t0) for all r ∈ [−1, 1]. Since ϕ|[−t0,t0] is now linear and hence smooth, it is smooth
everywhere by the usual argument (homogeneity).
Theorem 5.12. Let G, H be Lie groups and f : G → H a continuous group homomorphism
between them. Then f is smooth.
Proof. Pick a basis A1, . . . , Am in g and define a map ϕ : Rm
→ G by
(t1, . . . , tm) → exp(t1A1) · · · exp(tmAm)
Clearly ϕ is a diffeomorphism near 0. It is called a coordinate chart of a second kind (the first
kind is exp itself). The composition fϕ is the map
(t1, . . . , tm) → f(exp(t1A1)) · · · f(exp(tmAm))
which is smooth: each continuous one-parameter subgroup f(exp(tiAi)) is smooth by the previous
theorem and so is their product. Again we can globalize by homogeneity.
6. HOMOGENEOUS SPACES 13
Theorem 5.13 (The closed subgroup theorem). Let H ⊆ G be a subgroup (in the algebraic
sense) which is also closed as a subspace of a Lie group G. Then H is a submanifold and thus a
Lie subgroup.
Proof. We divide the proof into a few steps:
• Define
h = {˙γ(0) | γ : (R, 0) → (G, e) a smooth curve}
Then h is a linear subspace since ˙γ1(0) + ˙γ2(0) = d
dt t=0
(γ1(t) · γ2(t)) and k ˙γ(0) =
d
dt t=0
γ(kt).
• Let An ∈ g be a sequence converging to A and let tn > 0 converge to 0 ∈ R. We claim
that if exp(tnAn) ∈ H then exp(tA) ∈ H for all t ∈ R. We may suppose that t > 0.
Choose mn ∈ N in such a way that |t − mntn| is minimal. Then |t − mntn| → 0 and
consequently mntnAn → tA. But exp(mntnAn) = exp(tnAn)mn
∈ H and since H is
closed it follows that exp(tA) ∈ H too.
• We show that h = {A ∈ g | exp(tA) ∈ H ∀t ∈ R}. The inclusion ⊇ follows from the
definition of h. For the reverse inclusion let A ∈ g be ˙γ(0) for some curve γ : R → H.
For t small we write γ(r) = exp(A(t)). Then
A = ˙γ(0) = exp∗( ˙A(0)) = ˙A(0) = lim
n→∞
A( 1
n )
1
n
Setting An = nA 1
n → A and tn = 1
n we have
exp(tnAn) = exp A 1
n = γ 1
n ∈ H.
and by the previous point exp(tA) ∈ H for all t ∈ R.
• Let k ⊆ g be a linear subspace complementary to h. We claim that there exists a
neighbourhood 0 ∈ W ⊆ k such that exp(W) ∩ H = {e}. By contradiction let Bn → 0
be a sequence in k such that exp(Bn) ∈ H. With respect to some norm on k consider
An = Bn
|Bn| . By passing to a subsequence we may assume that An converges to some
A ∈ k. Putting tn = |Bn| we have exp(tnAn) = exp(Bn) ∈ H and thus exp(tA) ∈ H for
all t ∈ R. By the previous point A ∈ h, a contradiction to A ∈ k.
• Define ϕ : h × k → G by (A, B) → exp A · exp B. We will show that there exists a
neighbourhood 0 ∈ V ⊆ h for which the restriction
ϕ : V × W
∼=
−−→ U ⊆ G
is a diffeomorphism onto its image U (which is easy) and such that
U ∩ H = ϕ(V × {0}).
Therefore let x ∈ U ∩ H be in the image, x = exp A · exp B. As both x, exp A ∈ H, also
exp B ∈ H. By the previous point B = 0.
Thus we found a chart at e which flattens out H. Charts at other points are obtained by transla-
tion.
6. Homogeneous spaces
Definition 6.1. By a left action of a Lie group G on a smooth manifold M we understand
a smooth map : G × M → M satisfying e = id and a ◦ b = ab where we write a = (a, −).
The algebraic content is a homomorphism G → Diff(M).
The right action r : M × G → M has to satisfy re = id and ra ◦ rb = rba.
We will write a(x) = a · x and ra(x) = xa.
Remark. A right action of G is the same as a left action of the opposite group G∗
.
6. HOMOGENEOUS SPACES 14
Definition 6.2. The orbit of a point x ∈ M is the subset Gx = {ax | a ∈ G}. We call the
action transitive if there is only one orbit in M or equivalently if Gx = M for every x ∈ M.
The stabilizer subgroup of a point x ∈ M is the (closed) subgroup
Sx = {a ∈ G | ax = x}.
The action is called free if the stabilizer subgroup of each point is trivial, Sx = {e} for every x ∈ M.
The action is called effective if a = b implies a = b, i.e. if the homomorphism G → Diff(M) is
injective.
Set theoretically the action yields a diagram
G
(−,x)
GG
p
44
M
G/Sx
aSx → ax an injective map
``
and if the action is transitive then G/Sx → M is even a bijection. Naturally G/Sx is a topological
space, a quotient of G:
U ⊆ G/Sx is open ⇐⇒ p−1
(U) ⊆ G is open.
Theorem 6.3. Let H ⊆ G be a closed subgroup of a Lie group G. Then there exists a unique
smooth structure on G/H for which p : G → G/H is a submersion.
Proof. First we will demonstrate uniqueness in a more general context. The idea here is
that surjective submersions are quotient objects:
M
g
GG
f

P
N
h
bb
If f is a surjective submersion and g any smooth map which factors through f set-theoretically,
i.e. such that ker f ⊆ ker g (or more concretely f(x) = f(x ) implies g(x) = g(x )), then the unique
map h satisfying g = hf is smooth. This follows easily from the fact that f admits smooth local
sections (and h is thus a composition of g with such a section).
The uniqueness now follows formally since in the diagram
G
p
}}
p
33
G/H
id
DD
G/H
id
ll ←− possibly different smooth structures
the unique factorization maps are the identity maps and the fact that they are both smooth means
precisely that the two smooth structures coincide.
It remains to prove the existence. Let k ⊆ g be a linear subspace complementary to h. There
are neighbourhoods 0 ∈ V ⊆ k, 0 ∈ W ⊆ h and e ∈ U ⊆ G such that
ϕ : V × W −→ U
(A, B) −→ exp A · exp B
is a diffeomorphism and U ∩ H = ϕ({0} × W). Let 0 ∈ V ⊆ V be such that
(exp V )−1
· (exp V ) ⊆ U
which is possible by continuity of the operations. Suppose now that A1, A2 ∈ V are such that
(exp A1) · H = (exp A2) · H. Then (exp A1)−1
· exp A2 ∈ U ∩ H and is equal to exp B for a unique
B ∈ W. Multiplying back
ϕ(A2, 0) = ϕ(A1, B)
6. HOMOGENEOUS SPACES 15
which implies A1 = A2 and B = 0. This says that the map
f : V × H −→ G
(A, b) −→ (exp A) · b
is injective. Since it is also a local diffeomorphism on V ×(exp W) by translation it is so everywhere
and f is in fact a diffeomorphism onto its image.
We have now identified a neighbourhood of H ⊆ G with a product V × H and in such a way
that the cosets a · H lying in this “chart” are of the form {A} × H. Thus the map
ψ : V ∼= V × {e} → V × H → G
p
−−→ G/H
(sending A to (exp A) · H) embeds V as a neighbourhood of the coset eH ∈ G/H. We therefore
declare it a chart on G/H. In this way the map p becomes the projection V × H → V and thus
a submersion. To get a chart near arbitrary aH redefine f as
fa : V × H −→ G
(A, b) −→ a · (exp A) · b
and consequently ψa(A) = a · (exp A) · H. The transition map ψa a between the resulting charts
ψa and ψa is computed from
a · (exp(ψa aA)) · H = a · (exp A) · H
Multiplying by a−1
we obtain
exp(ψa aA) ∈ a−1
a · (exp A) · H
and thus ψa a is the composition
V
exp
−−−→ U
λa−1a
−−−−−−→ U
f−1
−−−−→ V × H −→ V
with all arrows smooth and λa−1a only locally defined.
Definition 6.4. The manifold G/H is called a homogeneous space.
Remark. In the lecture I mentioned AT THIS POINT what a bundle is and that p : G → G/H
is an important example.
Theorem 6.5. The orbit of each point is an immersed submanifold (i.e. image of an injective
immersion).
Proof. Consider the diagram
G
(−,x)
GG
p
44
M
G/Sx
f
``
with the map f smooth by the previous theorem. We need to show that it is an immersion (on the
other hand it is injective almost by the definition of Sx). Suppose first that for A ∈ g its image
p∗A is sent to 0 ∈ TxM by f∗. Then d
dt t=0
exp(tA)x = 0. On the other hand
d
dt t=t0
exp(tA)x = d
dt t=t0
exp(t0A) exp((t − t0)A)x
= ( exp(t0A))∗
d
dt t=t0
exp((t − t0)A)x
0
= 0
Thus exp(tA)x = x for all t ∈ R and exp(tA) ∈ Sx implying that A ∈ ker p∗ and p∗A = 0.
This finishes the proof that f is an immersion at eSx. At other points this is guaranteed by the
6. HOMOGENEOUS SPACES 16
homogeneity:
eSx_

G/Sx
f
GG
a
∼=

M
a
∼=

aSx G/Sx
f
GG M
Example 6.6. Fix v ∈ R2
and consider the following action of R on R2
R × R2
−→ R2
(t, u) −→ u + tv
Clearly the orbit of u is the line u + Rv. Passing to the torus T2
= R2
/Z2
we see that orbits need
not be embedded submanifolds.
Remark. In general every orbit is an initial submanifold.
Corollary 6.7. For a transitive action the map f : G/Sx → M is a diffeomorphism.
Proof. From Sard’s theorem it easily follows that smooth bijections exist only between manifolds
of the same dimension. Hence the immersion f is in fact a local diffeomorphism. Being also
bijective it is a diffeomorphism by the inverse function theorem.
Examples 6.8. Examples of homogeneous spaces:
• Let V be a vector space. Then GL(V ) acts transitively on V − {0} and thus V − {0} ∼=
GL(V )/Sv where v ∈ V − {0}.
• The sphere Sn−1
with the action of O(n) is a homogeneous space, Sn−1 ∼= O(n)/O(n−1)
where O(n − 1) is thought of as a subgroup of O(n) consisting of block matrices
O(n − 1) ∼=
A 0
0 1
∈ O(n) A ∈ O(n − 1)
• The n-dimensional affine space is acted upon by the group
GA(n) =
A v
0 1
∈ GL(n + 1) A ∈ GL(n), v ∈ Rn
of affine transformations, namely we identify a point x ∈ Rn
with a vector ( x
1 ) in Rn+1
and then
A v
0 1
x
1
=
Ax + v
1
The origin is preserved exactly by the subgroup
GL(n) =
A 0
0 1
∈ GA(n) A ∈ GL(n)
describing Rn
as GA(n)/ GL(n). Similarly with GL(n) replaced by O(n) we arrive at
Rn ∼= Euc(n)/O(n) with Euc(n) denoting the group of (not necessarily origin preserving)
isometries of Rn
.
• The Stiefel manifold (of orthonormal k-frames in V )
Sk(V ) = {(v1, . . . , vk) | vi, vj = δij}
has as examples S1(V ), the unit sphere in V , Sn(Rn
) = O(n). For general Sk(Rn
) there
is a natural action of O(n) componentwise:
A(v1, . . . , vk) = (Av1, . . . , Avk)
The stabilizer of the k-tuple (e1, . . . , ek) of the first k vectors of the standard basis is
clearly
O(n − k) ∼=
E 0
0 C
∈ O(n) C ∈ O(n − k)
7. THE ADJOINT REPRESENTATION 17
Thus Sk(Rn
) ∼= O(n)/O(n − k).
• The Grassmann manifold Gk(V ) of all k-dimensional subspaces of V is naturally a quotient
of Sk(V ), namely by the means of the map
Sk(V ) −→ Gk(V )
(v1, . . . , vk) −→ span(v1, . . . , vk)
The O(n)-action on Sk(Rn
) passes to Gk(Rn
) with the stabilizer of Rk
being
O(k) × O(n − k) ∼=
B 0
0 C
∈ O(n) B ∈ O(k), C ∈ O(n − k)
and thus providing Gk(Rn
) ∼= O(n)/O(k) × O(n − k).
• I have mentioned EXAMPLES of the homogeneous spaces of scalar products, complex
structures etc.
Theorem 6.9. Let N ⊆ G be a closed normal subgroup. Then G/N with its canonical smooth
structure is a Lie group.
Proof. The left vertical arrow in
G × G
µ
GG
p×p

G
p

G/N × G/N GG G/N
is a surjective submersion therefore the dotted arrow (the multiplication in G/N) is smooth.
We have already met an example. The additive group R admits a homomorphism to T = SU(1)
by t → e2πit
. Clearly the kernel is Z and thus we obtained an induced isomorphism R/Z
∼=
−→ T.
7. The adjoint representation
Definition 7.1. By a representation of G we understand a left action of G on a vector
space V by linear maps (automorphisms), i.e. for which each a : V → V is linear. Equivalently
ρ : G → GL(V ) is a (smooth) homomorphism of Lie groups.
Definition 7.2. A representation of a Lie algebra L on a vector space V is a homomorphism
π : L → gl(V ) of Lie algebras. More concretely π is a linear map for which π[X, Y ](v) =
πX ◦ πY (v) − πY ◦ πX(v).
Definition 7.3. A linear subspace W ⊆ V is called invariant with respect to a representation
ρ if ρ(a)(W) ⊆ W for all a ∈ G. Analogously it is called invariant with respect to a representation
π if π(X)(W) ⊆ W for all X ∈ L.
Theorem 7.4. Let G be a connected Lie group and ρ its representation on V , ρ∗ : g → V
the induced representation of g. Then W ⊆ V is invariant with respect to ρ if and only if it is
invariant with respect to ρ∗.
Proof. Consider the following subgroup of GL(V )
GL(V, W) = {ϕ ∈ GL(V ) | ϕ(W) ⊆ W}.
It is easy to show that
gl(V, W) = Lie(GL(V, W)) = {ϕ ∈ gl(V ) | ϕ(W) ⊆ W}.
The statement then becomes a special case of Lemma 5.10.
7. THE ADJOINT REPRESENTATION 18
Let : G × M → M be a left action and x ∈ M its fixed point (i.e. Sx = G). Then
ρ : G → GL(TxM) given by a → ( a)∗x is smooth by
TG × TM
∗
GG TM
G × TxM
c1
0×id
yy
ρ
WW
and consequently a representation of G on TxM. We apply these general considerations to the
action of G on itself via conjugation (inner automorphisms):
(a, b) −→ inta b = aba−1
Now e ∈ G is a fixed point and we define Ad : G → GL(g) as above
Ad(a)B = (inta)∗B
Moreover Ad(a) ∈ AutLie(g) since inta is a homomorphism of Lie groups. We denote the induced
representation by ad : g → gl(g) (in fact Der(g)).
Theorem 7.5. For each A, B ∈ g it holds ad(A)(B) = [A, B].
Proof. We compute
ad(A)(B) = ∂
∂s s=0
Ad(exp(sA))(B)
= ∂
∂s s=0
∂
∂t t=0
intexp(sA) exp(tB)
= ∂
∂s s=0
∂
∂t t=0
exp(sA) exp(tB) exp(−sA)
= ∂
∂s s=0
∂
∂t t=0
FlλA
−s FlλB
t FlλA
s (e)
= ∂
∂s s=0
(FlλA
−s)∗λB(FlλA
s (e)) = [λA, λB]e = [A, B]
Theorem 7.6. If H ⊆ G is a normal subgroup then h ⊆ g is an ideal, i.e. a linear subspace
such that [g, h] ⊆ h (meaning [A, B] ∈ h for all A ∈ g and B ∈ h).
Proof. Since aHa−1
⊆ H or inta H ⊆ H we differentiate to get Ad(a)(h) ⊆ h and finally
ad(g)h ⊆ h.
Theorem 7.7. Let H be a connected Lie subgroup of a connected Lie group G such that h ⊆ g
is an ideal. Then H is a normal subgroup.
Proof. We have ad : g → gl(g, h). Since G is connected Ad : G → GL(g, h). It is enough to
show that inta(exp tB) ∈ H for all B ∈ h since the subgroup generated by such elements is the
whole group H. But inta(exp tB) = exp(Ad(a)(tB)) ∈ H since Ad(a)(tB) ∈ h.
Theorem 7.8. Let ϕ : G → H be a homomorphism of Lie groups. Then its kernel is a closed
normal subgroup K ⊆ G and its Lie algebra k is the kernel of ϕ∗ : g → h.
Proof. A ∈ k iff exp tA ∈ K iff exp(t · ϕ∗A) = ϕ(exp tA) = e iff ϕ∗A = 0.
Definition 7.9. The centre C of a Lie group G is the set
C = {a ∈ G | ab = ba ∀b ∈ G}
In other words, C is the kernel of int : G → Aut(G).
Theorem 7.10. The centre of a connected Lie group G is the kernel of the adjoint representation
Ad.
Proof. a ∈ C iff inta(G) = e iff Ad(a)g = 0 iff Ad(a) = 0.
Definition 7.11. The centre of a Lie group L is the ideal
Z = {X ∈ L | [X, Y ] = 0 ∀Y ∈ L}
In other words, Z is the kernel of ad : L → gl(L).
8. FUNDAMENTAL VECTOR FIELDS 19
Theorem 7.12. For a connected Lie group G, the centre Z of g the Lie algebra of the centre
C of G.
Proof. Since C = ker(Ad), its Lie algebra Lie(C) = ker(ad).
Remark. If the centre of L is zero then L can be embedded into gl(L) via the representation
ad.
Theorem 7.13 (Ado). Every Lie algebra can be embedded into gl(V ) for some finite-dimensional
vector space V .
Corollary 7.14. Every Lie algebra is induced from some Lie group.
Proof. By Ado’s theorem L ⊆ gl(n). Since gl(n) = Lie(GL(n)) one can find a Lie subgroup
of GL(n) corresponding to L.
8. Fundamental vector fields
Consider a left action : G × M → M. To every vector A ∈ g we associate a vector field A
on M by A(x) = ( (−, x))∗A. As usual A is smooth and is called the fundamental vector field
on M corresponding to A ∈ g. Analogously we define fundamental vector fields for right actions.
Theorem 8.1. In the case of a left action of G on M it holds [ A, B] = −[A,B]. For the
right action we obtain [rA, rB] = r[A,B].
Proof. On M × G consider the vector field (0, λA)(x, a) = (0x, λA(a)).
r∗(x,a)(0, λA) = (r(x, −))∗aλA = (r(xa, −))∗eA = rA(xa)
says that (0, λA) is r-related to rA. As the same is true for B we obtain for the brackets that
[(0, λA), (0, λB)] is r-related to [rA, rB]. But
[(0, λA), (0, λB)] = ([0, 0], [λA, λB]) = (0, λ[A,B])
which is r-related to r[A,B]. Thus [rA, rB] = r[A,B].
The last theorem can be expressed by saying that r : g → XM, A → rA is a homomorphism
of Lie algebras. The left action gives an antihomomorphism.
Definition 8.2. By a right infinitesimal action of a Lie group G on a manifold M we understand
a homomorphism R : g → XM of Lie groups. A right infinitesimal action is called complete
if RA is a complete vector field for each A ∈ g. Analogously a left infinitesimal action is an
antihomomorphism.
Example 8.3. The fundamental vector fields are complete: r(x, exp tA) = x exp tA is an
integral curve through x defined for all t ∈ R.
Remark. A left action is a homomorphism of Lie groups G → Diff(M) (with infinite dimensional
target). The induced Lie algebra homomorphism is g → Lie(Diff(M)), the latter being
XM but with the opposite bracket. As for finite dimensional Lie groups we can “integrate” a
homomorphism of Lie groups but here under additional requirements - the completeness.
Theorem 8.4. For a complete right infinitesimal action R : g → XM of a simply connected
Lie group G on M there exists a unique right action r : M × G → M of G on M such that RA is
its fundamental vector field rA.
Remarks.
• The simple connectivity is necessary: for the action of G = R on itself by translations
the infinitesimal action rt = t passes to an infinitesimal action of the quotient R/Z on R
for which no action exists.
• The theorem holds locally without the assumptions of completeness and simple connec-
tivity.
• The usual translation between left and right yields an analogous statement for left actions.
8. FUNDAMENTAL VECTOR FIELDS 20
Proof. Let first r be an action of G on M. Let Sx denote the following submanifold
Sx = {(xa, a) | a ∈ G} ⊆ M × G
The tangent space of Sx is
TSx = {(rA(xa), λA(a)) | a ∈ G, A ∈ g}
Thus Sx is an integral submanifold of the distribution (rA, λA) | A ∈ g .
Let us now start the actual proof of the theorem by considering the distribution
D = (RA, λA) | A ∈ g
Then D is involutive since
[(RA, λA), (RB, λB)] = ([RA, RB], [λA, λB]) = (R[A,B], λ[A,B]).
Let Sx be the maximal integral submanifold of D through (x, e) ∈ M × G. We claim now that
px : Sx → M × G → G is a diffeomorphism.
First we show that it is a covering. Fix a ∈ G and consider an arbitrary (y, a) ∈ M × G. The
computation
d
dt t=t0
(FlRA
t (y), a exp tA)
γ(t)
= (RA(FlRA
t0
(y)), λA(a exp t0A)) ∈ D
shows that γ(t) is tangent to the distribution D. Let U ⊆ g be an open ball centered at 0 on
which exp is a diffeomorphism. If (y, a) ∈ Sx then also (FlRA
1 (y), a exp A) ∈ Sx for all A ∈ U
and such points form an open neighbourhood on which px is a diffeomorphism onto a exp U. If
(z, b) ∈ Sx is arbitrary with b ∈ a exp U then b = a exp A and thus the above subset considered
for (Fl
R−A
1 (z), b exp(−A)) contains (z, b). This finishes the proof that px is a covering and in fact
a diffeomorphism as G is simply connected.
We define for x ∈ M and a ∈ G the action by the requirement
(xa, a) ∈ Sx
By the previous part there is a unique choice for xa. We need to show that r is smooth but
first let us prove the axioms of an action. Clearly xe = x as Sx is an integral manifold through
(x, e). Consider now a left action of G on M × G by a(y, b) = (y, ab). The distribution D is
invariant under this action (as (id, λa)∗(RA, λA) = (RA, λA)) and thus also its maximal integral
submanifolds. The requirement for our action r can be then rewritten as
Sx = aSxa = a(bS(xa)b)) = (ab)S(xa)b
As also Sx = (ab)Sx(ab) the maximal integral submanifolds S(xa)b and Sx(ab) must also be equal
proving (xa)b = x(ab).
A word about smoothness...
Definition 8.5. Consider two actions r and r of a Lie group G on manifolds M and M . A
map f : M → M is called equivariant if f(xa) = f(x)a.
Theorem 8.6. If f : M → M is equivariant then rA is f-related to rA.
Proof. The requirement from the definition is f ◦r(x, −) = r (f(x), −). Applying the derivatives
of both sides to A we get f∗rA = rAf.
Theorem 8.7. Let f : M → M be a smooth map such that rA is f-related to rA. If G is
connected then f is equivariant.
Proof. Consider the set H ⊆ G of all a ∈ G for which f(xa) = f(x)a for all x ∈ M.
Then H is clearly a subgroup and thus we only need that it contains a neighbourhood of e. But
f(x exp tA) = f(FlrA
t (x)) = Fl
rA
t (f(x)) = f(x) exp tA, hence exp g ⊆ H and H is open and
therefore equal to G.
9. LOCALLY ISOMORPHIC LIE GROUPS 21
9. Locally isomorphic Lie groups
Let G be a connected Lie group. Recall that the universal covering of G is
˜G
p

{[γ] | γ : (I, 0) → (G, e)}
_

G γ(1)
with [γ] the homotopy class of γ relative to the boundary. ˜G is simply connected: firstly π1
˜G →
π1G is injective (this works for any covering) since we can lift homotopies and constant paths lift
to constant paths. The image consists exactly of the classes of loops that lift to loops. For ˜G if
γ : I → G lifts to a loop its endpoints must be equal ˜e = [γ] and the image is therefore trivial.
We give ˜G a structure of a Lie group: let γ, γ : (I, 0) → (G, e) be two paths. Define their
product to be the path
(γ · γ )(t) = γ(t)γ (t)
which easily passes to homotopy classes rel ∂I.
Theorem 9.1. The above multiplication on ˜G describes a structure of a Lie group for which
the projection p : ˜G → G is a local isomorphism (i.e. a homomorphism and a local diffeomorphism).
Proof. The unit and inverses are also pointwise. The diagram
˜G × ˜G GG

˜G
local diffeomorphism

G × G
smooth
GG G
shows that the dotted arrow (the multiplication in ˜G) is smooth. (This is cheating, one needs to
compute (γ ∗ δ) · (γ ∗ δ ) = (γ · γ ) ∗ (δ · δ ) and if both δ and δ were small then so is δ · δ . Add
more DETAILS.)
Remark. I would like to CHANGE the proceeding along this way: we know that π1G ⊂ ˜G
is the kernel of p : ˜G → G and as such is a discrete normal subgroup. It is therefore central
(this was before an exercise). We show that the multiplication coming from ˜G is the same as the
concatenation (and in fact the multiplication π1G× ˜G → ˜G may also be equivalently defined using
concatenation). The theorem may be deduced from lifting homomorphisms of Lie algebras to Lie
groups. The map ˜G → G is then automatically a (surjective) homomorphism and thus a quotient
by a subgroup Γ ⊆ π1G.
There is an action of π1G on ˜G, π1G × ˜G → ˜G given by
([α], [γ]) −→ [α] · [γ] = [α ∗ γ]
which respects the projection p : ˜G → G. Let Γ ⊆ π1G be a subgroup and consider
pΓ : ˜G/Γ → G
where ˜G/Γ is the space of orbits of the restriction of the action to Γ. Locally
π1G × U
1  GG

˜G
p

U
1  GG G
and the action of Γ is by left multiplication in π1G. Thus the projection pΓ from ˜G/Γ to G is
locally of the form
(π1G/Γ) × U → U
and in particular is a covering of G.
9. LOCALLY ISOMORPHIC LIE GROUPS 22
Theorem 9.2. Let G be a connected Lie group. Then the mapping
{subgroups Γ ⊆ π1G} −→ local isomorphisms ρ : G → G
with G any connected Lie group
/iso
Γ −→ (pΓ : ˜G/Γ → G)
is a bijection with inverse ρ −→ im(π1ρ : π1G → π1G).
Proof. The image of π1pΓ consists of those loops that lift to loops in ˜G/Γ. These are precisely
those in Γ. In the opposite direction any ρ fits into the diagram
˜G GG
88
p
&&
G
ρ
ÔÔ
˜G/Γ
∼=
VV
pΓ

G
with Γ = im(π1ρ). The top arrow exists by universality of ˜G. The dotted arrow exists since loops
in Γ lift to loops in G . It is an isomorphism of Lie groups.
Remark. We will show in the tutorial that π1G → ˜G is a homomorphism and the action of
π1G on ˜G is by left translations, i.e. ˜G/Γ is a quotient of ˜G by (a central subgroup) Γ.
Example 9.3 (The universal covering of a commutative connected Lie group G). Since Lie G =
Rn
with zero bracket it is also the Lie algebra of the simply connected Lie group Rn
(with vector
addition) and thus ˜G = Rn
. Therefore G ∼= Rn
/Γ where Γ is some discrete subgroup of Rn
. We
will show now that Γ = Zk
⊆ Rn
in some coordinates on Rn
.
First reduction is to the case n = k, namely we have span Γ = Rk
⊆ Rn
and Γ is still discrete
in Rk
. We must show that Γ = Zk
in some coordinates on Rk
.
We start an induction by k = 1 which we proved in the tutorial. For the induction step we may
assume that Γ ⊆ R × Rk
= Rk+1
is such that the intersection Γ ∩ R = 0 with the first coordinate
axis is nonzero. Since it is also discrete it is generated by some a0. In Rk
= Rk+1
/R consider its
subgroup Γ/ a0 . We show by contradiction that it is discrete. Namely assume the existence of a
sequence αn = (βn, γn) ∈ Γ with γn → 0 in Rk
. By adding a suitable multiple of a0 to each αn
we may assume that βn ∈ [−a0/2, a0/2] and by extracting a subsequence we may further assume
that αn converges. But then αn+1 − αn ∈ Γ converges to 0, a contradiction with Γ being discrete.
By the induction hypothesis Γ/ a0 = ˜a1, . . . , ˜ak . We choose for each ˜ai an element ai ∈ Γ
representing it. Then the suitable basis in which Γ = Zk+1
is formed by (a0, a1, . . . , ak).
0 GG Za0
GG
∼=

Z{a0, a1, . . . , ak} GG

Z{˜a1, . . . , ˜ak} GG
∼=

0
0 GG Γ ∩ R GG Γ GG Γ/(Γ ∩ R) GG 0
Corollary 9.4. The only compact connected commutative Lie group of dimension k is the
torus Tk
= (S1
)k
.
Example 9.5. For n ≥ 3 we have π1 SO(n) ∼= Z/2. Therefore SO(n) possesses a twosheeted
universal covering which is denoted by Spin(n) = SO(n). We will show geometrically
that π1 SO(3) = Z/2 in the tutorial. For higher n we have a fibration
SO(n) → SO(n + 1) → Sn
whose long exact sequence of homotopy groups contains the following portion
0 = π2(Sn
) → π1(SO(n))
∼=
−→ π1(SO(n + 1)) → π1Sn
= 0
10. PROBLEMS 23
10. Problems
Problem 10.1. An algebra is a vector space A together with a bilinear map · : A × A → A.
Let A be now an associative algebra and define [ , ] : A × A → A by [a, b] = a · b − b · a. Show
that with this operation A forms a Lie algebra.
A special case of the previous is the algebra End(V ) of endomorphisms of a vector space V
together with their compositions. The induced Lie algebra is denoted by gl(V ). The bracket of
two endomorphisms ϕ, ψ is
[ϕ, ψ] = ϕ ◦ ψ − ψ ◦ ϕ
Problem 10.2. Let A be an algebra. A linear map D : A → A is called a derivative if for all
a, b ∈ A
D(a · b) = D(a) · b + a · D(b)
Show that derivatives form a Lie subalgebra Der(A) ⊆ gl(A).
Problem 10.3. Let C∞
M = C∞
(M, R) denote the algebra of all smooth functions on M.
Then every vector field X on M determines a mapping
C∞
M −→ C∞
(M)
f −→ Xf = df(X)
Show that this mapping is a derivative (in the algebraic sense). Also show that all derivatives of
C∞
M are of this form.
Let us now describe the Lie bracket of vector fields from this point of view: [X, Y ] is simply
the vector field corresponding to the bracket of the two derivatives X and Y of C∞
M. This means
that [X, Y ]f = XY f − Y Xf and this formula determines a unique vector field [X, Y ].
It also holds that algebra homomorphisms C∞
N → C∞
M are in bijection with smooth maps
M → N. One may then rewrite the f-relatedness of vector fields X and Y as
C∞
N
f∗
GG
Y

C∞
M
X

C∞
N
f∗
GG C∞
M
It is then a simple matter to show that Xi ∼f Yi implies [X1, X2] ∼f [Y1, Y2].
Problem 10.4. Compute the Lie algebra of the additive Lie group Rn
.
Problem 10.5. Compute the Lie algebra of the Lie group GL(n, R) from the definition.
Problem 10.6. Compute the Lie algebra of the Lie group GL(n, R) from the formula [A, B] =
∂2
∂s∂t
(s,t)=(0,0)
ϕ(t)ψ(s)ϕ(t)−1
ψ(s)−1
.
Problem 10.7. Compute the Lie algebra of the Lie group S3
= Sp(1) of unit quaternions
and show that it is isomorphic to R3
with the vector product ×.
Problem 10.8. Let B : Rn
× Rn
→ R be a bilinear form and denote by
G(B) = {A ∈ GL(n, R) | AT
BA = B} ⊆ GL(n, R)
the closed subgroup of all automorphisms preserving the form B. Compute the Lie algebra of
G(B).
Problem 10.9. Compute the Lie algebra of SO(n, R).
Problem 10.10. Let A be an algebra and denote by Aut(A) the group of all algebra automorphisms
of A. Compute its Lie algebra.
Problem 10.11. Determine all Lie algebras of dimension 2 over R.
10. PROBLEMS 24
Problem 10.12. Prove that the element −2 0
0 −1 of GL(2, R) lies in the component of the
unit E but not in the image of exp.
Problem 10.13. Let
G =





1 a b
0 1 c
0 0 1

 ∈ GL(3, R) a, b, c ∈ R



denote the Heisenberg group. Show that the bracket on Lie(G) is non-trivial and exp is a global
diffeomorphism.
Problem 10.14. Show that for G = S3
= Sp(1) the map exp is not a local diffeomorphim at
all points of g.
Problem 10.15. Show that discrete subgroups of R are exactly those of the form Za for some
positive real number a. Deduce that the only Lie groups of dimension 1 are R and T = R/Z.
Problem 10.16. Show that a discrete normal subgroup of a connected Lie group must lie in
the centre. (Hint: inta : H → H for a ∈ G may be connected to inte = id. Since H is discrete
these must be equal hence inta = id and H is central.)
Problem 10.17. Let f : M → G be a smooth map from a manifold M to a Lie group G.
Denote by δlf the g-valued 1-form called the left logarithmic derivative of f given by
δlf(x, X) = (λf(x)−1 )∗f∗X
(with (x, X) denoting a tangent vector X ∈ TxM). For example
δlid(a, A) = (λa−1 )∗A = ω(A)
the Maurer-Cartan form. Compute δlλb, δlρb, δlµ, δlν and δl(f · g−1
).
As a corollary, for a connected manifold M two maps f, g : M → G satisfy δlf = δf g if and
only if f = c · g for some c ∈ G. There exists also a criterion for determining whether a g-valued
one-form is a left logarithmic derivative of a map into G. This generalizes the integral calculus of
functions.
Problem 10.18. Let ˜G be the universal covering of G. Show that π1G ⊆ ˜G is a discrete and
normal subgroup thus lying in the centre of ˜G.
Problem 10.19. Show that the image of the adjoint representation Ad : Sp(1) → GL(3, R) is
SO(3, R) and that its kernel is the subgroup {±1}. Thus Sp(1) is the 2-fold (universal) covering
of SO(3, R).
Problem 10.20. Let ϕ : Sp(1)×Sp(1) → SO(4, R) be the map sending (a, b) to the orthogonal
transformation of the quaternions x → axb−1
. Show that this map is a 2-fold (universal) covering.
Problem 10.21. Compute the centre of SO(n, R) or even better its centralizer in GL(n, R)+,
i.e. CSO(n,R) GL(n, R)+. Try to determine all connected Lie groups with Lie algebra so(n, R).
Problem 10.22. Try to determine the first few terms in the Baker-Campbell-Hausdorff formula
for
log(exp X · exp Y ) : g × g → g
where log is the (locally defined) inverse to exp in the case g = gl(n).
A semidirect product of groups is a split short exact sequence
1 GG K GG G
p
GG H GG
i
ii 1
The subgroup K ⊆ G is normal being a kernel of p. The map f : H
i
−→ G
int
−−→ Aut(K) given by
f(x)(y) = xyx−1
is a group homomorphism. For a ∈ G there are uniquely determined k ∈ K and
10. PROBLEMS 25
h ∈ H such that a = k · i(h). Namely h = p(a) and k = a · i(h)−1
. Therefore as sets G ∼= K × H
and the multiplication is given by
(k1, h1) · (k2, h2) = k1 · i(h1) · k2 · i(h2) = k1 · f(h1)(k2) · i(h1h2) = (k1 · f(h1)(k2), h1 · h2)
The resulting group is denoted by K H = K f H.
Problem 10.23. Show that GA(n, R) is a semidirect product GA(n, R) ∼= Rn
GL(n, R)
where the action of GL(n, R) on Rn
is the standard one.
Problem 10.24. Let G be a Lie group. Show that µ∗ : TG × TG → TG endows TG with a
structure of a Lie group.
Problem 10.25. Show that TG is a semidirect product TG ∼= g G and identify the involved
action of G on g.
Problem 10.26. Compute the Lie algebra of a semidirect product K f H.
Problem 10.27. Determine the Lie algebra of TG.
CHAPTER 2
Bundles
1. Bundles
The tangent bundle p : TM → M has the following property
(∀x ∈ M)(∃U x nbhd) : p−1
(U) ∼= U × Rm
Definition 1.1. By a bundle (or fibre bundle) we understand a triple (E, p, M) where E and
M are smooth manifolds and p : E → M is a smooth surjective1
map such that for each x ∈ M
there exists its neighbourhood U and a diffeomorphism ϕ : p−1
(U) ∼= U × F with F some smooth
manifold and such that
p−1
(U)
ϕ
GG
p
55
U × F
pr1
||
U
commutes. The space E is called the total space, M the base, p the projection, Ex = p−1
(x) the
fibre over x ∈ M and F the standard fibre.
Definition 1.2. The bundle pr1 : M × F → M is said to be trivial (or product). The map
ϕ : p−1
(U) ∼= U × F is referred to as a local trivialization.
Theorem 1.3. Let H ≤ G be a closed subgroup of a Lie group G. Then the projection
G → G/H is a bundle with standard fibre H.
Proof. This is exactly the proof of Theorem 6.3.
Examples 1.4.
• TS2
is not globally trivial (“nelze uˇcesati jeˇzka”).
• The M¨obius band R → L → S1
is also globally nontrivial.
• The Hopf bundle: let S3
⊆ H = C2
be the group of unit quaternions. The complex units
S1
form a subgroup of S3
and the Hopf bundle is
S1
→ S3
→ S3
/S1 ∼= CP1 ∼= S2
as S2
= C ∪ {∞}. Again the bundle is not trivial: π1S3
= 0 while
π1(S1
× S2
) ∼= π1S1
× π1S2 ∼= Z
More generally the Hopf bundle S1
→ S2n+1
→ CPn
is nontrivial.
Let us consider a bundle p : E → M, i.e. we have a cover Uα ⊆ M and local trivializations
ϕα : p−1
(Uα)
∼=
−→ Uα × F. Denoting Uαβ = Uα ∩ Uβ we obtain
Uαβ × F
88
p−1
(Uαβ)
ϕα
∼=
oo
ϕβ
∼=
GG

Uαβ × F
xx
Uαβ
1In principle surjectivity is not essential.
26
1. BUNDLES 27
composing to
ϕαβ : Uαβ × F
∼= GG
66
Uαβ × F
zz
Uαβ
Easily
• ϕαβ = (ϕβα)−1
,
• ϕβγ ◦ ϕαβ = ϕαγ over Uαβγ = Uα ∩ Uβ ∩ Uγ (the cocycle condition) and
• ϕαα = id.
On the other hand given a covering Uα and a collection of maps ϕαβ satisfying the above conditions
there exists a bundle p : Φ → M obtained from S = α Uα×F by passing to the quotient Φ = S/ ∼
by the relation
Uα × F (x, a) ∼ (x, ϕαβ(a)) ∈ Uβ × F
whenever x ∈ Uαβ.
Definition 1.5. A bundle p : E → M is called a vector bundle if each fibre Ex is given a
vector space structure and local trivializations ϕ : p−1
(U)
∼=
−→ U × Rk
could be chosen in such a
way that each Ex
∼=
−→ {x} × Rk ∼= Rk
is a linear isomorphism.
Examples 1.6.
• TM, T∗
M - the tangent and cotangent bundles,
• For a submanifold M ⊆ Rn
the normal bundle is
ν(M) = {(x, v) | x ∈ M, v ∈ TxM⊥
⊆ Rn
},
• Let p : E → M be any bundle. The vertical tangent bundle V E ⊆ TE is “the kernel of
p∗”, VyE = TyEp(y),
• Consider the Grassmann manifold
Gk(Rn
) = O(n)/ O(k) × O(n − k)
of linear subspaces of Rn
of dimension k. Over Gk(Rn
) we have a canonical vector bundle
γn
k → Gk(Rn
) where
γn
k = {(V, v) ∈ Gk(Rn
) × Rn
| v ∈ V }.
For example γ2
1 is the M¨obius band.
The transition maps ϕαβ : Uαβ × Rk
→ Uαβ × Rk
take form
(x, v) → (x, ψαβ(x) · v)
where ψαβ : Uαβ → GL(k) is a smooth map: the (i, j)-entry of ψαβ(x) is the i-th coordinate
of the second component of ϕαβ(x, ej). The cocycle condition ψβγψαβ = ψαγ is expressed via
multplication in GL(k).
Remark. GL(k) ⊆ Diff(Rk
). A general bundle has Diff(F) as a “structure group”.
Every bundle projection is a submersion: locally it is a projection. The converse is not true.
Theorem 1.7 (Ehresmann). If p : E → M is a proper surjective submersion then it is a
bundle.
Proof. Let us identify some neighbourhood of x ∈ M with a disc whose centre is x. By
properness, p−1
(Dm
) is compact and hence every vector field is complete (when we take care of
the boundary). Consider now ∂
∂xi
and lift it to a vector field Xi on p−1
(Dm
), i.e. Xi is such
that p∗(Xi) = ∂
∂xi
. This is possible locally by p being a submersion and globally is achieved by a
partition of unity. Define
ϕ : Dm
× p−1
(0) −→ p−1
(Dm
)
(t1, . . . , tm, y) −→ FlX1
t1
· · · FlXm
tm
(y)
2. BASIC OPERATIONS WITH BUNDLES 28
which is well-defined by the completeness - it lies over
Fl
∂/∂x1
t1
· · · Fl
∂/∂xm
tm
(0) = (t1, . . . , tm)
by the p-relatedness of Xi and ∂
∂xi
. It is easy to verify that ϕ is a local diffeomorphism at
{0} × p−1
(0), it is identity on {0} × p−1
(0) and ∂
∂ti
ϕ = Xi there. Since p−1
(0) is compact, ϕ is
a diffeomorphism onto its image on some neighbourhood U × p−1
(0). The surjectivity follows by
integrating backwards, namely y is the image of p(y), FlXm
−p(y)m
· · · FlX1
−p(y)1
(y) .
2. Basic operations with bundles
Definition 2.1. Let p : E → M and p : E → M be bundles. A pair of maps f : E → E
and f : M → M is called a morphism if the diagram
E
f
GG
p

E
p

M
f
GG M
commutes or in other words if f preserves fibres, f(Ex) ⊆ Ef(x). This determines f and is
automatically smooth when f is. If moreover M = M and f = idM then f is said to be base-
preserving.
Definition 2.2. A product of bundles p and p is p × p : E × E → M × M with standard
fibre F × F .
Definition 2.3. An induced bundle (or pullback) from p along a smooth map g : M → M
is the submanifold2
g∗
E = {(z, y) ∈ M × E | g(z) = p(y)} ⊆ M × E
together with the projection onto the first factor. We have a diagram
g∗
E
pr2
GG
pr1

E
p

M g
GG M
The universal property
E
33
33
33
g∗
E
pr2
GG
pr1

E
p

M g
GG M
can be expressed by saying that a morphism from E to E is the same as a base-preserving
morphism from E to the induced bundle f∗
E.
If i : N → M is a submanifold inclusion then i∗
E = E|N is the restriction of E to N,
i.e. i∗
E ∼= p−1
(N).
Definition 2.4. Let p : E → M and p : E → M be bundles over the same base. Their fibre
product is
E ×M E = ∆∗
(E × E ) = (E × E )|∆
2This is so since g is transverse to any submersion p.
2. BASIC OPERATIONS WITH BUNDLES 29
where ∆ : M → M × M is the diagonal.
E ×M E GG

E
p

E
p
GG M
It is the categorical product in the category of bundles over the fixed base M.
Theorem 2.5. If two maps g0, g1 : M → M are homotopic then the induced bundles g∗
0E
and g∗
1E are isomorphic.
Proof. See Differential topology lecture notes.
Theorem 2.6. Every bundle over Rn
is trivial.
Proof. The identity map idRn on Rn
is homotopic to the constant map 0. By the previous
theorem
E ∼= id∗
Rn E ∼= 0∗
E ∼= Rn
× p−1
(0)
giving a global trivialization.
Definition 2.7. A section of a bundle p : E → M is a smooth map s : M → E for which
p ◦ s = idM .
Examples 2.8.
• A section of TM is a vector field, a section of T∗
M is a 1-form.
• A section of a trivial bundle M × F → M is a smooth map M → F.
Definition 2.9. A local section is a smooth map s : U → E satisfying p ◦ s = idU where
U ⊆ M is an open subset.
Example 2.10. Local sections always exist (since F = ∅) global sections need not. Define
˚TM = TM − {(x, 0) | x ∈ M}, the space of all nonzero vectors. Easily ˚TM is a bundle over M
and a global section of ˚TM is a nowhere zero vector field which does not exist for example on S2
.
Theorem 2.11. If the standard fibre is diffeomorphic to Rk
then global sections always exist.
Proof. Local sections are glued together via a partition of unity (which has to be utilized
in a chart). More precisely one inductively extends a section, starting with a local section in a
bundle chart... FINISH!!!
Let s and s be sections of p : E → M and p : E → M respectively. They determine a section
(s, s ) of the fibre product E ×M E . A section s of p determines a section g∗
s of any induced
bundle g∗
E:
M s◦g
33
idM
33
g∗
s
44
g∗
E
pr2
GG
pr1

E
p

M g
GG M
g∗
s : z → (z, sf(z))
More generally any map t : M → E satisfying p ◦ t = g (a section of E along g) induces a section
of the induced bundle g∗
E. In fact this describes a bijection between sections along g and sections
of the induced bundle.
3. JET BUNDLES 30
Let now p : E → M and p : E → M be vector bundles. A morphism f : E → E is called
linear if every f|Ex
: Ex → Ef(x) is a linear map. Locally
U × Rk f
GG

V × Rl

U
f
GG V
f(x, v) = (f(x), g(x)v)
where g : U → hom(Rk
, Rl
) is a smooth map as g(x)ij = f2(x, ej)i.
Let p : E → M be a vector bundle, {Uα} a cover of M and ϕαβ(x, v) = (x, ψαβ(x)v) the
transition maps with ψαβ : Uαβ → GL(V ) smooth into the group of linear automorphisms of
the standard fibre V . Let there be given a homomorphism f : GL(V ) → GL(W) (e.g. W =
V ⊗k
, Sk
V, Λk
V ). The compositions f ◦ ψαβ : Uαβ → GL(W) then yield back a vector bundle with
standard fibre W which we denote f(E). In the construction of the dual vector bundle we obtain
from ϕαβ a linear map
Uαβ × V ∗
ϕ∗
αβ
←−− Uαβ × V ∗
going in the wrong direction. This is remedied by considering its inverse. In general we may pass
from a homomorphism f : GL(V )op
→ GL(W) to the composition GL(V )
f
−→ GL(W)op ν
−→ GL(W)
and apply the previous construction to get a vector bundle f(E) with standard fibre W. Examples
are E∗
, ¯E. The most general case is that of a homomorphism
f : GL(U1)op
× · · · × GL(Uk)op
× GL(V1) × · · · × GL(Vl) −→ GL(W)
which produces a vector bundle f(E1, . . . , Ek, F1, . . . , Fl) from arbitrary vector bundles E1 . . . , Ek,
F1, . . . , Fl with standard fibres U1, . . . , Uk, V1, . . . , Vl.
Example 2.12. The vector bundle hom(E, F) has as fibres hom(E, F)x = hom(Ex, Fx) and
as a special case hom(E, R) = E∗
where R here stands for the trivial bundle M × R → M. This
example is obtained from the general construction via the homomorphism
GL(U)op
× GL(V ) −→ GL(hom(U, V ))
(α, β) −→ (ϕ → β ◦ ϕ ◦ α)
3. Jet bundles
Let us consider the algebra C∞
(Rn
) of smooth maps on Rn
. By the inductive use of the
formula
g(x) = g(0) +
n
i=1
ai(x)xi
for a function g : Rn
→ R we derive
g = Trg + Rrg
a decomposition of g into its Taylor polynomial Trg of order r and a remainder lying in the ideal
mr+1
0 generated by the monomials xI
= xi1
1 · · · xin
n of degree |I| = i1 + · · · + in = r + 1. It is the
(r + 1)-st power of the ideal m0 generated by the coordinate functions. The association of the
Taylor polynomial or order r gives a surjective linear map
Tr : C∞
(Rn
) Pr(Rn
)
onto the vector space of all polynomials of order at most r on Rn
. Clearly the kernel is the
ideal mr+1
0 and hence Pr(Rn
) is naturally isomorphic to the quotient algebra C∞
(Rn
)/mr+1
0 . The
multiplication in this algebra is the truncated multiplication of polynomials. Let f : Rm
→ Rn
be
a smooth map sending 0 to 0. Then f induces by composition an algebra homomorphism
f∗
: C∞
(Rn
) → C∞
(Rm
)
3. JET BUNDLES 31
with the property f∗
(m0) ⊆ m0 and thus f∗
(mr+1
0 ) ⊆ mr+1
0 .
C∞
(Rn
)
f∗
GG
Tr

C∞
(Rm
)
Tr

g  GG
_

g ◦ f_

Pr(Rn
)
f∗
GG Pr(Rm
) Trg  GG Tr(g ◦ f)
Therefore Tr(g ◦f) only depends on Trg rather than on g. Since Pr(Rn
) is generated as an algebra
by the coordinate functions x1, . . . , xn we have f∗
(xi) = Tr(xi ◦f) = Tr(fi), the Taylor polynomial
of order r of the i-th component fi. Therefore if f and f have the same Taylor polynomial of
order r then f∗
= (f )∗
on Pr(Rn
) and thus Tr(g ◦ f) = Tr(g ◦ f ) only depends on Trf.
We have just proved that the Taylor polynomial of order r of a composition g ◦ f of maps g
and f depends only on their respective Taylor polynomials as long as they preserve the origin. In
particular we have
Theorem 3.1. The property of having the same Taylor polynomial of order r for maps
(Rm
, 0) → (Rn
, 0) does not depend on the coordinates (as long as their changes preserve the
origins).
Definition 3.2. Let M and N be two manifolds and f, f : M → N two maps defined in a
neighbourhood of x ∈ M. We say that f and f determine the same r-jet at x (with r ∈ N) if
f(x) = f (x) = y and for some (any) pair of charts ϕ on M cenetered at x and ψ on N centered
at y the maps ψ−1
fϕ and ψ−1
f ϕ have the same Taylor polynomial of order r at the origin. We
write jr
xf for the class determined by the map f and
Jr
(M, N) = {jr
xf | x ∈ M, f : M → N defined in a neighbourhood of x}.
For X = jr
xf we write αX = x for the source and βX = f(x) for the target of the r-jet X. Without
coordinates we can identify r-jets with source x and target y with algebra homomorphisms
C∞
(N)/mr+1
y −→ C∞
(M)/mr+1
x
There are obvious canonical projections πr
s : Jr
(M, N) → Js
(M, N) for 0 ≤ s ≤ r. For s = 0 we
have J0
(M, N) ∼= M × N via the map (α, β). Therefore πr
0 = (α, β). We denote
Jr
x(M, N) = α−1
(x), Jr
(M, N)y = β−1
(y), Jr
x(M, N)y = α−1
(x) ∩ β−1
(y)
the last being the fibre of Jr
(M, N) over (x, y) ∈ M × N via (α, β).
For X ∈ Jr
x(M, N)y and Y ∈ Jr
y (N, Q)z we define their composition Y ◦X ∈ Jr
x(M, Q)z either
as a composition of algebra homomorphisms or via representatives Y ◦ X = jr
x(g ◦ f) if X = jr
xf
and Y = jr
yg.
Definition 3.3. We say that X ∈ Jr
x(M, N)y is invertible if there exists X−1
∈ Jr
y (N, M)x
for which X−1
◦ X = jr
xidM and X ◦ X−1
= jr
yidN .
For r ≥ 1 we obtain X is invertible iff its linear part πr
1X is invertible. In particular for this
to happen we must have m = n.
Let us denote Lr
m,n = Jr
0 (Rm
, Rn
)0 which we know can be identified with homalg(Pr(Rn
), Pr(Rm
))
or with the set of polynomials of order at most r and without constant term, X = 1≤|I|≤r aIxI
.
Here aI ∈ Rn
are constant. The composition of jets
Lr
n,q × Lr
m,n → Lr
m,q
is the truncated composition of polynomials (i.e. the normal composition followed by ignoring all
the terms of order bigger than r). In particular it is smooth and
Gr
m = inv(Lr
m,m)
is therefore a Lie group with respect to the composition of jets, invertible jets forming an open
subset (they are those where a1, . . . , am are linearly independent). As a special case G1
m = GL(m).
3. JET BUNDLES 32
Let us consider now X ∈ Lr
m,n and consider a translation by v
λv : x → x + v
The following are mutually inverse diffeomorphisms
Rm
× Lr
m,n × Rn ∼=
−−→ Jr
(Rm
, Rn
)
(u, X, v) −→ jr
0λv ◦ X ◦ jr
uλ−u
(u = αY, jr
vλ−v ◦ Y ◦ jr
0λu, v = βY ) ←− Y
Now we can define on Jr
(M, N) a smooth structure so that the projection
(α, β) : Jr
(M, N) → M × N
becomes a bundle. We choose charts on U ⊆ M and V ⊆ N giving us an identification
α−1
U ∩ β−1
(V ) ∼= U × Lr
m,n × V
Declaring these to be diffeomorphisms we are left to show that the effect of another choice of
charts differs by a diffeomorphism preserving the projection onto U × V . But this is rather easy
to see using the concrete description of the involved maps.
A smooth map f : M → N induces a section jr
f : M → Jr
(M, N) sending x → jr
xf of the
bundle Jr
(M, N)
α
−→ M.
Example 3.4. For r = 1 we have J1
(M, N) ∼= hom(TM, TN) or rather hom(p∗
TM, q∗
TN)
with p : M × N → M and q : M × N → N the two projections. The map in one direction
is provided by j1
xf → Txf and is a diffeomorphism by an inspection in charts. As special cases
J1
0 (R, M) ∼= TM and J1
(M, R)0
∼= T∗
M.
We denote by Tr
k M = Jr
0 (Rk
, M)
β
−→ M the bundle which we call the bundle of k-dimensional
velocities of order r. In particular Tr
1 M is called the tangent bundle of order r. A smooth map
f : M → N induces a morphism of bundles Tr
k f : Tr
k M → Tr
k N via the composition jr
0g → jr
0(f ◦g)
Tr
k M
T r
k f
GG
β

Tr
k N
β

M
f
GG N
Dually Tr∗
k M = Jr
(M, Rk
)0, the bundle of k-dimensional covelocities of order r. In particular
Tr∗
1 M is called the cotangent bundle of order r. The bundle Tr∗
k M is a vector bundle with respect
to the addition jr
xϕ + jr
xψ = jr
x(ϕ + ψ) and multiplication λ · jr
xϕ = jr
x(λϕ), λ ∈ R. On the other
hand only local diffeomorphisms induce morphisms of bundles:
Tr∗
k M
T r∗
k f
GG
α

Tr∗
k N
α

M
f
GG N
jr
xϕ → jr
f(x)(ϕ ◦ f−1
)
Remark. For any smooth f we have a map on the section spaces
Γ(Tr∗
k N)
f∗
−−−→ Γ(Tr∗
k M)
Let Pr
M = invJr
0 (Rm
, M) ⊆ Tr
mM with m = dim M denote the “bundle of r-jets of maps
(Rm
, 0) → (M, x)”. The group Gr
m = invJr
0 (Rm
, Rm
)0 acts on Pr
M from the right via the jet
composition: for a map u : Rm
→ M and a change of coordinates ϕ : Rm
→ Rm
we have a new
map u ◦ ϕ : Rm
→ M
Pr
M × Gr
m −→ Pr
M
(jr
0u, jr
0ϕ) −→ jr
0(u ◦ ϕ)
4. PRINCIPAL AND ASSOCIATED BUNDLES 33
The situation is summarized in: Pr
M is a bundle, the action of Gr
m preserves the fibres and is
simply transitive on each of them: for jr
0u and jr
0v with u(0) = v(0) there exists a unique a ∈ Gr
m
for which jr
0v = jr
0u · a.
Example 3.5. For r = 1: P1
M = inv hom(Rm
, TM) which at the fibre over x ∈ M is the
same as a basis of TxM (namely the image of the standard basis in Rm
). We say that P1
M is the
bundle of frames in TM. We then think of Pr
M as a bundle of higher order frames.
Definition 3.6. Let p : E → M be a bundle. The r-th jet prolongation Jr
E is the space of
all jets of local sections of p. It is a manifold and bundle over M. One can either see this locally - a
local section is equivalent to a map U → F and thus Jr
E is locally in bijection with Jr
(U, F) but
it is not quite obvious what the transition maps look like. A global definition is via the pullback
diagram
Jr
E
1  GG

Jr
(M, E)
p∗

M 1 
jr
id
GG Jr
(M, M)
describing it as a restriction of Jr
(M, E) → Jr
(M, M) along jr
id. Locally
Jr
(M, E) ∼= Jr
(U, V ) ×U Jr
(U, F) −→ Jr
(U, V )
which is a bundle and the restriction “forgets the first component” to get Jr
(U, F).
A prolongation of sections: s : M → E induces jr
s : M → Jr
E but not every section of
Jr
E → M comes from a section of E → M.
Remark. A differential equation/inequation (relation) is a subset R ⊆ Jr
E. A solution of
R is a section s : M → E for which jr
xs ∈ R for all x ∈ M. A formal solution is a section
of Jr
E → M with image in R. The jet prolongation restricts by definition to a map sol → fsol
between the space of solutions and the space of formal solutions with fsol being much bigger.
Nevertheless this map is quite often a homotopy equivalence.
4. Principal and associated bundles
Definition 4.1. Let us consider a bundle π : P → M and a Lie group G having a right action
r : P × G → P on P. We say that P is a principal bundle with a structure group G if
• the action r preserves fibres, π(u · a) = π(u) and
• G acts on each fibre Px simply transitively, u, v ∈ Px ⇒ ∃!a ∈ G : v = u · a.
We write P(M, G) to mean that P is a principal bundle over M with structure group G. We also
say that P is a principal G-bundle.
Theorem 4.2. Let H ≤ G be a closed subgroup of a Lie group G. Then the projection
G → G/H is a principal H-bundle.
Proof. This is contained in the proof of Theorem 6.3.
Examples 4.3.
• The frame bundle Pr
M(M, Gr
m).
• Consider a vector bundle E → M with standard fibre Rk
. Denote by PE → M the
following bundle over M
PE = inv hom(Rk
, E) ⊆ hom(Rk
, E) ∼= E ×M · · · ×M E
k times
In the last isomorphism we identify (u1, . . . , uk) with a unique linear map sending ei to
ui. Clearly this map is invertible iff u1, . . . , uk are linearly independent. The right action
of GL(k) is either via composition u · a = u ◦ a or as (u · a)i = j ujaji. We obtain a
principal bundle PE(M, GL(k)) of frames in the vector bundle E.
4. PRINCIPAL AND ASSOCIATED BUNDLES 34
A local section s : U → P determines a trivialization π−1
(U) ∼= U × G in the following way
U × G −→ π−1
(U)
(x, a) −→ s(x) · a
This is easily a smooth bijection. We need to verify that it is a local diffeomorphism. This is so
because the restriction to U ×{a} is a section and hence an immersion. The restriction to {x}×G
is an immersion by Theorem 6.5. The images of the respective derivatives are complementary.
Another feature of this trivialization is that it is equivariant.
Alternatively we may thus characterize principal G-bundles as right G-spaces P for which
there exists in a neighbourhood of every point an equivariant diffeomorphism with Rm
× G.
Theorem 4.4. A principal bundle is trivial if and only if it admits a global section.
Proof. Obvious from the preceding arguments.
Definition 4.5. A manifold Mm
is called parallelizable if it admits an m-tuple of linearly
independent (pointwise) vector fields.
Examples 4.6.
• S2
is not parallelizable since it does not admit even one linearly independent (i.e. nowhere
zero) vector field.
• Every Lie group is parallelizable via left translations: G × g → TG is given by (a, A) →
(λa∗)A.
Remark. Obviously M is parallelizable if and only if P1
M is trivial.
Theorem 4.7. The bundle Pr
M is trivial if and only if M is parallelizable.
Proof. A section of Pr
M determines by composition M → Pr
M
πr
1
−→ P1
M a section of
P1
M and hence M is parallelizable. Assume on the other hand P1
M admits a global section.
The projection Pr
M → P1
M is a bundle with standard fibre Rk
, the polynomials of degree at
most r with zero linear part which is easily seen locally as the canonical projection πr
1 : Gr
m → G1
m
is a surjective homomorphism of Lie groups hence isomorphic to a projection Gr
m → Gr
m/ ker πr
1
which is a bundle by Theorem 4.2. We know that such bundles always admit sections. The
composition M → P1
M → Pr
M is then a section of Pr
M and hence it is trivial.
The local description of principal bundles via charts and transition maps simplifies as follows
ϕαβ : Uαβ × G −→ Uαβ × G
(x, a) = (x, e)a −→ ϕαβ(x, e)a = (x, ψαβ(x)a)
with ψαβ : Uαβ → G smooth. In other words the transition map is a left multiplication by the map
ψαβ. Again we have ψαα = e and ψβγψαβ = ψαγ, the maps form a so-called G-valued cocycle. In
the opposite direction from a G-valued cocycle one can construct a principal G-bundle.
We will now address the question of when two principal G-bundles P, P are isomorphic. Let
they be given by transtion maps ψαβ and ψαβ respectively. Then f : P
∼=
−→ P is locally given by
fα : Uα × G −→ Uα × G
(x, a) = (x, e)a −→ fα(x, e)a = (x, gα(x)a)
For a different chart ϕβ we have a comparison diagram
Uαβ × G
ϕα

fα
GG
ϕαβ

Uαβ × G
ϕα

ϕαβ

Uαβ × G
ϕβ
{{
fβ
GG Uαβ × G
ϕβ
66
P
f
GG P
4. PRINCIPAL AND ASSOCIATED BUNDLES 35
In the small square we see that (x, a) at top left is mapped to (x, gα(x)ψαβ(x)a) at bottom right via
bottom left corner and to (x, ψαβ(x)gβ(x)a) via top right corner. Thus we have ψαβ = gαψαβg−1
β .
Theorem 4.8. Let {Uα} be a cover of M such that both P and P are trivialized over each
Uα. Then P ∼= P if and only if there exist gα : Uα → G such that ψαβ = gαψαβg−1
β (in this case
we say that the cocycles are equivalent).
Definition 4.9. Let p : E → M be a bundle. A subbundle of E is a subspace E ⊆ E for
which there exist local trivializations of E which also trivialize E :
p−1
(U) ∼= U × F
E ∩ p−1
(U) ∼=
⊆
U × F
⊆
Definition 4.10. Let H ⊆ G be a Lie subgroup. A subbundle Q ⊆ P of a principal bundle
P is called a reduction of P to the subgroup H if for each u ∈ Q we have u · a ∈ Q ⇐⇒ a ∈ H.
Examples 4.11.
• A reduction to the trivial subgroup {e} ⊆ G is the same as a section of P, that is a
trivialization of P.
• Consider a Riemannian manifold (M, g). Then P1
M = PTM is a principal GL(m)bundle
possessing a reduction to O(m):
PTM = inv hom(Rm
, TM) ⊇ iso(Rm
, TM),
the subspace of isometries. They are clearly closed under the action of O(m) and more
over the action is transitive so that we obtain a reduction to O(m).
In the opposite direction let Q ⊆ inv hom(Rm
, TM) be a reduction to O(m). It
defines a metric on M in the following way: every u ∈ Qx is an isomorphism u : Rm
→
TxM and we declare it an isometry or in other words we transport by u the standard
metric from Rm
. The result does not depend on q.
More generally metrics on a vector bundle p : E → M are in bijection with reductions
of PE to O(k).
• Consider an arbitrary Lie subgroup G ≤ GL(m). A G-structure on a manifold M is a
reduction of P1
M to the subgroup G. Similarly for subgroups G ≤ Gr
m of higher order
frame bundles. A reduction is then called a G-structure of r-th order.
Definition 4.12. Let P(M, G) and Q(N, H) be two principal bundles. A bundle morphism
f : P → Q is called a morphism of principal bundles with respect to a homomorphism ϕ : G → H
of Lie groups if
(∀u ∈ P)(∀a ∈ G) : f(u · a) = f(u) · ϕ(a)
If ϕ = id then we speak simply of a morphism of principal bundles or a G-morphism.
Examples 4.13.
• A reduction Q ⊆ P can be equivalently described as follows: the embedding Q → P is a
morphism of principal bundles with respect to the embedding H → G.
• Let f : M → N be a local diffeomorphism. Then
Jr
0 (Rm
, M) = Tr
mM
T r
mf
−−−−→ Tr
mN
restricts to f∗ : Pr
M → Pr
N, a morphism of principal bundles.
Let P(M, G) be a principal bundle and consider a left action : G × F → F of G on F.
Definition 4.14. A bundle p : E → M with a standard fibre F is said to be an associated
bundle to P if to each u ∈ Px there is given a diffeomorphism ˜u : F → Ex (a so-called frame map
determined by the frame u on E) such that the total frame map
ρ : P × F −→ E
(u, z) −→ ˜u(z)
is smooth and u · a = ˜u ◦ a. In terms of the total frame map ρ(u · a, z) = ρ(u, a · z).
4. PRINCIPAL AND ASSOCIATED BUNDLES 36
Remark. The idea is that we think of the principal bundle as consisting of coordinates choices
each of which gives us an identification of the standard fibre F with the geometric fibre Ex. Hence
P parametrizes these possible identifications allowing us to make constructions in coordinates in
such a way that they automatically do not depend on the choice. EXPLAIN BETTER!
Remark. We will use later ρ to denote a representation. We should therefore CHANGE the
above map to q.
Example 4.15. Let p : E → M be a vector bundle and PE = inv hom(Rm
, E) the frame
bundle of E, a principal GL(m)-bundle. We will show that E is associated to PE. For that we
need an action of GL(m) on the standard fibre of E. This being Rm
we will use the standard
action of GL(m). Each u ∈ (PE)x is by definition an invertible map Rm
→ Ex and this is our
frame map ˜u. The equivariancy condition is then obvious since
u · a = u ◦ a = ˜u ◦ a.
Also the total map PE × Rm
→ E is smooth since it sends (u, v) → u(v).
Example 4.16. The bundle β : Jr
(M, N) → N is associated to Pr
N. The standard fibre is
Jr
(M, Rn
)0 and the left action of Gr
n = invJr
0 (Rn
, Rn
)0 is by composition. The total frame map
is (as Pr
M = invJr
0 (Rn
, N))
invJr
0 (Rn
, N) × Jr
(M, Rn
)0 −→ Jr
(M, N)
(u, X) −→ u ◦ X
Again the equivariancy is verified easily.
Example 4.17. Analogously α : Jr
(M, N) → M is associated to Pr
M via the action of Gr
m
on Jr
0 (Rm
, N), a · X = X ◦ a−1
and (α, β) : Jr
(M, N) → M × N is associated to Pr
M × Pr
N.
Theorem 4.18. For a given principal bundle P(M, G) and a G-space F there exist an associated
bundle. Any two such are canonically isomorphic.
Proof. Let us start with any associated bundle E and its total frame map
ρ : P × F → E
By definition ρ factors through (P × F)/ ∼ with ∼ denoting the equivalence relation (u · a, z) ∼
(u, a · z). It is a simple matter to show that the resulting map
˜ρ : P × F/ ∼→ E
is a bijection: ρ(u, z) = ρ(u , z ) implies that π(u) = π(u ) and hence u = u · a so that ρ(u , z ) =
ρ(u, a · z ) and hence z = a · z since ˜u is a diffeomorphism.
We denote the quotient space P[F] = P ×G F the latter expressing a similarity to the tensor
product of modules over a ring. Now we will verify that P[F] bears a canonical smooth structure
(as a quotient of P × F) for which the projection P[F] → M is a bundle with standard fibre F.
This is done locally:
π−1
(Uα)[F]
∼=
−−→ (Uα × G) ×G F
∼=
−−→ Uα × F
[(x, a), z] −→ (x, az)
[(x, e), z] ←− (x, z)
the first arrow being the trivialization ϕα × id. We use these to put a smooth structure on P[F].
We are left to exhibit the effect of changing a trivialization:
(x, z)
_

Uαβ × F GG
∼=
Uαβ × F
∼=
(x, ψαβ(x) · z)
[(x, e), z]
 PP
(Uαβ × G) ×G F GG (Uαβ × G) ×G F [(x, ψαβ(x)), z]
_
yy
4. PRINCIPAL AND ASSOCIATED BUNDLES 37
These are clearly smooth but we see how the associated bundle P[F] is constructed from local
charts using the transition maps Uαβ
ψαβ
−−−−→ G −−→ Diff(F).
It remains to show that P[F] is really associated to P. But this is provided by the quotient
map P × F −→ P ×G F = P[F].
Remark. From now on when we speak about “the associated bundle” we mean the canonical
bundle P[F] constructed in the proof.
A particular case is that of a bundle associated to a principal G-bundle P via a representation
ρ : G → GL(W) of G on a vector space W. In this case P[W] is canonically a vector bundle with
standard fibre W.
Let us consider two principal bundles P(M, G) and Q(N, G) and a G-morphism
P
f
GG

Q

M
f
GG N
with respect to ϕ : G → H. Let E → M be associated to P and D → N associated to Q with the
same fibre F.
Definition 4.19. We say that a bundle morphism g : E → D over the same f as above is a
morphism associated to f if for each u ∈ P the diagram
F
˜u
~~
f(u)
22
Ex gx
GG Dx
commutes.
Theorem 4.20. A morphism g : P[F] → Q[F] associated to f is unique,
g = f[F] : [u, z] → [f(u), z]
Remark. In a similar way one can consider a morphism f × h : P[F] → Q[L] with respect
to a homomorphism ϕ : G → H of groups and a G-map h : F → L between a G-space F and an
H-space L.
Definition 4.21. By a natural bundle E over m-dimensional manifolds we understand a rule
(a functor) which associates to each m-dimensional manifold M a bundle pM : EM → M and to
each local diffeomorphism f : M → N a morphism of bundles Ef : EM → EN over f in such a
way that
• localization: for any open subset U ⊆ M we have EU = EM|U = p−1
M (U),
• functoriality: EidM = idEM and E(g ◦ f) = Eg ◦ Ef.
Remark. From the two properties it follows that Ef is also a local diffeomorphism. The
association f → Ef is called a lifting of local diffeomorphisms.
Examples 4.22.
• The tangent and the cotangent bundles.
• Tr
k , Tr∗
k or more generally Jr
(−, N) and Jr
(M, −).
• For a left action of the group Gr
m on a manifold F we can construct a natural bundle
over m-dimensional manifolds as
EM = Pr
M[F] → M (f : M → N) → (Ef = Pr
f[F])
Theorem 4.23 (Palais-Terng). For every natural bundle there exists r ≥ 0, a smooth manifold
F and a left action : Gr
m × F → F so that EM = Pr
M[F] and Ef = Pr
f[F].
5. FURTHER PROPERTIES OF PRINCIPAL AND ASSOCIATED BUNDLES 38
5. Further properties of principal and associated bundles
Let P(M, G) be a principal bundle and F a left G-space. A map σ : P → F is called
equivariant if σ(u · a) = a−1
· σ(u).
Consider a section s : M → P[F] = P ×G F of the associated bundle. For each u ∈ P there
is a unique z = σ(u) ∈ F so that s(x) = [u, z] where x = π(u). This defines a smooth map
σ : P → F which is equivariant by
[u, σ(u)] = s(x) = [u · a, σ(u · a)] = [u, a · σ(u · a)]
Another point of view is that each u ∈ Px gives an identification ˜u : F → Ex and σ(u) is simply
(˜u)−1
s(x). This also explains why σ should be equivariant.
If on the other hand σ : P → F is equivariant then in the diagram
u
 GG [u, σ(u)]
P GG

P ×G F
M
YY
there exists a (unique) factorization since M = P/G and u, u · a are carried both to the same
point in P ×G F. This factorization is a section of P[F].
Theorem 5.1. The above construction describes a bijection between sections of the associated
bundle P[F] and equivariant maps P → F.
Example 5.2. Let P = P1
M and F = Rm
with the standard action of GL(m). Hence
P1
M[Rm
] = TM and a section X : M → TM (i.e. a vector field) determines an equivariant map
ξ : P1
M → Rm
, the so-called frame form. It sends a basis (u1, . . . , um) of TxM to the coordinates
of X(x) in this basis, u · ξ(u) = X(x).
Example 5.3. Morphisms of principal bundles P → Q are exactly equivariant maps. By the
preceding they are in bijection with sections of P[Q] → M.
Let H ≤ G be a closed subgroup. The action of G on itself via left translations passes to the
quotient G/H. The associated bundle is
P[G/H] = P ×G G/H
∼=
−−→ P/H
[u, aH] −→ (ua)H
[u, eH] ←− uH
Theorem 5.4. There is a canonical bijection between sections of P[G/H] and reductions of
P to H.
Proof. Let a section s : M → P[G/H] determine an equivariant map σ : P → G/H. Easily
σ is a submersion on every fibre and thus Q = σ−1
(eH) is the desired reduction.
Let, on the other hand, Q ⊆ P be a reduction to H. Then in the diagram
Q 1  GG

P GG P/H
M
TT
the dotted factorization exists, since M = Q/H, providing a section. DETAILS!
Example 5.5. Let G ≤ GL(m) be the stabilizer of e1 ∈ Rm
, the group of matrices of the
form ( 1 ∗
0 ∗ ). Then GL(m)/G ∼= Rm
− {0} and thus reductions of P1
M to G are in bijection with
sections of ˚TM = TM − 0, the tangent bundle with the zero section removed. These are clearly
nowhere zero vector fields.
6. PROBLEMS 39
6. Problems
Problem 6.1. Determine P[∗] and P[G].
Problem 6.2. Let P be a principal G-bundle that admits a reduction Q to the subgroup
H ⊆ G. Show that P ∼= Q ×H G as principal G-bundles where the right G-action on Q ×H G is
[u, a]b = [u, ab].
Problem 6.3. Bundles associated to P are precisely those associated to Q via an action of
G.
Problem 6.4. Show that GL(m)/ O(m) ∼= R
m(m−1)
2 and apply this to the case of reductions
to O(m) ⊆ GL(m).
One possibility is to note that the mapping exp induces a diffeomorphism between the manifold
of all symmetric matrices and all positively definite matrices (regardless of the fact that these are
not Lie algebra/group pair).
Problem 6.5. Show that πr
r−1 : Jr
(M, N) → Jr−1
(M, N) is an affine bundle.
This may be solved on the models: Lr
m,n → Lr−1
m,n is an affine bundle (with a fibre-preserving
affine action of Gr
m × Gr
n).
Problem 6.6. Show that T(G/H) ∼= G ×H g/h where the action of H on g/h is induced by
the adjoint action of H on g.
Problem 6.7. Show that each sphere Sm
is stably parallelizable, i.e. that there exists an
isomorphism TSm
⊕ Rk ∼= Rm+k
for k 0.
Problem 6.8. Show that TRPm
is stably isomorphic to the direct sum of m copies of the
canonical line bundle over RPm
.
Problem 6.9. Show that the canonical bundle over the Stiefel manifold Sk(Rn
) of orthonormal
k-frames in Rn
is associated to the trivial representation of O(n − k) on Rk
while its orthogonal
complement is associated to the standard representation of O(n − k) on Rn−k
.
Problem 6.10. Show that the Stiefel manifold Sk(Rn
) is parallelizable for k > 2.
The main idea is that TSk(Rn
) ∼= O(n)×O(n−k) o(n)/o(n−k) and the O(n−k)-representation
o(n)/o(n − k) is a direct sum of a trivial representation of dimension k(k − 1)/2 and k copies of
the standard representation Rn−k
. Then one observes that the sum of a trivial representation of
dimension k and the standard representation induces a trivial bundle. Similarly for the Grassmann
manifold Gk(Rn
) but this time none of the two bundles is trivial.
Problem 6.11. Let E → M be a vector bundle associated to a principal GL(k)-bundle P.
Define the orientation bundle (a 2-sheeted covering) P[GL(k)/ GL+(k)] (which is isomorphic to
(Λk
E − 0)/R+). Show that if M is connected E possesses an orientation if and only if this
orientation covering is trivial.
CHAPTER 3
Connections
1. Connections
Let f : M → N be a smooth map which we think of as a section (id, f) of the trivial bundle
M × N → M. The derivative of f is obtained by differentiating the section and composing with
the canonical projection TM × TN → TN. For a bundle which is not trivial there is no obvious
way of projecting onto the tangent space of the fibre. This projection is the content of a connection
on the bundle.
Definition 1.1. Let p : E → M be a bundle. A connection on p is a smooth linear projection
v : TE → V E onto the vertical subbundle V E = x∈M TEx = ker(p∗ : TE → TM).
We call v the vertical projection. An associated horizontal projection is h = id − v. There is
a short exact sequence of bundles over E
0 → V E → TE → p∗
TM → 0
A vertical projection, i.e. a retraction of TE onto V E, is equivalent to a section of the projection
TE → p∗
TM. This is our second definition of a connection.
Definition 1.2. A connection on p : E → M is a “lifting map” Γ : E×M TM = p∗
TM → TE
which is smooth, linear and satisfies p∗(Γ(y, X)) = X.
Equivalently Γ(y, −) is a 1-jet of a section M → E. The mapping y → Γ(y, −) is then a
section E → J1
E.
Definition 1.3. A connection on p : E → M is a smooth section Γ : E → J1
E of the jet
prolongation J1
E → E.
Remark. The bundle J1
E → E is affine since J1
(M, E) → M × E is a vector bundle, hence
so is its pullback along (p, id) : E → M × E and the condition j1
yp ◦ j1
xs = j1
xid is affine.
Theorem 1.4. Every bundle admits (globally) a connection.
For our next formulation observe that the lifting map is completely determined by its image,
a subbundle of TE.
Definition 1.5. A connection on p : E → M is a smooth distribution Γ on E which at each
point y ∈ E is complementary to the vertical distribution VyE.
Definition 1.6. A vector field ξ : E → TE is called projectable is there exists a vector field
ξ : M → TM such that the diagram
TE
p∗
GG TM
E p
GG
ξ
yy
M
ξ
yy
commutes, i.e. such that ξ is p-related to ξ. Loosely speaking from the top one sees only one
vector over each point x ∈ M. In coordinates xi
on M and yp
on the fibre
ξ = ξi
(x) ∂
∂xi
ξ
+ ξp
(x, y) ∂
∂yp
40
1. CONNECTIONS 41
Definition 1.7. Let X : M → TM be a vector field and ˜X : E → TE given by ˜X(y) =
Γ(y, X) using the lifting map of a connection. Then ˜X is a projectable vector field on E over
X. We call this vector field the Γ-lift of X (or the horizontal lift when Γ is understood from the
context).
When the section E → J1
E is given by
dyp
= Fp
i (x, y)dxi
the horizontal lift is ˜X = Xi ∂
∂xi + Fp
i (x, y)Xi ∂
∂yp
Definition 1.8. Let p : E → M be a vector bundle. Then so is J1
E → M. A connection
Γ : E → J1
E is called linear if it is a linear morphism of vector bundles.
In coordinates the function Fp
i (x, y) must be linear in y. We write1
Fp
i (x, y) =
q
Γp
qi(x)yq
.
Thus in this case dyp
= i,q Γp
qi(x)yq
dxi
. The functions Γp
qi are almost exactly the classical
Christofell symbols.
We are now able to write formally the definition of the derivative of a section. Consider an
arbitrary connection Γ on a bundle p : E → M and a section s : M → E. We define
Γs(x) : TxM → Vs(x)E
X → s∗(X) − ˜X(s(x))
The result lies in the vertical subbundle since both s∗X and ˜X(s(x)) are lifts of X. In the first
case this follows from the section property. Equivalently Γs(x) is the vertical projection v(s∗X)
of the derivative s∗X. Using an easy adjunction
Γs(x) ∈ Vs(x)E ⊗ T∗
x M = (V E ⊗ p∗
(T∗
M))s(x)
For short we write V E ⊗T∗
M instead of V E ⊗p∗
(T∗
M). It is a bundle over E and by composing
with p also over M.
Definition 1.9. The section Γs : M → V E ⊗ T∗
M is called the covariant derivative of s
with respect to the connection Γ.
In coordinates for s given by yp
= sp
(x) we have s∗x = ∂sp
∂xi · dxi ∂
∂yp and further
Γ(s(x), −) = Fp
i (x, s(x)) · dxi ∂
∂yp
yielding
Γs(x) = ∂sp
∂xi (x) − Fp
i (x, s(x)) dxi ∂
∂yp
Definition 1.10. Let γ : R → M be a path on M defined in some neighbourhood of 0. A
section of E along γ is a map s : R → E for which p(s(t)) = γ(t)
E
p

R
s
bb
γ
GG M
≡
γ∗
E GG

E
p

R
s
ii
γ
GG M
or equivalently a section of the pullback bundle.
Definition 1.11. We say that the section s(t) along a path γ(t) is parallel if ˙s(t) ∈ Γ(s(t))
for all t. We will see shortly that there is an induced connection on γ∗
E and the condition says
that the covariant derivative is 0.
1THE QUESTION IS WHAT IS WRONG WITH iq???
1. CONNECTIONS 42
In coordinates for γ given by xi
(t) and s(t) by (xi
(t), yp
(t))
˙s(t) = dxi
dt
∂
∂xi + dyp
dt
∂
∂yp
s is parallel if and only if
dyp
dt = Fp
i (x(t), y(t))dxi
dt
From the theory of differential equations we know that for each yp
(0) there exists locally a
unique solution, i.e. every choice of s(0) extends to a unique parallel section along γ(t). Moreover
this notion does not depend on reparametrization of γ - if s(t) is parallel along γ(t) then s(t(τ))
is parallel along γ(t(τ)).
Lemma 1.12. A connection on a vector bundle is linear if and only if a linear combination of
parallel sections is again a parallel section.
Let us consider now a vector bundle p : E → M. We know that for a vector space W we have
TW = W × W. For the vertical bundle V E this means V E ∼= E ×M E. An isomorphism from
E ×M E to V E is given by (u, v) → d
dt t=0
(u + tv). Further V E ⊗ T∗
M ∼= (E ×M E) ⊗ T∗
M ∼=
E ×M (E ⊗ T∗
M) and for a section s : M → E we write
Γs = (s, Γ
s)
where Γ
s is now a section of E ⊗ T∗
M → M.
Definition 1.13. The section Γ
s is called the covariant differential of s.
In coordinates for a linear connection as above Γ
s(x) is
∂sp
∂xi − Γp
qisq
dxi ∂
∂yp
For a vector field X : M → TM we might evaluate the covariant differential on X to obtain
Γ
Xs(x) = ( Γ
s(x))(X(x)) : M → E
Definition 1.14. We call this section the covariant derivative of the section s with respect to
the vector field X.
Γ
Xs = ∂sp
∂xi − Γp
qisq
Xi
· ∂
∂yp
In this way we obtain a map
Γ
: XM × C∞
E −→ C∞
E
(X, s) −→ Γ
Xs
Theorem 1.15. The following equalities hold
(1) Γ
X(s1 + s2) = Γ
Xs1 + Γ
Xs2,
(2) Γ
X(f · s) = (Xf) · s + f · Γ
Xs, (the Leibniz rule)
(3) Γ
X1+X2
s = Γ
X1
s + Γ
X2
s,
(4) Γ
f·Xs = f · Γ
Xs.
Proof. We compute (2) from the coordinate expression
Γ
X(f · s)(x) = ∂
∂xi (f · sp
) − Γp
qifsq
Xi
· ∂
∂yp
= ∂f
∂xi · sp
+ f · ∂sp
∂xi − f · Γp
qisq
Xi
· ∂
∂yp
= (Xf) · s + f · Γ
Xs
Theorem 1.16 (The Koszul principle). Let : XM × C∞
E → C∞
E be a map satisfying the
conditions (1)-(4). Then there exists a unique linear connection Γ on E for which = Γ
.
1. CONNECTIONS 43
Proof. Locally E ∼= U × V where V is a vector space, C∞
E = C∞
(U, V ). Let v ∈ V and
we think of it as a constant map U → V , i.e. a section x → (x, v) whose derivative at X ∈ TxU is
(X, 0). Thus we are forced to put
˜X(x, v) = (id, v)∗X − (0, Xv) = (X, − Xv)
in order to ensure at least Xv = Γ
Xv. This formula on the other hand describes a bilinear map
E ×M TM → TE, i.e. a linear connection Γ on E. It remains to show = Γ
. But a general
section is locally of the form
s(x) = ai
(x)vi
and thus the formula (2) yields
Γ
Xs(x) = (Xai
)vi + ai Γ
Xvi
which reduces the general case to v.
Remark. Let U × V
∼=
−−→ U × V be an isomorphism of the trivial vector bundle over U. It
is given by a smooth map A : U → GL(V ) as (x, v) → (x, A(x) · v). The ordinary derivative ds of
a map s : U → V is changed to
d(A · s) = A · ds + dA · s
with the first part being the ordinary derivative transformed by the vector bundle morphism and
the second term amounts to a map E ×M TM → E,
((x, v), (x, X)) → dA(x, X) · v
a linear connection. We will see now that only certain connections (so-called flat ones) arise in
this way.
Let us investigate now for an arbitrary bundle p : E → M whether a given connection in the
form of a distribution is integrable (i.e. involutive). For vector fields X, Y : M → TM we consider
their horizontal lifts ˜X, ˜Y : E → TE. Since ˜X and ˜Y are p-related to X and Y , also [ ˜X, ˜Y ] is
p-related to [X, Y ]. In other words [ ˜X, ˜Y ] is a lift of [X, Y ]. Is Γ is to be involutive it is necessary
that [ ˜X, ˜Y ] = [X, Y ]. As also the vector fields of the form ˜X generate Γ it is at the same time a
sufficient condition. We have proved
Theorem 1.17. A connection Γ (considered as a distribution) is involutive if and only if
[ ˜X, ˜Y ] = [X, Y ].
Definition 1.18. The mapping CΓ : E×M Λ2
TM → V E given by the formula CΓ(y, X, Y ) =
([X, Y ] − [ ˜X, ˜Y ])(y) is called the curvature of the connection Γ. By a dualization we think of it as
a section CΓ : E → V E ⊗ Λ2
T∗
M.
Remark. To make this definition correct we have to prove that the defining expression does
not depend on the extension of X and Y to local vector fields. We will do this in the coordinates
X = Xi ∂
∂xi , Y = Y i ∂
∂xi , [X, Y ] = Xj ∂Y i
∂xj − Y j ∂Xi
∂xj
∂
∂xi
The horizontal lifts are given by
˜X = Xi ∂
∂xi + Fp
i Xi ∂
∂yp , ˜Y = Y i ∂
∂xi + Fp
i Y i ∂
∂yp
and finally
[X, Y ] = (Xj ∂Y i
∂xj − Y j ∂Xi
∂xj ) ∂
∂xi + Fp
i (Xj ∂Y i
∂xj − Y j ∂Xi
∂xj ) ∂
∂yp .
On the other hand
[ ˜X, ˜Y ] = Xj ∂Y i
∂xj − Y j ∂Xi
∂xj
∂
∂xi +
∂F p
i
∂xj (Xj
Y i
− Xi
Y j
) ∂
∂yp
+ Fp
i Xj ∂Y i
∂xj − Y j ∂Xi
∂xj
∂
∂yp + Fq
j
∂F p
i
∂yq (Xj
Y i
− Y j
Xi
) ∂
∂yp
2. PRINCIPAL CONNECTIONS 44
giving our final formula
CΓ(y, X, Y ) =
∂F p
i
∂xj (Xj
Y i
− Xi
Y j
) ∂
∂yp + Fq
j
∂F p
i
∂yq (Xj
Y i
− Y j
Xi
) ∂
∂yp .
Using the convention dxi
∧ dxj
= dxi
⊗ dxj
− dxj
⊗ dxi
we rewrite it as
CΓ(y) =
∂F p
j
∂xi + Fq
i
∂F p
j
∂yq dxi
∧ dxj ∂
∂yp
This computation shows that CΓ indeed depend only on the values of the vector fields X and Y
at the point p(y) and is thus correctly defined.
For a linear connection on E = TM we get the classical theory of connections on a manifold.
The curvature is in this case a tensor of type (1, 3), i.e. a section M −→ TM ⊗ (T∗
M)⊗3
(or in
fact M −→ TM ⊗ T∗
M ⊗ Λ2
T∗
M). The classical definition is X Y Z − Y XZ − [X,Y ]Z.
One can verify that this agrees with our (more general) definition up to the change of sign and
indices of the Christoffel symbols Γk
ij as mentioned before.
Theorem 1.19. A connection Γ is involutive if and only if the parallel transport does not
locally depend on the path.
Proof. When Γ is involutive there is an integral manifold Ly through each point y ∈ E. The
composition ϕy : Ly → E
p
−→ M is a local diffeomorphism and the parallel transport of γ is simply
obtained by composition ˜γ = ϕ−1
y ◦ γ the endpoint depending only on γ(1). The converse is also
true.
The integrability of Γ says that locally in E one can find charts of the form U × V such that
the projection p becomes the projection U × V → U and such that the distribution is TxU × {0}.
To extend this trivialization globally we need the following notion.
Definition 1.20. A connection Γ is called complete if the parallel transport exists globally.
A sufficient condition is for example that the fibre is compact. Also a linear connection is
always complete (a proof in the tutorial).
Theorem 1.21. If a connection Γ is complete and involutive then there exist local trivializations
p−1
(U) ∼= U × F such that Γ(x, y) = TxU × {0}.
Proof. The trivialization is given by the following construction. Choose a basis X1, . . . , Xm
of the base M and use their lifts ˜X1, . . . , ˜Xm to define
Rm
× F −→ E
(t = (t1
, . . . , tm
), y) −→ Flt ˜X
1 (y) = PtFltX (x)(y, 1)
where we denote for simplicity tX = t1
X1 +· · ·+tm
Xm. The right hand side is only defined when
FltX
(x) is defined on the interval [0, 1] but such t form a neighbourhood of 0, independently of
y.
2. Principal connections
Let us consider a principal bundle P(M, G). We take A ∈ g which we may express as A =
d
dt t=0
exp(tA). The fundamental vector field on P is
A∗
(u) = (r(u, −))∗(A) = d
dt t=0
(u · exp(tA)) ∈ VuP
The reason it lies in the vertical subbundle is that u · exp(tA) is a curve in Pπ(u). Globally we get
a map
P × g −→ V P
(u, A) −→ A∗
(u)
and it is clearly an isomorphism of vector bundles, i.e. a trivialization of V P.
A connection on P thought of as a vertical projection v : TP → V P then yields a 1-form
ωΓ : TP
v
−→ V P ∼= P × g → g. The defining equation is ωΓ(X)∗
= vX and the vertical projection
2. PRINCIPAL CONNECTIONS 45
is obtained uniquely from a g-valued 1-form ω provided that ω(A∗
) = A for all A ∈ g (expressing
that the map v is really a projection onto the vertical subbundle, vA∗
= A∗
).
Theorem 2.1. The following conditions are equivalent for a connection Γ on a principal
bundle, where we abbreviate Xa = (ra)∗(X) for a vector X ∈ TP (this in fact defines an action
of G on TP)
(1) v(Xa) = (vX)a,
(2) h(Xa) = (hX)a,
(3) ˜X(ua) = ˜X(u)a,
(4) the horizontal distribution is equivariant, Γ(ua) = Γ(u)a,
(5) ωΓ(Xa) = Ad(a−1
)ωΓX,
(6) the section Γ : P → J1
P satisfies Γ(u) = j1
xs =⇒ Γ(ua) = j1
x(sa).
A connection satisfying these conditions is called principal.
Proof. (1) and (2) are equivalent since v +h = id and id is equivariant. (2) is also equivalent
to (3) since they both say that the action of G preserves horizontal vectors (it preserves lifts by
definition). For point (4) note that the condition (1) is automatically satisfied on vertical vectors
and on horizontal ones (those in the kernel) it is plainly (4).
The most interesting is (5), we compute
v(Xa) = (ω(Xa))∗
(vX)a = (ωX)∗
a = d
dt t=0
u · exp(t · ωX) · a = d
dt t=0
u · exp(t · ωX)a
= d
dt t=0
(ua) · (a−1
exp(t · ωX)a) = d
dt t=0
(ua) exp(Ad(a−1
)(t · ωX))
= d
dt t=0
(ua) exp(t · Ad(a−1
)ωX) = (Ad(a−1
)ωX)∗
Thus v(Xa) = (vX)a iff ω(Xa) = Ad(a−1
)ωX.
For (6) observe that the lift ˜X(u) can be expressed as ˜X(u) = s∗X. Therefore ˜X(u)a =
(s∗X)a = (sa)∗X and this equals ˜X(ua) iff sa represents Γ(ua).
Remark. In the lecture I have used quite a lot parallel sections in the explainations. It might
be worth to start already here. A connection is principal if and only if the action preserves parallel
sections.
Corollary 2.2. For every g-valued 1-form ω on P satisfying
(1) ω(Xa) = Ad(a−1
)ω(X)
(2) ωA∗
= A
there exists a unique principal connection Γ on P whose connection form is ω.
Proof. Γ = ker ω.
Let us consider a left G-space F and the associated bundle E = P[F] = P ×G F. Let Γ be a
principal connection on P.
Definition 2.3. An associated connection ΓF : E → J1
E is defined as follows. Suppose that
Γ(u) = j1
xs. Then
ΓF ([u, y]) = j1
x[s, y]
where [s, y] : M → P ×G F is the mapping x → [s(x), y].
We have to verify that the definition does not depend on the choice of the representatives.
Firstly [s, y] is the composition
M
(s,y)
−−−−→ P × F
proj
−−−−→ P ×G F
and so it only depends on the 1-jet of s. It remains to verify that starting with [ua, y] or [u, ay]
yields the same results. But Γ(ua) = j1
x(sa) by principality and thus the two jets in question are
j1
x[sa, y] and j1
x[s, ay].
2. PRINCIPAL CONNECTIONS 46
We will now describe the associated connection in terms of the horizontal lifts. Let X ∈ TxM
and compute
˜X[u, y] = [s, y]∗X = [s∗X, 0] = [ ˜X(u), 0]
The bracket is not meant to be the Lie bracket. To explain the notation:
M
(s,y)
GG P × F
proj
GG P ×G F
TM
(s∗,0)
GG TP × TF
“[ , ]”
GG T(P ×G F)
X
 GG ( ˜X(u), 0)
 GG [ ˜X(u), 0]
CHANGE the bracket to q∗ where q is the canonical map P × F → P ×G F.
A further description of the associated connection uses parallel sections. Let s be a parallel
section along a curve γ. Then [s, y] is again a parallel section where y is a constant map at y ∈ F.
We will now bring the equivalence of vector bundles and principal GL(k)-bundles further. We
now know that a principal connection induces a (linear as we will see shortly) connection on the
vector bundle. To get back consider a vector bundle E → M and a linear connection Γ : E → J1
E
on it. The total space of the frame bundle PE is naturally an open subset
PE ⊆ E ×M · · · ×M E
in the k-fold fibre product of E with itself. Let u = (u1, . . . , uk) ∈ PE be a frame in Ex and let
us represent Γ(ui) = j1
xsi. Define ˜Γ(u) = j1
x(s1, . . . , sk). Easily ˜Γ is a connection on PE. We will
verify now that it is principal. The GL(k)-action on PE is given by the matrix multiplication-like
formula
ua = (u1, . . . , uk) · (aij) = ( uiai1, . . . , uiaik)
By linearity Γ( uiaij) = j1
x( siaij) and thus
˜Γ(ua) = (j1
x( siai1), . . . , j1
x( siaik)) = j1
x((s1, . . . , sk) · (aij))
where (s1, . . . , sk) represents ˜Γ(u). We have proved
Theorem 2.4. The connection ˜Γ is principal.
If on the other hand ˜Γ is a principal connection on PE then we will show that the associated
connection ˜ΓRk is linear: let u ∈ E ∼= PE×GL(k)Rk
be represented by u = [(u1, . . . , uk), (α1
, . . . , αk
)],
i.e. u = αi
ui. Then
˜ΓRk (u) = j1
x[s, (α1
, . . . , αk
)] = j1
x( αi
si)
where s = (s1, . . . , sk) and this expression is clearly linear in the αi
.
Theorem 2.5. The associations Γ → ˜Γ and ˜Γ → ˜ΓRk give a bijection between linear connections
on E and principal connections on PE.
There is an alternative description using parallel transport. For a path γ through x and
a frame u = (u1, . . . , uk) ∈ PE in Ex let si(t) be a parallel transport of ui along γ. Then
s(t) = (s1(t), . . . , sk(t)) is a path in PE covering the path γ and we declare it to be parallel. Thus
we define the horizontal lift of X = ˙γ(0) to u by
˜X(u) = d
dt t=0
s(t) = ( ˜X(u1), . . . , ˜X(uk))
Since linear combinations of parallel sections are parallel it is easy to see that this connection is
principal.
In the opposite direction a parallel section in PE[Rk
] are given by [s, α] where s is a parallel
section in PE and α = (α1
, . . . , αk
) ∈ Rk
. Under the identification PE[Rk
] ∼= E this becomes
[s, α] ∼ αi
si(t). Clearly linear combinations of such sections are again of the same form thus
parallel and the associated connection is linear.
3. THE COVARIANT DIFFERENTIAL ON ASSOCIATED BUNDLES 47
3. The covariant differential on associated bundles
A section s of the associated bundle P[F] → M can be described via an equivariant map
σ : P → F using the diagram
F
P × F
yy

P
(id,σ)
UU
σ
PP
[id,σ]
GG
π

P ×G F
M
s
UU
Now for X ∈ TxM we have s∗X = [id, σ]∗
˜X = [ ˜X, σ∗
˜X] and so
Γs(X) = v(s∗X) = [ ˜X, σ∗
˜X] − [ ˜X, 0] = [0, σ∗
˜X]
since [ ˜X, 0] is the horizontal lift of X. The moral is that the covariant differential is no more than
“a derivative in the direction of horizontal vectors”.
Remark. Let σ : P → F be equivariant and Y : P → TP an invariant vector field, i.e. we
require Y (ua) = Y (u)a, the most important example being Y = ˜X. Then the composition
σ∗Y : P
a
−−→ TP
σ∗
−−−→ TF
is equivariant since σ∗Y (ua) = σ∗(Y (u)a) = a−1
(σ∗Y (u)) where again the action is via ( a−1 )∗.
Schematically
G × TF GG
 •
0×id

TF
TG × TF
∗
GG TF
Therefore by the general theory σ∗Y determines a section of the associated bundle
P[TF] = P ×G TF ∼= V (P[F]) −→ M
[u, ˙γ] ∼ d
dt t=0
[u, γ]
where V denotes the vertical subbundle. This section is exactly the covariant differential when
Y = ˜X which is seen from
(0u, σ∗
˜X)
 GG [0, σ∗
˜X] = Γs(X)
TP × TF GG T(P ×G F) = TE
P GG P × TF
c1
yy
GG P ×G TF = V E
c1
the inclusion of the vertical subbundle
yy
u  GG (u, σ∗
˜X)
The part of the diagram on the right is the restriction of T(P ×G F) ∼= TP ×T G TF from the base
TM to the zero section M.
In fact σ∗ : TP → TF is already appropriately equivariant and hence determines a section of
TP[TF] = TE → TM which is “surprisingly” just s∗. Here TP → TM is a principal TG-bundle
3. THE COVARIANT DIFFERENTIAL ON ASSOCIATED BUNDLES 48
and TF is naturally a TG-space. To summarize we have explained the following passages between
equivariant maps and sections of associated bundles.
σ : P → F ←→ s : M → E
σ∗ : TP → TF ←→ s∗ : TM → TE
σ∗
˜X : P → TF ←→ Γs(X) : M → V E
We have expressed the covariant derivative as an ordinary derivative in the direction of horizontal
vectors. The derivative along vertical vectors is already determined by equivariancy.
Lemma 3.1. σ∗A∗
(u) = − A(σ(u)), where A is the fundamental vector field corresponding to
A ∈ g on the G-space F. In particular the derivative along vertical vectors does not depend on
σ∗u but only on σ(u).
Note. As σ∗A∗
is not equivariant it does not induce a section of V E.
Proof. This is an easy computation
σ∗A∗
(u) = d
dt t=0
σ(u · exp(tA)) = d
dt t=0
exp(−tA) · σ(u) = − A(σ(u))
Now we will specialize to vector bundles. Let ρ : G → GL(W) be a linear representation so
that P[W] is a vector bundle. We replace σ∗ by dσ, i.e. by the composition TP
σ∗
−−−→ TW
ωW
−−−→ W
where ωW is the Maurer-Cartan form on W (or more simply just translation to 0, TW ∼= W ×
W
pr2
−−−→ W). Evaluating at ˜X we obtain dσ( ˜X) : P → W which is again equivariant and thus
induces a section of P[W], namely the covariant derivative Γ
Xs.
Remark. The differential dσ is not TG-equivariant but merely G-equivariant. Hence one has
to pass to a G-reduction of TP → TM which is plainly HP thought of as a horizontal subbundle
(any principal connection produces a choice of such).
dσ|HP : HP → V ←→ (id, Γ
s) : TM → HE = TM ×M E
dσ( ˜X) : P → V ←→ Γ
Xs : M → E
We will now generalize this form of the covariant differential to forms of higher degree. We
start a bit more generally with a smooth manifold M and a vector space W.
Definition 3.2. A W-valued k-form on M is a smooth antisymmetric multilinear map
ϕ : TM ×M · · · ×M TM −→ W or ϕ : Λk
TM −→ W
We write ϕ ∈ Ωk
(M; W).
Let ϕ = ϕj
ej where ϕj
∈ Ωk
(M) and (ej) a basis of W. We define
dϕ = (dϕj
)ej
which is a W-valued (k + 1)-form that does not depend on the choice of the basis since a change
of basis is linear as is the differential.
Let ρ : G → GL(W) be a representation and P(M, G) a principal bundle.
Definition 3.3. We say that ϕ ∈ Ωk
(P; W) is of type ρ if
ϕ(A1a, . . . , Aka) = ρ(a−1
)ϕ(A1, . . . , Ak)
If this is the case we write ϕ ∈ Ω(P; ρ). Observe that the left hand side is simply
ϕ(ra∗A1, . . . , ra∗Ak) = r∗
aϕ(A1, . . . , Ak)
Therefore the condition may be rewritten simply as r∗
aϕ = ρ(a−1
)ϕ.
Example 3.4. The form ωΓ of a principal connection Γ is of type Ad
3. THE COVARIANT DIFFERENTIAL ON ASSOCIATED BUNDLES 49
Definition 3.5. We say that ϕ is horizontal if ϕ(A1, . . . , Ak) = 0 whenever one of Ai is
vertical. In this way ϕ can be thought of as a map
Λk
HP = Λk
(TP/V P) −→ W
Theorem 3.6. Horizontal k-forms of type ρ are in bijection with P[W]-valued k-forms on M,
i.e. vector bundle morphisms
Λk
TM
ϕ
GG
00
P[W]
ÑÑ
M
Proof. A horizontal k-form ϕ : Λk
TP → W of type ρ induces, as we observed, a G-map
Λk
HP → W or equivalently a G-map ˜ϕ : P ×M Λk
TM → W. We have seen how to identify any
equivariant map P → W with a section of P[W] and in the present situation we just carry Λk
TM
over2
to obtain ϕ : Λk
TM → P[W]:
ϕ(X1, . . . , Xk) = [u, ˜ϕ(u, X1, . . . , Xk)] = [u, ϕ( ˜X1(u), . . . , ˜Xk(u))]
wehre u ∈ P is any point lying over the same point as all the vectors Xi. This formula says
that for vector fields X1, . . . , Xk the section ϕ(X1, . . . , Xk) of P[W] corresponds to the equivariant
map ϕ( ˜X1, . . . , ˜Xk) : P → W which may be interpreted as: having a correspondence for
sections/equivariant maps and vector fields/equivariant horizontal vector fields gives a correspondence
for forms.
Remark. When the representation ρ is trivial, ∀a ∈ G : ρ(a) = id, then ϕ( ˜X1(u), . . . , ˜Xk(u))
does not depend on the choice of u over x and defines a map Λk
TM → W. This corresponds to
Λk
TM → P[W] ∼= M × W
pr
−→ W.
Let ϕ ∈ Ωk
(P, ρ) then dϕ ∈ Ωk+1
(P, ρ) is of the same type since
r∗
adϕ = dr∗
aϕ = d(ρ(a−1
) ◦ ϕ) = ρ(a−1
) ◦ dϕ
by linearity of the map ρ(a−1
). The horizontality on the other hand needs not be preserved by d.
Definition 3.7. An exterior covariant differential of a W-valued k-form on P is a (k+1)-form
Dϕ(X0, . . . , Xk) = dϕ(hX0, . . . , hXk)
Clearly Dϕ is horizontal. If ϕ is moreover of type ρ then so is Dϕ since both the horizontal
projection h and dϕ are equivariant. Therefore we get a diagram
Ωk
(P, ρ)
d GG Ωk+1
(P, ρ)
h∗

Ωk
hor(P, ρ)
c1
yy
D GG
yy
∼=

Ωk+1
hor (P, ρ)
yy
∼=

Ωk
(M, P[W]) GG Ωk+1
(M, P[W])
with h∗
ψ(X0, . . . , Xk) = ψ(hX0, . . . , hXk).
2Another possibility is to view P = P ×M ΛkTM as a pullback bundle of P along the projection ΛkTM → TM.
Then ˜ϕ is a G-map P → W and these are in bijection with sections of P [W] which are exactly the maps as in the
statement of the theorem.
4. THE STRUCTURE EQUATION 50
Remark. The dotted arrow can be described explicitly: either write locally ϕ = sidfi,1 ∧
· · · ∧ dfi,k and then Dϕ = si ∧ dfi,1 ∧ · · · ∧ dfi,k or
Dϕ(X0, . . . , Xk) =
i
(−1)i
Xi
ϕ(X0, . . . , ˆXi, . . . , Xk)
+
i<j
(−1)i+j
ϕ([Xi, Xj], X0, . . . , ˆXi, . . . , ˆXj, . . . , Xk)
Consider a principal connection Γ on P and its curvature
CΓ : P ×M Λ2
TM −→ V P
defining a g-valued 2-form Ω on P by the formula
Ωu(Y, Z) := ωCΓ(u, π∗Y, π∗Z)
Theorem 3.8. Ω = Dω, i.e. the curvature Ω is a covariant derivative of the form of the
connection.
Proof. We first express Ω using the defining equation for ω:
Ωu(Y, Z)∗
= CΓ(u, π∗Y, π∗Z) = −v[π∗Y , π∗Z] = −v[hY, hZ]
Therefore Ωu(Y, Z) = −ω[hY, hZ]. Now we compute
Dω(Y, Z) = dω(hY, hZ) = (hY ) ω(hZ)
0
−(hZ) ω(hY )
0
−ω[hY, hZ] = Ω(Y, Z)
4. The structure equation
Let U, V, W be vector spaces and f : U ⊗ V → W a linear map. Let ϕ : TM → U and
ψ : TM → V be 1-forms. Consider the antisymmetrization of
TM ⊗ TM
ϕ⊗ψ
−−−−→ U ⊗ V,
a 2-form ϕ∧ψ : Λ2
TM → U ⊗V . By composing with f we obtain a 2-form f(ϕ, ψ) : Λ2
TM → W,
explicitly
f(ϕ, ψ)(X, Y ) = f(ϕ(X) ⊗ ψ(Y )) − f(ϕ(Y ) ⊗ ψ(X))
Applying this construction to the Lie algebra bracket g ⊗ g → g and ω ∈ Ω1
(M, g)
[ω, ω](X, Y ) = [ωX, ωY ] − [ωY, ωX] = 2[ωX, ωY ]
On a principal bundle with a principal connection Γ we have the form of the connection ω ∈
Ω1
(P, Ad).
Theorem 4.1 (The structure equation). dω + 1
2 [ω, ω] = Ω.
Corollary 4.2 (The second Bianchi identity). dΩ = [Ω, ω]. In particular DΩ = 0.
Proof. Applying a linear map f to d(ϕ ∧ ψ) = dϕ ∧ ψ − ϕ ∧ dψ one obtains
d(f(ϕ, ψ)) = f(dϕ, ψ) − f(ϕ, dψ)
Thus using the structure equation
dΩ = d(dω + 1
2 [ω, ω]) = 1
2 [dω, ω] − 1
2 [ω, dω] = [dω, ω]
since ω ∧ dω = tw ◦ dω ∧ ω and [ , ] is anticommutative, [ , ] ◦ tw = −[ , ]. Using the structure
equation again
[dω, ω] = [Ω − 1
2 [ω, ω], ω] = [Ω, ω]
since [[ω, ω], ω] = 0 by the Jacobi identity:
[[ω, ω], ω](X, Y, Z) = [[ω, ω](X, Y ), ωZ] − [[ω, ω](X, Z), ωY ] + [[ω, ω](Y, Z), ωX]
= 2([[ωX, ωY ], ωZ] − [[ωX, ωZ], ωY ] + [[ωY, ωZ], ωX]) = 0.
The last part follows from DΩ = h∗
dΩ = [h∗
Ω, h∗
ω] and h∗
ω = 0.
4. THE STRUCTURE EQUATION 51
The proof of the structure equation. First we deal with a Lie group G thought of as
a principal G-bundle over a point. The vertical projection in this case is the identity and thus
there exists a unique connection. Since
A∗
(a) = d
dt t=0
a · exp(tA) = (λa)∗A
the unique connection form is ωG(a, X) = (λa−1 )∗X. This is the canonical g-valued 1-form on G
called the Maurer-Cartan form. The structure equation reduces in this case to
Theorem 4.3 (Maurer-Cartan equation). dωG + 1
2 [ωG, ωG] = 0.
Note. The curvature must be zero since TM = 0.
Proof. Observe that any X ∈ TaG extends to a left-invariant vector field λωGX and denote
for short ωGX = A and ωGY = B. Thus
dωG(X, Y ) = dωG(λA, λB) = λA(ωGλB) − λB(ωGλA) − ωG[λA, λB]
= λA(const) − λB(const) − ωGλ[A,B] = −[A, B] = −1
2 [ωG, ωG](X, Y ).
Let us now proceed to the general case with P a principal G-bundle over M and a principal
connection on P. For A ∈ g the fundamental vector field A∗
: P → V P is given by
A∗
(u) = d
dt t=0
u · exp(tA)
The derivative at a general t0 is
d
dt t=t0
u · exp(tA) = d
dt t=t0
u · exp(t0A) · exp((t − t0)A) = A∗
(u · exp(t0A))
In particular FlA∗
t = u · exp(tA) or in other words FlA∗
t = rexp(tA).
Lemma 4.4. For arbitrary horizontal vector field Y on P, [A∗
, Y ] is also horizontal.
Proof. We determine the Lie bracket by
[A∗
, Y ](u) = d
dt t=0
(FlA∗
−t )∗Y (FlA∗
t (u)) = d
dt t=0
(rexp(tA))∗Y (u · exp(tA))
Here Y (u · exp(tA)) is horizontal by assumption and the action preserves horizontality. Therefore
the curve lies in HuP and so does its derivative.
The proof of the structure equation splits into three cases by bilinearity
• both X and Y vertical: then Ω(X, Y ) = 0 as Ω is horizontal. The restriction ω|T Px
of
the connection form to the fibre is the Maurer-Cartan form ωG and the Maurer-Cartan
equation finishes this case.
• X = A∗
vertical and Y horizontal: still Ω(X, Y ) = 0 by horizontality. The left hand side
is
dω(A∗
, Y ) = A∗
( ωY
0
) − Y (ωA∗
const.
) − ω [A∗
, Y ]
horizontal
= 0
1
2 [ω, ω](A∗
, Y ) = [ωA∗
, ωY
0
] = 0
• both X and Y horizontal: then 1
2 [ω, ω](X, Y ) = 0 and the structure equation says
dω(X, Y )
?
= Dω(X, Y ) = dω(hX, hY )
which is satisfied by the horizontality of X and Y .
Lemma 4.5. A differential k-form ϕ on P projects to a k-form ψ on M (i.e. ϕ = π∗
ψ) if and
only if the following two conditions are satisfied
(1) ϕ is horizontal and
(2) ϕ(X1a, . . . , Xka) = ϕ(X1, . . . , Xk), i.e. ϕ is of the type given by the trivial representation
e : G → GL(1), a → e = id.
4. THE STRUCTURE EQUATION 52
Proof. This is a special case of the bijection Ωk
hor(P, ρ) ∼= Ωk
(M, P[W]) for W = R with the
trivial action so that P[W] = M × R.
Now we will construct k-forms on M from the curvature Ω. We denote by Ik
(G) the set of
all symmetric multilinear maps
f : g × · · · × g −→ R
satisfying f(Ad(a)A1, . . . , Ad(a)Ak) = f(A1, . . . , Ak). In other words f is equivariant with respect
to the trivial action of G on R. The curvature form Ω : Λ2
TP → g then induces
Λ2
TP ⊗ · · · ⊗ Λ2
TP
Ω⊗···⊗Ω
−−−−−−−→ g ⊗ · · · ⊗ g
f
−−→ R
Antisymmetrizing we obtain f(Ω) : Λ2k
TP → R.
Theorem 4.6. The (2k)-form f(Ω) on P projects to a (2k)-form f(Ω) on M.
Proof. Easily f(Ω) is horizontal since Ω is and
f(Ω)(X1, . . . , X2k) = f(Ω(Xσ(1), Xσ(2)), . . . , Ω(Xσ(2k−1), Xσ(2k)))
the sum being taken over (2, . . . , 2)-shuffles. The equivariancy follows from Ω being of type Ad
and f being equivariant.
Definition 4.7. The form f(Ω) is called the Chern-Weil form.
Lemma 4.8. When an r-form ϕ on P projects to an r-form ϕ on M then dϕ = Dϕ for any
connection on P.
Proof. First we express
Dϕ(X0, . . . , Xr) = Dϕ( ˜X0, . . . , ˜Xr) = dϕ( ˜X0, . . . , ˜Xr)
To compute dϕ we use ϕ(X0, . . . , Xr) = ϕ( ˜X0, . . . , ˜Xr) and differentiate
dϕ(X0, . . . , Xr) = (−1)i
Xi · ϕ(X0, . . . , Xi, . . . , Xr)
+ (−1)i+j
ϕ([Xi, Xj], X0, . . . , Xi, . . . , Xj . . . , Xr)
= (−1)i ˜Xi · ϕ( ˜X0, . . . , ˜Xi, . . . , ˜Xr)
+ (−1)i+j
ϕ([Xi, Xj], ˜X0, . . . , ˜Xi, . . . , ˜Xj . . . , ˜Xr)
This is exactly dϕ( ˜X0, . . . , ˜Xr) when [Xi, Xj] is replaced by [ ˜Xi, ˜Xj]. But since the difference is a
vertical vector and ϕ is horizontal this makes no difference.
Theorem 4.9. All Chern-Weil forms f(Ω) are closed.
Proof. By the previous lemma df(Ω) = df(Ω) = Df(Ω) and
Df(Ω) = D(f ◦ (Ω ∧ · · · ∧ Ω)) = f ◦ (Ω ∧ · · · ∧ Ω ∧ DΩ
0
∧Ω ∧ · · · ∧ Ω) = 0
with DΩ = 0 by the Bianchi identity.
Lemma 4.10. Let ϕ be a horizontal 1-form of type ρ. Then
Dϕ(X, Y ) = dϕ(X, Y ) + ω(X) · ϕ(Y ) − ω(Y ) · ϕ(X)
where the dot stands for the infinitesimal action A · w = ρ∗(A)(w) with ρ∗ : g → gl(W) the
derivative of ρ. We may write simply
Dϕ = dϕ + ω · ϕ
Proof. Again we split the proof into three cases.
• both X and Y horizontal. Then Dϕ(X, Y ) = dϕ(X, Y ) and ω(X) = 0 = ω(Y ).
4. THE STRUCTURE EQUATION 53
• both X = A∗
, Y = B∗
vertical. Then Dϕ(X, Y ) = 0 and
dϕ(A∗
, B∗
) = A∗
ϕ(B∗
)
0
−B∗
ϕ(A∗
)
0
−ϕ[A∗
, B∗
] = −ϕ[A, B]∗
= 0
As ϕ(A∗
) = 0 = ϕ(B∗
) the equality holds trivially.
• X = A∗
vertical and Y = ˜Z horizontal. Still Dϕ(X, Y ) = 0 and
dϕ(A∗
, ˜Z) = A∗
ϕ( ˜Z) − ˜Z ϕ(A∗
)
0
−ϕ[A∗
, ˜Z]
In the last term
[A∗
, ˜Z] = d
dt t=0
(rexp(−tA))∗
˜Z(u · exp(tA)) = d
dt t=0
˜Z(u) = 0
so that
dϕ(A∗
, ˜Z) = A∗
ϕ( ˜Z) = d
dt t=0
ϕ( ˜Z(u · exp(tA)) = d
dt t=0
ϕ( ˜Z(u) · exp(tA))
= d
dt t=0
ρ(exp(−tA))ϕ( ˜Z(u)) = −ρ∗(A) · ϕ( ˜Z(u))
Since A = ω(X) this equals −ω(X) · ϕ(Y ). As ω(Y ) · ϕ(X) = 0 the equality holds.
We are now aiming at the independence of the cohomology class of the Chern-Weil form under
the choice of the principal connection. Therefore let Γ0 and Γ1 be two principal connections with
associated forms ω0 and ω1. Put α = ω1 − ω0 ∈ Ω1
hor(P, Ad), horizontal by ω1(A∗
) = A = ω0(A∗
).
Then a covariant derivative with respect to some principal connection ω is
Dα = dα + [ω, α]
since α is of type ρ = Ad and ρ∗ = ad = [ , ]. We consider a 1-parameter family of connections
ωt = ω0 + tα = (1 − t)ω0 + tω1
(note that principal connections form an affine space in Ω1
(P, g) as both conditions - being of type
Ad and the reproduction of vertical vector fields - are affine). We denote by Ωt the curvature
associated to ωt and Dt the covariant differential.
Lemma 4.11. d
dt Ωt = Dtα.
Proof. To explain the formula Ωt(u) ∈ hom(Λ2
TuP, g) and the derivative is taken in this
vector space. Differentiate the structure equation
Ωt = dωt + 1
2 [ωt, ωt]
to obtain
d
dt Ωt = d
dt d(ωt) + 1
2 [ d
dt ωt, ωt] + 1
2 [ωt, d
dt ωt]
= d( d
dt ωt) + 1
2 [α, ωt] + 1
2 [ωt, α] = dα + [ωt, α] = Dtα
Definition 4.12. Define a horizontal (2k − 1)-form on P
f(α, Ωt, . . . , Ωt
k−1
) = f ◦ (α ∧ Ωt ∧ · · · ∧ Ωt) : Λ2k−1
TP −→ R
It projects onto a (2k − 1)-form f(α, Ωt, . . . , Ωt) ∈ Ω2k−1
(M). Let
Φ = k ·
1
0
f(α, Ωt, . . . , Ωt) ∈ Ω2k−1
(M)
Theorem 4.13. It holds dΦ = f(Ω1) − f(Ω0) so that the forms f(Ω0) and f(Ω1) determine
the same class in the de Rham cohomology.
5. THE CANONICAL FORM ON P 1
M (SOLDER FORM) 54
Proof. Since f(Ω1) − f(Ω0) =
1
0
d
dt f(Ωt) dt we compute
d
dt f(Ωt, . . . , Ωt) = f(Ωt, . . . , d
dt Ωt, . . . , Ωt) = k · f( d
dt Ωt, Ωt, . . . , Ωt)
= k · f(Dtα, Ωt, . . . , Ωt) = k · Dtf(α, Ωt, . . . , Ωt)
= k · d(f(α, Ωt, . . . , Ωt))
Thus
f(Ω1) − f(Ω0) =
1
0
k · d(f(α, Ωt, . . . , Ωt)) dt = d
1
0
k · f(α, Ωt, . . . , Ωt) dt = dΦ
Theorem 4.14 (reformulation). For each f ∈ Ik
(G) the de Rham class of the Chern-Weil
form f(Ω) does not depend on the connection Γ.
Example 4.15. Consider the example G = GL(k), i.e. the example of vector bundles of
dimension k. Here g = gl(k) and Ad(a)(A) = aAa−1
. The trace of a matrix is a map tr : gl(k) → R
satisfying tr(aAa−1
) = tr(A). Therefore tr ∈ I1
(GL(k)) and yields a class [ tr(Ω)] ∈ H2
(M). In
the tutorial we will show that I1
(GL(k)) = tr . There exist higher traces
tr
j
: gl(k) ⊗ · · · ⊗ gl(k) −→ R
X1 ⊗ · · · ⊗ Xj −→ tr(X1 · · · Xj)
which exhibits a cyclic symmetry. Fully symmetrizing we get sym(trj) ∈ Ij
(GL(k)). “A bit
of representation theory” implies that all Chern-Weil forms are generated by trj via the wedge
product and linear combinations.
We will show now that for j odd these classes are zero. This will follow from the fact that
every principal GL(k)-bundle P admits a reduction Q to O(k). This means that for the connection
induced from a principal connection on Q the curvature Ω takes values in o(k) (this will be shown
at the tutorial), the algebra of anti-symmetric matrices. Since an odd power of an anti-symmetric
matrix is again anti-symmetric the trace tr2i−1(Ω) must be zero.
For complex vector bundles there are non-zero classes in all even dimensions up to the dimension
of the vector bundle.
5. The canonical form on P1
M (solder form)
Let π : P1
M → M be the bundle of frames on M, P1
M = PTM u = (u1, . . . , um)
a basis of TxM, x = π(u). It is a principal GL(m)-bundle whose fibre can be described as
inv hom(Rm
, TxM). The action of GL(m) on P1
M is then given as precomposition. Alternatively
(u1, . . . , um)a = ( uiai1, . . . , uiaim).
Definition 5.1. The canonical form on P1
M is the Rm
-valued 1-form θ defined by
π∗X = θ1
(X)u1 + · · · + θm
(X)um
In other words, the components of θ(X) are the coordinates of π∗X in the basis u. Using the
frame map ρ : P1
M × Rm
→ TM, (u, α) → αi
ui the definition becomes
ρ(u, θ(X)) = π∗(X)
Theorem 5.2. θ ∈ Ω1
hor(P, id) where id : GL(m) → GL(m) is the standard representation of
GL(m) on the vector space Rm
.
Proof. The horizontality is obvious since π∗X = 0 for a vertical vector X. Since the action
preserves fibres we have
(ua) · θ(Xa) = π∗(Xa) = π∗X = u · θ(X)
implying aθ(Xa) = θ(X) as both are coordinates of π∗X in the basis u. Finally θ(Xa) = a−1
θ(X).
6. THE SECOND TANGENT SPACE T T M 55
Lemma 5.3. Under the identification
XM = C∞
TM = C∞
P1
M[Rm
] ∼= mapGL(m)(P1
M, Rm
)
a vector field X corresponds to θ( ˜X).
Proof. The section of P1
M[Rm
] corresponding to u → θ( ˜X(u)) sends x to
[u, θ( ˜X(u))] ∼ u · θ( ˜X(u)) = X(x)
where u ∈ P1
Mx is arbitrary and ∼ denotes the identification P1
M[Rm
] ∼= TM.
Definition 5.4. Let Γ be a principal connection on P1
M. The covariant differential Dθ is
called the torsion form of the connection Γ.
Theorem 5.5. Dθ = 0 if and only if the connection Γ has no torsion.
Proof. We will show that the section corresponding to Dθ( ˜X, ˜Y ) is
XY − Y X − [X, Y ]
for any vector fields X and Y on M. But
Dθ( ˜X, ˜Y ) = ˜X(θ ˜Y ) − ˜Y (θ ˜X) − θ[ ˜X, ˜Y ]
where θ ˜Y = Y by the last lemma and hence ˜X(θ ˜Y ) = XY . By horizontality of θ the last term
can be simplified θ[ ˜X, ˜Y ] = θ[X, Y ] = [X, Y ].
For the next theorem denote by dot the following pairing
− · − : gl(m) ⊗ Rm
→ Rm
Theorem 5.6 (The first structure equation). It holds Dθ = dθ+ω·θ where ω ∈ Ω1
(P1
M, gl(m))
is the form of the connection and θ ∈ Ω1
(P, Rm
) is the canonical form.
Proof. We have shown this more generally, Dϕ = dϕ + ρ∗ω · ϕ.
Remark. By differentiating covariantly once more we obtain
D2
θ = h∗
d(dθ + ω · θ) = h∗
(dω · θ − ω · dθ) = h∗
dω · h∗
θ = Dω · θ = Ω · θ
We have not used any specific property of θ and thus for ϕ ∈ Ωk
hor(P, ρ) it holds generally that
D2
ϕ = ρ∗Ω · ϕ. In particular the covariant differential does not in general square to zero.
6. The second tangent space TTM
What is TTM = T(TM)? Locally one has
TTRm
= T(Rm
× Rm
) = Rm
× Rm
base
× Rm
× Rm
fibre
Let us write the coordinates as (x, X, Y, α). For σ : R2
→ Rm
with coordinates s and t on R2
we
have
∂
∂t t=0
σ : R −→ TRm
s −→ (σ(s, 0), ∂
∂t σ(s, 0))
Differentiating again we obtain ∂
∂s s=0
∂
∂t t=0
σ ∈ TTRm
with coordinates
(σ(0), ∂
∂t σ(0), ∂
∂s σ(0), ∂2
∂s∂t σ(0))
We have a well defined map TTRm
→ TTRm
by
∂
∂s s=0
∂
∂t t=0
σ −→ ∂
∂t t=0
∂
∂s s=0
σ
which plainly swaps the middle coordinates but in this form it clearly does not depend on coordinates
and thus induces a map
κ : TTM → TTM
on the second tangent bundle of any manifold M.
7. MORPHISMS OF CONNECTIONS 56
Let ˜ : TM ×M TM → TTM be a lifting map of a linear connection on TM. Locally
((x, X), (x, Y )) → (x, X, Y, (Γk
ijXi
Y j
)). We can use κ to introduce a new lifting map ˆ via the
diagram
(x, Y, X, (Γk
ijY i
Xj
)) (x, X, Y, (Γk
jiXi
Y j
))
TTM
κ GG TTM
TM ×M TM
˜
yy
TM ×M TM
exoo
ˆ
yy
((x, Y ), (x, X)) ((x, X), (x, Y ))
with ex denoting exchanging the two factors.
Theorem 6.1. ˆ = κ ◦˜◦ ex prescribes a linear connection on TM, the so-called conjugate
connection ¯Γ. In coordinates ¯Γk
ij = Γk
ji.
Let us denote the “translation” map by
tr : V TM ∼= TM ×M TM
pr2
−−−→ TM
(x, X, 0, Z) −→ (x, Z)
Lemma 6.2. For any vector fields X, Y ∈ XM
tr(TY ◦ X − κ ◦ TX ◦ Y ) = [X, Y ]
Proof. In coordinates X : x → (x, X), Y : x → (x, Y ) so that
TY ◦ X : x −→ (x, Y, X, Xj ∂Y i
∂xj
∂
∂xi )
TX ◦ Y : x −→ (x, X, Y, Y j ∂Xi
∂xj
∂
∂xi )
TY ◦ X − κ ◦ TX ◦ Y : x −→ (x, Y, 0, [X, Y ])
tr
−−→ (x, [X, Y ])
where e.g. the last coordinate of TY ◦ X is the derivative of Y along X.
Theorem 6.3. The following holds
(1) ¯ XY = Y X + [X, Y ],
(2) the section corresponding to Dθ( ˜X, ˜Y ) is XY − ¯ XY .
Proof. Once (1) is proved, (2) follows from Dθ( ˜X, ˜Y ) = XY − Y X − [X, Y ]. To prove
(1) we observe that by definition
Y X = tr(v(TX ◦ Y )) = tr(TX ◦ Y − ˜Y (X)) = tr(κ ◦ TX ◦ Y − κ( ˜Y (X)))
using tr ◦κ = tr and analogously
¯ XY = tr(TY ◦ X − ˆY (X)) = tr(TY ◦ X − κ( ˜Y (X)))
Subtracting the two formulas reduces the theorem to the previous lemma.
7. Morphisms of connections
Let pi : Ei → M, i = 1, 2 be two bundles and also qi : Di → N. Further let fi : Ei → Di be
bundle morphisms over the same base map f : M → N. We obtain
f1 × f2 = f1 ×f f2 : E1 ×M E2 → D1 ×N D2
the so-called fibre product of f1 and f2.
Let now˜: E×M TM → TE be a lifting map for a connection Γ on E → M andˆ: D×N TN →
TD a lifting map for a connection ∆ on D → N, f : D → E a bundle morphism over f : N → M.
7. MORPHISMS OF CONNECTIONS 57
Definition 7.1. Connections ∆ and Γ are calles f-related if the diagram
TD
f∗
GG TE
D ×N TN
ˆ
yy
f×f∗
GG E ×M TM
˜
yy
commutes. In other words f is required to preserve horizontal vectors, i.e. f∗
ˆX(y) = f∗X(f(y))
or f∗∆(y) ⊆ Γ(f(y)).
We also say that f is a morphism of connections ∆ and Γ.
Definition 7.2. An induced connection g∗
Γ on the pullback bundle g∗
E
g∗
E
¯g
GG

E

N g
GG M
is determined by the requirement that the horizontal distribution g∗
Γ is the preimage of the
horizontal distribution Γ, i.e. g∗
Γ(x, y) = (¯g∗(x,y))−1
Γ(y).
Theorem 7.3. The distribution g∗
Γ gives a connection on g∗
E.
Proof. By the diagram
(X, Y )  GG
_

Y_

X  GG g∗X = p∗Y
for (X, Y ) ∈ g∗
Γ(x, y) necessarily Y = g∗X(y) so that for each X ∈ TxN there is a unique
Y ∈ TyE with (X, Y ) ∈ g∗
Γ(x, y).
Another characterization of g∗
Γ is via the jets of sections: a section s : M → E representing
the horizontal subspace of Γ, i.e. Γ(sg(x)) = j1
g(x)s, induces a section g∗
s : N → g∗
E
N
id
33
s◦g
55
g∗
s
GG g∗
E GG

E

N g
GG M
and g∗
Γ(x, sg(x)) = j1
x(g∗
s). In this way one obtains the horizontal spaces at all points.
Theorem 7.4. The connections g∗
Γ and Γ are ¯g-related.
Proof. Follows immediately from the definition.
Theorem 7.5. In the diagram
D
˜f
GG
f
AA

f∗
E GG

E

N N
f
GG M
the connections ∆ and Γ are f-related if and only if ∆ and f∗
Γ are ˜f-related.
Proof. Easy.
8. PROBLEMS 58
Theorem 7.6. If ∆ and Γ are f-related then the following diagram commutes
V D
f∗
GG V E
D ×N Λ2
TN
f×Λ2
f∗
GG
C∆
yy
E ×M Λ2
TM
CΓ
yy
We say that C∆ and CΓ are f-related.
Proof. Let y ∈ D, X, Y ∈ Tq(y)N where q : D → N is the bundle projection. Suppose first
that it is possible to extend X and Y to vector fields X and Y that are f-related to vector fields
X and Y on M. Then
CΓ(f(y), X , Y ) = −v[ ˜X , ˜Y ](f(y))
But ˆX and ˆY are f-related to ˜X and ˜Y so that [ ˆX, ˆY ] is also f-related to [ ˜X , ˜Y ] and also [X, Y ]
is f-related to [X , Y ]). Subtracting we obtain that C∆(−, X, Y ) is f-related to CΓ(−, X , Y )
which is exactly the commutativity of the diagram from the theorem.
In general the extensions X and Y might not exist. They do exist for f an immersion and a
submersion. But it is possible to decompose f into a composition of such, namely
D
˜f
GG

f∗
E GG

pr∗
2E GG

E

N N
(id,f)
GG N × M pr2
GG M
The graph (id, f) of f is obviously an immersion while the projection pr2 is a submersion. The
extensions are easy to construct.
8. Problems
Problem 8.1. Show that a linear connection is complete.
Problem 8.2. Show that for a vector bundle E → M the vector space C∞
E of all smooth
sections of E is naturally a bundle over C∞
M. Further show that if E and F are two vector
bundles over M then there is a bijection between linear morphisms E → F and C∞
M-linear
homomorphisms C∞
E → C∞
F.
Problem 8.3. Show that if ϕ : C∞
E1 × C∞
E2 −→ C∞
F is bilinear over C∞
M then the
value of ϕ(s1, s2) at x depends only on s1(x) and s2(x) and this dependence describes a bilinear
morphism E1 ×M E2 −→ F of bundles over M.
One may reduce to the previous problem by showing that
C∞
E1 ⊗C∞M C∞
E2
∼= C∞
(E1 ⊗M E2)
Problem 8.4. Apply the previous problem to the curvature
CΓ : C∞
(p∗
TM) × C∞
(p∗
TM) −→ C∞
V E
Problem 8.5. Show that an exterior derivative of a 1-form ϕ ∈ Ω1
(M) satisfies
dϕ(X, Y ) = Xϕ(Y ) − Y ϕ(X) − ϕ[X, Y ]
for any two vector fields X, Y ∈ XM. Generalize to higher degrees.
Problem 8.6. Show that I1
(GL(k)) = tr .
Decomposition into symmetric and antisymmetric matrices yields an easy reduction to linear
forms on symmetric matrices which are O(k)-invariant. Since each is equivalent to a diagonal one
this gives the result.
8. PROBLEMS 59
Problem 8.7. Let Q ⊆ P be a reduction of a principal G-bundle P to H ⊆ G. Prove the
following two characterizations of principal connections on P induced from principal connections
on Q:
• Γ is a principal connection tangent to Q, i.e. Γ|Q ⊆ TQ,
• ω is a principal connection whose restriction ω|Q to Q takes values in h.
Problem 8.8. Show that the canonical form θ : TP1
M → Rm
corresponds to id : TM → TM.
Problem 8.9. Show that under the identification of sections s ∈ C∞
(P[W]) and equivariant
maps σ : P → W we get that Xs corresponds to Dσ( ˜X). Maybe not a good problem. . .
Problem 8.10. Let E, F be two vector bundles associated to P(M, G) and let s ∈ C∞
E and
t ∈ C∞
F be two sections. Then X(s ⊗ t) = Xs ⊗ t + s ⊗ Xt.
Problem 8.11. Let P → M be a principal GL(k)-bundle and Q ⊆ P a reduction to O(k)
which is equivalent to a scalar product g ∈ C∞
(E ⊗ E)∗
. Let Γ be a principal connection of P.
Show that the following conditions are equivalent.
• Γ reduces to Q,
• g = 0 (g is then called covariantly constant),
• the parallel transport on E preserves the scalar product.
Problem 8.12. Let P(M, G) be a principal bundle with a principal connection ω and let
ρ : G → GL(W) be a representation, E = P ×G W the associated vector bundle. Describe the
curvature CΓW of the associated bundle in terms of Ω.
Problem 8.13. Describe for X, Y ∈ XM and s ∈ C∞
E the expression
X Y s − Y Xs − [X,Y ]s
in terms of the equivariant map σ : P → W corresponding to s.
Problem 8.14. Let ι : Q → P be an inclusion of a reduction Q of P to a subgropu H ⊆ G.
Let ΓQ be a reduction of a principal connection ΓP on P. Show that ΓQ and ΓP are ι-related and
that ΩQ is a restriction of ΩP to Q. Deduce that ΩP |Q takes values in h.
As a consequence for a principal GL(k)-connection ω the curvature is trace-free, tr Ω = 0,
when the connection reduces to O(k). Namely any element of P may be expressed as u · a with
u ∈ Q and a ∈ GL(k). Then
tr Ω( ˜X(u · a), ˜Y (u · a)) = tr(a−1
Ω( ˜X(u), ˜Y (u))a) = tr Ω( ˜X(u), ˜Y (u)) = 0
since Ω( ˜X(u), ˜Y (u)) is an antisymmetric matrix and as such has zero trace.
CHAPTER 4
Riemannian geometry
1. Interpretation of Riemannian geometry
Let us start with a motivation which I presented in the tutorials. Let P → M be a principal
GL(k)-bundle, Q ⊆ P a reduction to O(k) and Γ a principal connection on P. The question arises
how to recognize whether Γ is associated to a principal connection on Q where we think of P as
Q ×O(k) GL(k) to make sense of this. In such a situation we say that the connection Γ reduces to
Q. In this special case a reduction to O(k) is the same as a choice of a scalar product
g : E ⊗ E → R
on the associated bundle E = P ×GL(k) Rk
.
Theorem 1.1. The following conditions are equivalent
(1) Γ reduces to Q,
(2) g = 0 (we say that g is covariant constant),
(3) the parallel transport on E preserves the scalar product.
Lemma 1.2. Let E1 and E2 be two vector bundles associated to P(M, G), s1 ∈ C∞
E1, s2 ∈
C∞
E2 two sections. Then
X(s1 ⊗ s2) = Xs1 ⊗ s2 + s1 ⊗ Xs2
Proof. Let si be associated to an equivariant map σi : P → Wi, i.e. si(π(u)) = [u, σi(u)].
We have an isomorphism
(P ×G W1) ⊗ (P ×G W2)
∼=
−−→ P ×G (W1 ⊗ W2)
[u, v1] ⊗ [u, v2] −→ [u, v1 ⊗ v2]
under which the section s1 ⊗ s2 becomes σ1 ⊗ σ2 : P → W1 ⊗ W2 since
(s1 ⊗ s2)(π(u)) = [u, σ1(u)] ⊗ [u, σ2(u)] = [u, (σ1 ⊗ σ2)(u)]
In our correspondence Xsi becomes dσi( ˜X) and thus X(s1 ⊗ s2) corresponds to
d(σ1 ⊗ σ2)( ˜X) = dσ1( ˜X) ⊗ σ2 + σ1 ⊗ dσ2( ˜X)
because the coordinates of σ1 ⊗ σ2 are products of coordinates of σ1 and of σ2. The right hand
side then corresponds to the formula from the statement.
Proof of the theorem. We start with “(1) ⇒ (2)”. The scalar product section g of (E ⊗
E)∗ ∼= Q ×O(k) (Rk
⊗ Rk
)∗
corresponds to an equivariant map γ : Q → (Rk
⊗ Rk
)∗
which we now
identify. Let u ∈ Qx be an orthonormal basis of Ex thought of as a map u : Rk
∼=
−→ Ex. Then
g(π(u)) = [u, γ(u)] : Ex ⊗ Ex
u−1
⊗u−1
−−−−−−−→ Rk
⊗ Rk γ(u)
−−−−→ R
Thus γ(u) is the scalar product g expressed in the orthonormal basis u, i.e. γ(u) is the standard
scalar product (independently of u) and γ is constant. Thus dγ = 0 and hence g = 0.
We now reformulate (2) slightly. Let ev : (Rk
⊗ Rk
)∗
⊗ Rk
⊗ Rk
−→ R be the evaluation map
h ⊗ v ⊗ w → (h(v ⊗ w). Then ev induces a (linear) morphism of induced vector bundles
P ×GL(k) ((Rk
⊗ Rk
)∗
⊗ Rk
⊗ Rk
) −→ P ×GL(k) R = M × R
60
1. INTERPRETATION OF RIEMANNIAN GEOMETRY 61
For sections g, s1 and s2 we therefore obtain
X(g(s1 ⊗ s2)) = X ev(g ⊗ s1 ⊗ s2) = ev X(g ⊗ s1 ⊗ s2)
= ev( Xg ⊗ s1 ⊗ s2 + g ⊗ Xs1 ⊗ s2 + g ⊗ s1 ⊗ Xs2)
= Xg(s1 ⊗ s2) + g( Xs1 ⊗ s2) + g(s1 ⊗ Xs2)
In other words
X s1, s2 = Xg(s1 ⊗ s2) + Xs1, s2 + s1, Xs2
Now we are ready to prove “(2) ⇒ (3)”. For vectors v1, v2 ∈ Ex and a path γ : R → M
through x let us denote the parallel transport of vi along γ by ˜γi. The definition gives ˙γ ˜γi = 0
and thus
d
dt ˜γ1, ˜γ2 = ˙γ ˜γ1, ˜γ2 = ˙γ ˜γ1, ˜γ2 + ˜γ1, ˙γ ˜γ2 = 0
The scalar product ˜γ1, ˜γ2 is therefore constant which is exactly what (3) asserts.
Finally we prove “(3) ⇒ (1)”. Let x ∈ M and represent X ∈ TxM as the velocity ˙γ(0) of a path
γ : R → M. Choose an orthonormal basis u = (u1, . . . , uk) of Ex. By (3) the parallel transports ˜γi
of ui along γ form an orthonormal basis at the points γ(t) on the path. The derivatives d
dt t=0
˜γi
are the horizontal lifts ˜X(ui) and they constitute a horizontal lift
˜X(u) = ( ˜X(u1), . . . , ˜X(uk)) = d
dt t=0
(˜γ1, . . . , ˜γk)
of X at u. Since the path takes place in Q, ˜X(u) ∈ TuQ and therefore the connection reduces to
Q.
A second exercise was to identify the curvature of the associated connection. The idea is that
this should be determined by the curvature of the principal connection which is equivalent to the
curvature form Ω. Thus one should be able to express the curvature CΓW translated from V E to
to the zero section E using Ω. More precisely tr CΓ is a bundle morphism
tr CΓW : E ×M Λ2
TM −→ W
and as such is induced by an equivariant map
P ×M Λ2
TM −→ map(W, W)
In fact we will see that the curvature is linear and map(W, W) can be replaced by hom(W, W) =
gl(W).
Theorem 1.3. Let P(M, G) be a principal bundle equipped with a principal connection ω,
ρ : G → GL(W) a linear representation, E = P ×G W the associated vector bundle. Then
the curvature tr CΓW (−, X, Y ) : E → E of the associated connection is induced by the map
P → gl(W), u → ρ∗Ω( ˜X, ˜Y )(u).
Proof. As we do not want to confuse the Lie bracket with points in E = P ×G W (i.e. classes
[u, v] of pairs (u, v) ∈ P × W) we will use the quotient map
q : P × W −→ P ×G W = E
to denote the latter. The horizontal lifts ˆX on E are given by
ˆX(q(u, v)) = q∗( ˜X(u), 0v)
using the horizontal lift ˜X for P. We need to compute their Lie bracket. For this we observe that
we have ( ˜X, 0) ∼q
ˆX and ( ˜Y , 0) ∼q
ˆY . Therefore
[( ˜X, 0), ( ˜Y , 0)] ∼q [ ˆX, ˆY ]
Since q is a submersion this determines [ ˆX, ˆY ]:
[ ˆX, ˆY ](q(u, v)) = q∗(u,v)[( ˜X, 0), ( ˜Y , 0)] = q∗(u,v)([ ˜X, ˜Y ], 0)
Subtracting from [X, Y ](q(u, v)) = q∗(u,v)([X, Y ], 0) we get
CΓW (q(u, v), X, Y ) = q∗(CΓ(u, X, Y ), 0v) = q∗(Ω( ˜X, ˜Y )∗
(u), 0v)
2. THE CURVATURES OF A RIEMANNIAN SPACE 62
The last step is to use A∗
(u) = d
dt t=0
u · exp(tA) to simplify to
d
dt t=0
q(u · exp(t · Ω( ˜X, ˜Y )(u)), v) = d
dt t=0
q(u, exp(t · Ω( ˜X, ˜Y )(u)) · v)
When composed with the translation this can be written as
q(u, d
dt t=0
exp(t · Ω( ˜X, ˜Y )(u)) · v)) = q(u, ρ∗(Ω( ˜X, ˜Y )(u)) · v)
The last exercise was to express the section X Y s − Y Xs − [X,Y ]s via an equivariant
map
Theorem 1.4. The following formula holds.
X Y s − Y Xs − [X,Y ]s = tr CΓ(s, X, Y )
Proof. If s corresponds to σ then Xs corresponds to ˜Xσ and the whole formula to
( ˜X ˜Y − ˜Y ˜X − [X, Y ])σ = ([ ˜X, ˜Y ] − [X, Y ])σ = −CΓW (−, X, Y )σ
Now we express the result using Ω to get
−CΓW (u, X, Y )σ = − Ω( ˜X, ˜Y )(u)
∗
σ(u) = Ω( ˜X, ˜Y )(u)(σ(u)) = ρ∗(Ω( ˜X, ˜Y )(u)) · σ(u)
By the previous theorem this corresponds to the section
tr CΓW (q(u, σ(u)), X, Y ) = tr CΓW (s, X, Y )
2. The curvatures of a Riemannian space
For a smooth manifold M a Riemannian structure is a section g : M → S2
+T∗
M of the bundle
of symmetric positive definite bilinear forms. We say that (M, g) is a Riemannian manifold (a
manifold M equipped with a Riemannian metric g).
Definition 2.1. A Levi-Civita connection on TM is characterized by its three properties
(1) it is linear,
(2) torsion-free, i.e. Dθ = dθ + ω · θ = 0,
(3) g = 0, i.e. comes from a connection on the subbundle Q1
M ⊆ P1
M of orthonormal
frames.
Let us consider the curvature of the Levi-Civita connection
R(X, Y )Z = X Y Z − Y XZ − [X,Y ]Z
a section R of TM ⊗ Λ2
T∗
M ⊗ T∗
M with factors corresponding to the value of R, the X and Y
entries, and the Z entry. We have shown that the equivariant map inducing R is
Q1
M −→ Rm
u −→ Ω( ˜X, ˜Y )(u) · θ ˜Z(u)
where θ ˜Z is the map corresponding to Z. The map corresponding to g is the constant map with
value the standard scalar product.
Theorem 2.2. For X, Y, Z, U ∈ XM the following holds
g(R(X, Y )Z, U) = −g(R(X, Y )U, Z)
Proof. By what we have said the left hand side corresponds to
Ω( ˜X, ˜Y ) · θ ˜Z, θ ˜U
As Ω takes values in the Lie algebra o(m) of all skew-symmetric matrices (anti-self-adjoint maps)
the result follows.
2. THE CURVATURES OF A RIEMANNIAN SPACE 63
The tensor field R of type (0, 4) sending
(X, Y, Z, U) −→ R(X, Y, Z, U) = −g(R(X, Y )Z, U)
is called the covariant form of the curvature tensor field R of type (1, 3). In coordinates
R = Rijkl · dxi
⊗ dxj
⊗ dxk
⊗ dxl
and we have so far proved
Rijkl = −Rjikl Rijkl = −Rijlk
Theorem 2.3 (The first Bianchi identity). Rijkl + Rjkil + Rkijl = 0.
Proof. Our previous (more general) first Bianchi identity claimed D2
θ = Ω · θ. Since in our
case Dθ = 0 we have
0 = (Ω · θ)( ˜X, ˜Y , ˜Z) = Ω( ˜X, ˜Y ) · θ ˜Z + Ω( ˜Y , ˜Z) · θ ˜X + Ω( ˜Z, ˜X) · θ ˜Y
Multiplying by θ ˜U and converting to the section form yields the result.
Theorem 2.4. As an algebraic consequence of the previous identities
Rijkl = Rklij
Proof. Consider two instances of the first Bianchi identity
Rijkl + Rjkil + Rkijl = 0
Rijlk + Rjlik + Rlijk = 0
Subtracting we obtain
2Rijkl + Rjkil + Rkijl − Rjlik − Rlijk = 0
Changing the indices according to i j k l
k l i j one gets
2Rklij + Rlikj + Riklj − Rljki − Rjkli = 0
and finally subtracting the last two equalities one obtains
2Rijkl − 2Rklij = 0
Definition 2.5. For a linear connection on M we define its Ricci tensor field of type (0, 2)
by the formula
Rij =
k
Rk
ijk
i.e. R(X, Y ) is the trace of R(−, X)Y : TM → TM.
Theorem 2.6. The Ricci tensor field of the Levi-Civita connection is symmetric
Proof. Let ui be an orthonormal basis of TxM. Then tr(R(−, X)Y ) equals
i
g(R(ui, X)Y, ui) = g(R(Y, ui)ui, X) = g(R(ui, Y )X, ui)
which is tr(R(−, Y )X).
Definition 2.7. The function s = i,j ˜gij
Rij is called the scalar curvature of the Riemannian
space.
s : M
Ricci
−−−−→ T∗
M ⊗ T∗
M
∼=
−−→ TM ⊗ T∗
M
eval
−−−→ R
Using an orthonormal frame ui we can write
s =
i,j
g(R(ui, uj)uj, ui) =
i,j
R(ui, uj, ui, uj)
2. THE CURVATURES OF A RIEMANNIAN SPACE 64
Observe that the covariant form of the curvature is a section
M → Λ2
T∗
M ⊗ Λ2
T∗
M
For a pair of vectors u, v consider R(u, v, u, v). Changing the basis to (u , v ) = (u, v)·A we obtain
u ∧ v = det A · u ∧ v and thus
R(u , v , u , v ) = (det A)2
· R(u, v, u, v)
Theorem 2.8. Let p ⊆ TxM be a two-dimensional linear subspace, u, v ∈ p a basis. Then the
number
K(p) =
R(u, v, u, v)
g(u, u)g(v, v) − g(u, v)2
does not depend on the choice of the basis u, v of p.
Definition 2.9. The number K(p) is called the sectional curvature of (M, g) in the direction
of the two-dimensional subspace p.
Proof. We know that
g(u, u)g(v, v) − g(u, v)2
=
g(u, u) g(u, v)
g(u, v) g(v, v)
equals the square of the volume of the parallelpiped determined by u, v. In particular by passage
to (u , v ) = (u, v) · A this expression gets multiplied by (det A)2
.
Remark. If u, v ∈ p is an orthonormal basis then K(p) = R(u, v, u, v) since the denominator
is 1.
Information 2.10. (without proof) Using geodesics to transport the disc D(r, p) cetred at
0 ∈ p and of radius r to M. We obtain a two-dimensional submanifold V (r, p) = exp(D(r, p)) ⊆ M.
It holds
K(p) = 12 lim
r→0
πr2
− vol V (r, p)
πr4
The normalization is chosen so that for the unit sphere K(p) = 1.
Definition 2.11. We say that a Riemannian space (M, g) has constant curvature if its sectional
curvature is the same at all points and in all directions.
Theorem 2.12 (Schur). Let (M, g) be a connected Riemannian space of dimension at least 3.
If K(p) depends only on the point x then M has a constant curvature.
Proof. Later.
Remark (The Cartan’s viewpoint). The curvature
Ω ∈ Ω2
hor(P, gl(m)) ∼= Ω2
(M, End(TM))
corresponds to an equivariant map
P → hom(Λ2
Rm
, gl(m))
with its image in fact lying in the GL(m)-submodule homB(Λ2
Rm
, o(m)) of tensors satisfying
the Bianchi identities. This submodule happens to decompose into three irreducible components
and thus the curvature decomposes correspondingly. The components are respectively the scalar
curvature, the traceless Ricci and Weyl curvature.
Theorem 2.13. Let (M, g) be a Riemannian manifold, Q1
M the principal O(m)-bundle of
orthonormal frames. Then on Q1
M there exists a unique torsion-free principal connection. It is
called the Levi-Civita connection.
3. NORMAL COORDINATES 65
Proof. We are searching for ω : T(Q1
M) → o(m), it is already determined on the vertical
subbundle V (Q1
M). To determine the horizontal distribution we need to solve
0 = Dθ = dθ + ω · θ
expressing that ω is torsion-free. As Dθ is horizontal independently of ω this condition is automatically
satisfied for vertical vectors. We use θ to make an identification of some complementary
subspace with Rm
, i.e. Hu(Q1
M) ∼= Rm
. In this way the above equation becomes
Λ2
Rm ∼=
←−− Λ2
Hu(Q1
M)
dθ+ω·θ
−−−−−−→ Rm
The mapping ω → dθ + ω · θ is affine with the associated linear map ω → ω · θ. We will now show
that it is bijective. Then there exists a unique ω for which ω · θ = −dθ verifying the theorem. At
each u ∈ Q1
M the map ωu → ωu · θu becomes under our identification the map
hom(Rm
, o(m)) −→ hom(Λ2
Rm
, Rm
)
α −→ (β : v ∧ w → α(v)w − α(w)v)
Since the vector spaces have the same dimension it is sufficient to prove injectivity. Denoting
α(ei)ej = ak
ijek we have the antisymmetry relation ak
ij = −aj
ik and the coordinates of the image
β are simply
bk
ij = β(ei ∧ ej)k
= (α(ei)ej)k
− (α(ej)ei)k
= ak
ij − ak
ji
Clearly the kernel consists precisely of those ak
ij symmetric in the lower indices. Thus ak
ij = ak
ji =
−ai
jk and repeating this cyclic permutation three times
ak
ij = −ai
jk = +aj
ki = −ak
ij
In the end ak
ij = 0 and the map is injective.
The equivariancy of ω follows by uniqueness from
0 = r∗
a(dθ + ω · θ) = d(r∗
aθ) + r∗
aω · r∗
aθ = a−1
dθ + r∗
aω · a−1
θ
(where we use equivariancy of θ) and
0 = a−1
(dθ + ω · θ) = a−1
dθ + Ad(a−1
)ω · a−1
θ
3. Normal coordinates
Let be a linear connection on M. A path γ : I → M, where I ⊆ R is an interval, is called
geodesic if the tangent vector field ˙γ along γ is parallel. We will now express this condition in
coordinates using the classical Christoffel symbols
∂xi ∂xj
= Γk
ij∂xk
If γ(t) is given in coordinates by functions xi
(t) then ˙γ is given by dxi
dt (t) and the differential
equation describing geodesic paths becomes
d2
xk
dt2 + Γk
ij
dxi
dt
dxj
dt = 0
We see immediately from the shape of the equation that the geodesic paths are preserved by affine
reparametrizations t = aτ + b, a = 0. For X ∈ TxM there exists a unique geodesic path γ with
˙γ(0) = X which we denote by γX.
Theorem 3.1. The rule expx(X) = γX(1) defines on some neighbourhood Ux of 0 ∈ TxM a
smooth map expx : Ux → M which is a local diffeomorphism at 0.
Proof. Let us denote the unit ball in TxM by Bx and observe that by compactness the
geodesic paths γX are defined on [−ε, ε] for X ∈ Bx. For X ∈ εBx they are defined on [−1, 1] by
affine reparametrization. Since
(expx)∗0(X) = d
dt t=0
expx(tX) = d
dt t=0
γX(t) = X
the derivative at 0 is (expx)∗0 = id.
3. NORMAL COORDINATES 66
Definition 3.2. This map is called the exponential map of the connection .
Remark.
• expx needs not be defined globally. For Rm
with the classical connection (the LeviCivita
connection of the standard metric) exp0 = idRm and thus also for any open
subset U ⊆ Rm
. In this case exp0 is not defined globally. For compact manifolds
expx : TM → M is always defined globally.
• expx needs not be a global diffeomorphism, e.g. for Sm
with the standard connection the
whole sphere of radius π centred at 0 ∈ TxSm
is mapped to the opposite point −x.
Definition 3.3. The local coordinate chart determined by expx for a linear torsion-free
connection on M is called the normal coordinate chart.
Theorem 3.4. In the normal coordinate chart at x it holds Γk
ij(x) = 0.
Proof. The geodesic going through x = 0 with speed X has a coordinate expression ai
t.
Then the differential equation for the geodesic becomes
0 = d2
xk
dt2 + Γk
ij(x)dxi
dt
dxj
dt = Γk
ij(tai
)ai
aj
For t = 0 we get Γk
ij(0)ai
aj
= 0 for arbitrary ai
. Since Γk
ij is symmetric in the lower indices
Γk
ij(0) prescribes a symmetric bilinear form with vanishing associated quadratic form. Hence it
must be zero itself.
Let us compute the coordinate expression of the curvature tensor
R(X, Y )Z = X Y Z − Y XZ − [X,Y ]Z
We first compute
∂xi ∂xj ∂xk
= ∂xi Γh
jk∂xh
=
∂Γl
jk
∂xi ∂xl
+ Γh
jkΓl
ih∂xl
which in the normal coordinates is simply
∂Γl
jk
∂xi ∂xl
. Thus
Rl
ijk =
∂Γl
jk
∂xi −
∂Γl
ik
∂xj
Theorem 3.5 (First Bianchi identity). For any torsion-free connection it holds Rl
ijk + Rl
jki +
Rl
kij = 0 or equivalently R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0.
Proof. We compute in the normal coordinates where
Rl
ijk =
∂Γl
jk
∂xi −
∂Γl
ik
∂xj
Rl
jki =
∂Γl
ki
∂xj −
∂Γl
ji
∂xk
Rl
kij =
∂Γl
ij
∂xk −
∂Γl
kj
∂xi
Adding up all terms cancel by symmetry.
We will now explain what is meant by XR. Here R is a tensor of type (1, 3), i.e. a section
of P1
M ×GL(M) hom(⊗3
Rm
, Rm
). Using the evaluation map
ev : hom(⊗3
Rm
, Rm
) ⊗ Rm
⊗ Rm
⊗ Rm
−→ Rm
the corresponding linear map on the associated bundles is
ev : hom(⊗3
TM, TM) ⊗ TM ⊗ TM ⊗ TM −→ TM
The covariant derivative commutes with linear maps thus
X(R(Y, Z)U) = ( XR)(Y, Z, U) + R( XY, Z)U + R(Y, XZ)U + R(Y, Z) XU
Theorem 3.6 (Second Bianchi identity). For every torsion-free linear connection it holds
( XR)(Y, Z, U) + ( Y R)(Z, X, U) + ( ZR)(X, Y, U) = 0.
3. NORMAL COORDINATES 67
Proof. In the normal coordinates we write the components of R as Rl
ijk;h. To compute
them we plug
X = ∂xh
Y = ∂xi
Z = ∂xj
U = ∂xk
into our formula for ( XR)(Y, Z, U). At the origin
XY (0) = Γl
hi(0)∂xl
= 0
and similarly for the other covariant derivatives. Thus
Rl
ijk;h(0) = ( XR)(Y, Z, U)(0) = X(R(Y, Z)U)(0) = ∂
∂xh
∂Γl
jk
∂xi −
∂Γl
ik
∂xj (0)
Summing with the cyclic permutations we obtain the result.
Remark. To relate this version of the second Bianchi identity to the more general one DΩ = 0
we will show that the tensor1
( XR)(Y, Z, U) + R(T(X, Y ), Z)U + cyclic permutations in X, Y, Z
corresponds to an equivariant map DΩ( ˜X, ˜Y , ˜Z) · θ ˜U. We have
R(Y, Z)U ∼ Ω( ˜Y , ˜Z) · θ ˜U
which when differentiated along ˜X yields
X(R(Y, Z)U) ∼ ˜XΩ( ˜Y , ˜Z) · θ ˜U + Ω( ˜Y , ˜Z) · ˜Xθ ˜U
R( XY, Z)U ∼ Ω( XY , ˜Z) · θ ˜U
R(Y, XZ)U ∼ Ω( ˜Y , XZ) · θ ˜U
R(Y, Z) XU ∼ Ω( ˜Y , ˜Z) · ˜Xθ ˜U
Subtracting one obtains
( XR)(Y, Z, U) ∼ ˜XΩ( ˜Y , ˜Z) · θ ˜U − Ω( XY , ˜Z) · θ ˜U − Ω( ˜Y , XZ) · θ ˜U
For torsion one has T(X, Y ) = XY − Y X − [X, Y ] so that
R(T(X, Y ), Z)U ∼ Ω( XY , ˜Z) · θ ˜U + Ω( ˜Z, Y X, ˜Z) · θ ˜U − Ω([ ˜X, ˜Y ]), ˜Z) · θ ˜U
Adding up all the cyclic permutations (with respect to X, Y, Z only) of the last two correspondences
one gets
( XR)(Y, Z, U) + R(T(X, Y ), Z)U ∼ ˜XΩ( ˜Y , ˜Z) − Ω([ ˜X, ˜Y ]), ˜Z) · θ ˜U
On the right it is easy to recognize the formula for DΩ( ˜X, ˜Y , ˜Z) · θ ˜U.
Remark. We have derived a general form of the second Bianchi identity
( XR)(Y, Z, U) + R(T(X, Y ), Z)U + cyclic permutations in X, Y, Z = 0
Analogously one can prove for a general connection that
R(X, Y )Z + cyclic = T(T(X, Y ), Z) + ( XT)(Y, Z) + cyclic
by showing that the right hand side corresponds to D2
θ = Ω · θ which we know that corresponds
to the left hand side.
Lemma 3.7. Let V be a finite dimensional vector space and R0, R1 : ⊗4
V → R to linear maps
satisfying
(a) Ri(x ⊗ y ⊗ z ⊗ u) = −Ri(y ⊗ x ⊗ z ⊗ u),
(b) Ri(x ⊗ y ⊗ z ⊗ u) = −Ri(x ⊗ y ⊗ u ⊗ z),
(c) Ri(x ⊗ y ⊗ z ⊗ u) + Ri(y ⊗ z ⊗ x ⊗ u) + Ri(z ⊗ x ⊗ y ⊗ u) = 0.
If R0(x ⊗ y ⊗ x ⊗ y) = R1(x ⊗ y ⊗ x ⊗ y) then R0 = R1.
1Thinking of R as a section of T∗M ⊗ Λ2T∗M ⊗ T∗M ⊗ TM the summation over cyclic permutations
corresponds to the antisymmetrization in the first three variables.
3. NORMAL COORDINATES 68
Proof. We set R = R1 − R0. Multiplying out 0 = R(x ⊗ (y + u) ⊗ z ⊗ (y + u)) we obtain
0 = R(x ⊗ y ⊗ x ⊗ y) + R(x ⊗ u ⊗ x ⊗ u) + R(x ⊗ y ⊗ x ⊗ u) + R(x ⊗ u ⊗ x ⊗ y)
where the first two terms are zero hence so must be the sum of the last two. According to
Theorem 2.4 this sum equals 2R(x ⊗ y ⊗ x ⊗ u). Thus
R((x + tz) ⊗ y ⊗ (x + tz) ⊗ u) = 0
and taking derivative we obtain that
d
dt t=0
R((x + tz) ⊗ y ⊗ (x + tz) ⊗ u) = R(x ⊗ y ⊗ z ⊗ u) + R(z ⊗ y ⊗ x ⊗ u) = 0
Thus R is fully anti-symmetric in the first three indices and thus
R(x ⊗ y ⊗ z ⊗ u) = R(y ⊗ z ⊗ x ⊗ u) = R(z ⊗ x ⊗ y ⊗ u)
We can now rewrite (c) as 3R(x ⊗ y ⊗ z ⊗ u) = 0 and R = 0, i.e. R0 = R1.
Theorem 3.8 (Schur). Let (M, g) be a connected Riemannian space of dimension at least 3
such that K(p) depends only on the point x for which p ⊆ TxM. Then M has constant curvature.
Proof. The tensor field R1(X, Y, Z, U) = g(X, Z)g(Y, U) − g(Y, Z)g(X, U) satisfies (a), (b)
and (c). At each point x ∈ M it holds
K(x) =
R(X, Y, X, Y )
R1(X, Y, X, Y )
where we denote by K(x) the common value of K(p) for all p ⊆ TxM. By the previous lemma
R = K(x)R1 since they agree on tensors of type X ⊗ Y ⊗ X ⊗ Y . We want to show that K(x) is
a constant function.
To determine in what sense is R1 constant we get back to the curvature tensor of type (1, 3).
R1(X, Y )Z = g(Y, Z)X − g(X, Z)Y ∼ θ ˜Y , θ ˜Z θ ˜X − θ ˜X, θ ˜Z θ ˜Y
This is summarized in the diagram
P1
M
θ ˜X⊗θ ˜Y ⊗θ ˜Z
GG Rm
⊗ Rm
⊗ Rm

x ⊗ y ⊗ z_

Rm
y, z x − x, z y
showing that the map form of R1 is the constant map
P1
M → hom(⊗3
Rm
, Rm
)
sending everything to the above x ⊗ y ⊗ z → y, z x − x, z y. In particular XR1 = 0 and thus
XR = X(K · R1) = XK · R1 + K XR1 = XK · R1
Now we use the second Bianchi identity
( XR)(Y, Z, U) + ( Y R)(Z, X, U) + ( ZR)(X, Y, U) = 0
which in our case takes form
XK · (g(Z, U)Y − g(Y, U)Z) + Y K · (g(X, U)Z − g(Z, U)X)
+ZK · (g(Y, U)X − g(X, U)Y ) = 0
Take an orthonormal system X, Y, Z = U and substitute to obtain
XK · Y − Y K · X = 0
Since X and Y are linearly independent XK = 0 = Y K. As they were also arbitrary the derivative
of K is zero and K is locally constant. By connectedness it is globally constant.
Theorem 3.9. For an arbitrary Riemannian space with constant curvature K
R(X, Y )Z = K(g(Y, Z)X − g(X, Z)Y )
Proof. See the proof of the last theorem.
4. THE SECOND FUNDAMENTAL FORM OF A HYPERSURFACE 69
4. The second fundamental form of a hypersurface
Theorem 4.1. Let (N, g) be a Riemannian manifold, M ⊆ N its submanifold, N
and M
the Levi-Civita connections on N and M. Then for all X ∈ TxM and Y ∈ XM and any extension
¯Y ∈ XN of Y the following holds
M
X Y ≡ N
X
¯Y mod νx
where νx is the orthogonal complement of TxM in TxN. In other words M
X Y is the orthogonal
projection of N
X
¯Y onto TxM.
Proof. Let us define by the formula from the statement, i.e. XY is the orthogonal
projection of N
X
¯Y onto TxM. We will show that is metric and torsion-free2
. This will imply
M
= by uniqueness.
X(g(Y, Z)) = X(g(Y, Z)) = X(g( ¯Y , ¯Z))
and the other terms of ( Xg)(Y, Z) are
−g( XY, Z) = −g( N
X
¯Y , ¯Z)
since the difference of XY and N
X ¯Y is orthogonal to ¯Z. Similarly
−g(Y, XZ) = −g(Y, N
X
¯Z)
and adding these three equalities we obtain
( Xg)(Y, Z) = ( N
Xg)( ¯Y , ¯Z) = 0
so that is metric.
Projecting the equality N
¯X
¯Y − N
¯Y
¯X = [ ¯X, ¯Y ] onto TxM we obtain
XY − Y X = [X, Y ]
since [ ¯X, ¯Y ] is already tangent to M: in effect X is ι-related to ¯X (for ι : M → N the inclusion),
analogously for Y and thus the same holds for their bracket.
The normal projection of N
X
¯Y only depends on the value of X and Y at x. Before we start
with the verification let us denote the normal projection by π and B(X, Y ) = π N
X
¯Y . Then we
compute
B(X, fY ) = π N
X( ¯f ¯Y ) = π(X ¯f · ¯Y
tangent
+ ¯f · N
X
¯Y ) = f(x) · π( N
X
¯Y ) = f(x)B(X, Y )
Thus B is tensorial. Moreover it is symmetric as
B(X, Y ) − B(Y, X) = π([ ¯X, ¯Y ]) = π[X, Y ] = 0
so that B : S2
TM → νM with the target being the normal bundle of M.
Definition 4.2. The tensor B is called the second fundamental form of the submanifold
M ⊆ N.
Let us consider a special case M ⊆ Em+1, a hypersurface in a Euclidean space with its
standard metric and orientation. An orientation of a manifold M is a continuous choice of an
orientation of TxM for all x ∈ M. The normal bundle ν is one-dimensional and the orientation
of M determines an orientation of ν by declaring u ∈ νx positive iff (u, e1, . . . , em) is positive in
Em+1 with (e1, . . . , em) positive in TxM. The unique unit positive vector nx thus provides a global
trivialization of ν by
M × R −→ ν
(x, t) −→ t · nx
Let us denote by D the covariant derivative on Em+1 and the covariant derivative on M. Then
B(X, X)x = DXX, nx · nx
2Verifying that it is a linear connection is easy.
4. THE SECOND FUNDAMENTAL FORM OF A HYPERSURFACE 70
where X ∈ TxM. We represent X by a path γ : R → M with ˙γ(0) = X and extend X to a vector
field in such a way that X(γ(t)) = ˙γ(t). Then
DXX = d
dt t=0
X(γ(t)) = d
dt t=0
˙γ(t) = ¨γ(0)
and consequently B(X, X)x = ¨γ(0), nx · nx.
Theorem 4.3. The normal acceleration ¨γ(0), nx · nx depends only on ˙γ(0) = X and equals
B(X, X).
Definition 4.4. In the case of an oriented hypersurface M in Em+1 by the second fundamental
form we understand the map
h : S2
TM → R h(X, Y ) = B(X, Y ), nx
or in other words B(X, Y ) = h(X, Y )nx.
In a local coordinate chart f(u1, . . . , um) : Rm
→ M the basis of TxM is formed by fi = ∂f
∂ui
.
We denote fij = ∂2
f
∂ui∂uj
. A path γ(t) on M ha a coordinate expression ui = ui(t), we obtain
˙γ(t) ∈ Tγ(t) the velocity, ¨γ = d2
γ
dt2 the acceleration and d2
γ
dt2 , nx the normal acceleration. Since
γ(t) = f(u(t)) we may write
dγ
dt = fi(u(t))dui
dt
and thus
¨γ(t) = (fij u(t)) · dui
dt
duj
dt + fi(u(t))d2
ui
dt2
tangent to M
so that
¨γ, nx = fij(u(t)), nx
hij
·dui
dt
duj
dt
Remark. The first fundamental form is gij = fi, fj or more geometrically the scalar product
g on M.
Let X, Y be vector fields on M ⊆ Em+1 and ¯X, ¯Y their extensions to vector fields on Em+1.
Then D ¯X
¯Y = XY + h(X, Y ) · n where n is the (choice of a) unit normal vector field on M. The
curvature of Em+1 is zero while the curvature for the hypersurface M is
g(R(X, Y )Z, U) = g( X Y Z − Y XZ − [X,Y ]Z, U).
Theorem 4.5 (Gauss formula). For a hypersurface M ⊆ Em+1 it holds
g(R(X, Y )Z, U) = h(Y, Z)h(X, U) − h(X, Z)h(Y, U)
Proof. By the metricity of the connection
g( X Y Z, U) = Xg( Y Z, U) − g( Y Z, XU)
= ¯X D¯Y
¯Z, ¯U − D¯Y
¯Z, D ¯X
¯U + h(Y, Z)h(X, U)
and similarly
g( [X,Y ]Z, U) = D[X,Y ]
¯Z, ¯U = D[ ¯X, ¯Y ]
¯Z, ¯U .
Therefore g(R(X, Y )Z, U) equals
R( ¯X, ¯Y ) ¯Z, ¯U + h(Y, Z)h(X, U) − h(X, Z)h(Y, U)
with the first term zero since the curvature of Em+1 vanishes.
The sectional curvature in the direction of a plane p ⊆ TxM spanned by v1 and v2 is defined
by
K(p) =
−g(R(v1, v2)v1, v2)
g(v1, v1)g(v2, v2) − g(v1, v2)2
5. THE GEODESIC CURVES OF A RIEMANNIAN SPACE 71
Consider now a surface M ⊆ E3 with a local parametrization f(u1, u2) : R2
→ M and compute
the sectional curvature K(x) = K(TxM) by substituting ∂
∂u1
, ∂
∂u2
into the Gauss formula
K(x) =
h( ∂
∂u1
, ∂
∂u1
)h( ∂
∂u2
, ∂
∂u2
) − h( ∂
∂u1
, ∂
∂u2
)2
g( ∂
∂u1
, ∂
∂u1
)g( ∂
∂u2
, ∂
∂u2
) − g( ∂
∂u1
, ∂
∂u2
)2
=
h11h22 − h2
12
g11g22 − g2
12
=
det h
det g
This is the classical Gauss curvature from the differential geometry of curves and surfaces.
Corollary 4.6 (Theorema Egregium). The Gauss curvature belongs to the inner geometry
of a surface, i.e. it does not depend on the isometric embedding M → E3.
Remark. The Gauss curvature is a product of the curvatures in the principal directions - the
eigenvectors of h.
5. The geodesic curves of a Riemannian space
Definition 5.1. Let f : N → M be a smooth map. A vector field along f is a smooth map
F for which the diagram
TM

N
F
aa
f
GG M
commutes. In other words it is a section of the pullback bundle f∗
TM → N.
For a linear connection on M the induced connection on f∗
TM will be also denoted by .
Then F : TN → f∗
TM and for a vector field X ∈ XN, XF : N → f∗
TM, i.e. XF : N →
TM is again a vector field along f.
TN
F∗
GG
F
AA
TTM
v GG V TM

∼= TM ×M TM
pr2
ww
N
X
yy
X F
GG TM
Definition 5.2. Let γ : R → M be a path and v : R → TM a vector field along γ. It is said
to be transported parallelly along γ if v = 0 or equivalently
˙γv := d
dt
v = 0
Definition 5.3. A path γ(t) is geodesic if ˙γ(t) is parallel along γ.
Remark (on Cartan’s point of view). Consider the principal bundle P1
M → M of frames on
M. There are two forms on P1
M, the connection form ω and the canonical form θ. Combined
together they provide
(ω, θ) : TP1
M → ga(m) = Rm
gl(m)
a trivialization of the tangent bundle TP1
M. Here ga(m) is the Lie algebra of matrices of the
form
0 0
v A
,
i.e. the Lie algebra of the Lie group GA(m) of all affine isomorphisms of Rm
. This is an example
of a Cartan connection of type (ga(m), gl(m)). Taking v ∈ Rm
and thinking of it as the matrix
( 0 0
v 0 ) in ga(m) we obtain a vector field (ω, θ)−1
(v) on P1
M, horizontal by definition. Let ˜γ(t) be
its integral curve and γ(t) = π ◦ ˜γ(t). Then d
dt ˜γ is a horizontal vector field along γ and hence is
transported parallelly.
We will now show that γ is a geodesic. By definition θ(˙˜γ) = v, the coordinates of the projection
π∗
˙˜γ(t) = ˙γ(t) in the basis ˜γ(t) = (ui(t)). Since ui(t) are parallel along γ(t) so is their constant
linear combination ˙γ(t) and thus ˙γ ˙γ = 0.
5. THE GEODESIC CURVES OF A RIEMANNIAN SPACE 72
Now we draw some consequences of the geodesicity of γ. Firstly
d
dt |˙γ(t)|2
= d
dt g(˙γ(t), ˙γ(t)) = 2g( ˙γ ˙γ(t), ˙γ(t)) = 0
implying that |˙γ(t)| is constant. By a reparametrization we may assume that |˙γ(t)| = 1. In this
case we say that γ is parametrized by the arc length and use s for the parameter instead of t.
Let now C be a curve, i.e. a 1-dimensional submanifold. Locally we parametrize C by the arc
length as γ : R → C. The geodesic curvature of C is defined as
Kg(C) = | ˙γ ˙γ|.
Definition 5.4. A curve C is called a geodesic if its parametrization by the arc length is a
geodesic curve, i.e. Kg(C) = 0.
Remark (the Frenet’s formulas). For a planar curve we define e1 = ˙γ(s) the tangent unit
vector field along C and e2 (a choice of) the unit normal vector field. Then ˙γ ˙γ = ±Kg · e2 since
g( ˙γ ˙γ, ˙γ) = d
ds g(˙γ, ˙γ) = d
ds 1 = 0
and thus ˙γ ˙γ is a vector field perpendicular to ˙γ and of length Kg.
For a connected Riemannian manifold (M, g) we define
d(x, y) = inf{ (γ) | γ : [0, 1] → M, γ(0) = x, γ(1) = y}
where (γ) =
b
a
|˙γ(t)| dt is the length of a (piecewise) smooth curve γ. Easily d(x, y) ≥ 0 and
d(x, z) ≤ d(x, y) + d(y, z) (when considering smooth curves only one needs to use smoothing of
the concatenation).
Let us choose x ∈ M and using the scalar product gx on TxM we denote by N(x, r) the open
ball centred at 0x of radius r. For small r the exponential map expx is defined on N(x, r) and is
a diffeomorphism onto U(x, r) ⊆ M.
Theorem 5.5. For any r > 0 for which expx : N(x, r) → U(x, r) is a diffeomorphism the
following holds
(a) Every point y ∈ U(x, r) may be joined with x by a unique geodesic inside U(x, r).
(b) The length of the geodesic from (a) is exactly d(x, y).
(c) U(x, r) is the set of all y ∈ M for which d(x, y) < r.
Remark. It follows that d(x, y) = 0 iff x = y and d is a metric on M, U(x, r) being the ball
in this metric.
Proof. Firstly (a) follows from the fact that geodesics emanating from x are exactly the
images under expx of the rays from 0x. For (b) we will need the following lemma in which we
denote by g0
the Riemannian metric on TxM given by the scalar product gx at each v ∈ TxM.
Lemma 5.6 (Gauss lemma). Let v ∈ TxM lie in the domain of expx. Then for arbitrary
w ∈ TxM
g0
((v, v), (v, w)) = g(expx∗(v, v), expx∗(v, w))
i.e. expx∗ preserves the scalar product whenever one of the vectors is radial.
We will prove the lemma later. Let us denote by pr : TTxM → TTxM the radial projection,
pr(v, w) = v,
v, w
v, v
· v .
Let γ : [0, 1] → N(x, r) be a path and δ = expx ·γ its image in M. The length is
(δ) =
1
0
| ˙δ| dt
Decomposing ˙γ(t) into the radial part and the complement the orthogonality is preserved by expx∗
by Gauss lemma. In particular
| ˙δ(t)|2
= | expx∗ ˙γ(t)|2
= | expx∗ pr ˙γ(t)|2
+ | expx∗(˙γ(t) − pr ˙γ(t))|2
≥ | expx∗ pr ˙γ(t)|2
= | pr ˙γ(t)|2
5. THE GEODESIC CURVES OF A RIEMANNIAN SPACE 73
with equality only for ˙γ(t) radial. Therefore
(δ) ≥
1
0
| pr ˙γ(t)| dt ≥
1
0
| pr ˙γ(t)|or dt
where we write | pr ˙γ(t)|or for the oriented length (the sign being that of w/v)
|(v, w)|or = | pr(v, w)|or = dn(v, w)
where n : N(x, r) − {0x} → R+ is the norm | · |. Thus
(δ) ≥
1
0
dn(˙γ(t)) dt = |n(γ(1)) − n(γ(0))| = |γ(1)|
The equality occurs iff γ is radial and positively oriented hence a reparametrization of a linear
path in N(x, r). The path δ is then a reparametrization of a geodesic taking care of paths staying
inside U(x, r). But if δ left U(x, r) then its beginning would be a path from x to a point z of the
same geodesic distance from x as that of y. The length of this part of δ would then be at least
this geodesic distance proving (b). The very same argument proves (c).
Definition 5.7. A space with a linear connection, i.e. a manifold M togetherwith a linear
connection on TM, is called complete if every geodesic path γ : I → M extends to the whole R.
Remark. Equivalently the vector fields (ω, θ)−1
(v) are complete.
Theorem 5.8. If (M, g) is complete as a metric space then it is complete with respect to the
Levi-Civita connection.
Proof. Let γ : (a, b) → M be a geodesic path parametrized by the arc length and let bn be
a sequence in (a, b) converging to b. By the previous theorem d(γ(bn), γ(bm)) ≤ |bn − bm| and
thus γ(bn) is Cauchy. Let x ∈ M be its limit point. In a neighbourhood of x every geodesic
parametrized by the arc length is defined on an interval of a uniform radius by compactness. Thus
γ can be prolonged.
We will later prove the reverse implication.
Let M be an oriented 2-dimensional Riemannian manifold. The sectional curvature is a
function K : M → R, K(x) = K(TxM). Further there is a volume 2-form volg = e∗
1 ∧ e∗
2 where
e∗
1, e∗
2 is an oriented orthonormal basis of T∗
M.
Definition 5.9. The 2-form κ = K · volg is called the curvature 2-form on M.
Consider on M a oneparameter family of curves γ : I × J → U ⊆ M for which
• γ is a diffeomorphism I × J
∼=
−→ U,
• for each s ∈ J the curve γ(−, s) is parametrized by the arc length, | ∂
∂t γ(−, s)| = 1.
Let us denote ˙γ(t, s) = ∂
∂t γ(t, s), a vector field on U. Then g( ˙γ ˙γ, ˙γ) = 0. We denote by ν
the unit vector field orthogonal to ˙γ, namely that for which (˙γ, ν) is a positive basis. On U define
a 1-form ω = g( ˙γ, ν), i.e. ω(X) = g( X ˙γ, ν).
Lemma 5.10. dω = −κ.
Proof. It is enough to verify on the basis, dω(˙γ, ν) = −κ(˙γ, ν). To determine the right hand
side volg(˙γ, ν) = 1 and
K = R(˙γ, ν, ˙γ, ν) = −g(R(˙γ, ν)˙γ, ν).
Putting together −κ(˙γ, ν) = g(R(˙γ, ν)˙γ, ν) while dω(˙γ, ν) is
˙γω(ν) − νω(˙γ) − ω[˙γ, ν]
= ˙γg( ν ˙γ, ν) − νg( ˙γ ˙γ, ν) − g( [ ˙γ,ν] ˙γ, ν)
= g( ˙γ ν ˙γ, ν) − g( ν ˙γ ˙γ, ν) − g( [ ˙γ,ν] ˙γ, ν) + g( ν ˙γ, ˙γν) − g( ˙γ ˙γ, νν)
= g(R(˙γ, ν)˙γ, ν) + g( ν ˙γ, ˙γν) − g( ˙γ ˙γ, νν)
5. THE GEODESIC CURVES OF A RIEMANNIAN SPACE 74
and the last two terms are zero by the following argument. Since g(˙γ, ˙γ) = 1 the derivative X ˙γ
is orthogonal to ˙γ and thus X ˙γ ν. Similarly Y ν ˙γ and so
g( X ˙γ, Y ν) = 0
for arbitrary vectors X, Y .
Let us denote by B(r) the open disc in R2
of radius r and by S(r) the circle of radius r both
centred at the origin.
Definition 5.11. We say that a curve C is simple closed if there exists a diffeomorphism
ϕ : B(1 + ε) → U ⊆ M onto a neighbourhood U of C such that ϕ(S(1)) = C. The set ϕ(B(1)) is
called the interior of the curve C.
Notation. For a curve C we have the curve integral C
f ds and for a 2-dimensional region
D we have D
f dσ both defined by multiplying a function f by the respective volume form
associated to the induced metric.
The oriented geodesic curvature is Kg = g( ˙γ ˙γ, ν). This depends on the choice on ν which
we make in such a way that (˙γ, ν) is positively oriented.
Theorem 5.12 (Gauss-Bonet). Let C be a simple closed curve with the oriented geodesic
curvature Kg and let D be its interior. Then
C
Kg ds = 2π −
D
K dσ
Proof. Let us choose ϕ : B(1 + δ)
∼=
−→ U with ϕ(S(1)) = C and ϕ(B(1)) = D. We may
assume3
that in a small neighbourhood of the origin ϕ = expϕ(0). Around the origin we consider
a small circle Cε and on the annulus Dε we construct the 1-form ω corresponding to the (local)
parametrization of C by the arc length
Dε = S1
× [ε, 1] → U
By the Stokes theorem
C
ω −
Cε
ω =
Dε
dω = −
Dε
κ = −
Dε
K dσ
and also
C
ω =
S1
ω(˙γ) ds =
S1
g( ˙γ ˙γ, ν) ds =
C
Kg ds
Clearly limε→0 Dε
K dσ = D
K dσ and thus it remains to show that
lim
ε→0 Cε
Kg(Cε) ds = 2π
The rough idea is that in the Euclidean plane Kg(Cε) = 1/ε and thus
Cε
Kg(Cε) ds =
2π·ε
0
1/ε dt = 2π.
As ε → 0 the geometry approaches the Euclidean geometry and thus the limit formula holds. Now
for a more precise proof.
First we need a lemma about describing the geodesic curvature when the parametrization is
not by the arc length.
Lemma 5.13. Let γ : S1
→ M be an embedding. Then
Kg ◦ γ = g( ˙γ ˙γ, ν)/|˙γ|2
3This is the classical disc isotopy theorem which we probably want to avoid.
5. THE GEODESIC CURVES OF A RIEMANNIAN SPACE 75
Proof. By definition
Kg ◦ γ = g( ˙γ/| ˙γ|(˙γ/|˙γ|), ν) = g( ˙γ(˙γ/|˙γ|), ν)/|˙γ|
= g(1/|˙γ| · ˙γ ˙γ + d
dt (1/|˙γ|) · ˙γ, ν)/|˙γ|
and the proof is finished by observing that g(˙γ, ν) = 0.
Then we can compute Cε
Kg(Cε) ds in the coordinate chart given by ϕ and using the
parametrization γε : S1
→ R2
, γ(z) = ε · z
Cε
Kg(Cε) ds =
S1
g( ˙γε
˙γε, ν)/|˙γε|2
· |˙γε| ds
=
S1
gij · ¨γi
ε/|˙γε| · νj
ds +
S1
gijΓi
kl ˙γk
ε ˙γl
ενj
/|˙γε| ds
Easily the second term tends to zero while the first tends to the situation where4
gij = δij is
constant and thus the integrand tends to 1, the limit being 2π.
We will now interpret geometrically C
Kg ds. Let γ : [a, b] → M be a path parametrized by
the arc length and (u(t), v(t)) be a positive orthonormal basis at γ(t) obtained by transporting
u(a) and v(a) parallelly along γ(t). Express ˙γ(t) in this basis as
˙γ(t) = cos ϕ(t) · u + sin ϕ(t) · v
Then ν(t) = − sin ϕ(t) · u + cos ϕ(t) · v and we may compute
˙γ ˙γ = ˙γ(cos ϕ(t) · u) + ˙γ(sin ϕ(t) · v)
= d
dt (cos ϕ(t)) · u + cos ϕ(t) · ˙γu
0
+ d
dt (sin ϕ(t)) · v + sin ϕ(t) · ˙γv
0
= ˙ϕ(t) · (− sin ϕ(t) · u + cos ϕ(t) · v) = ˙ϕ(t) · ν
Therefore Kg = g( ˙γ ˙γ, ν) = ˙ϕ and finally
C
Kg ds =
C
˙γ dt = ϕ(1) − ϕ(0) = ∠(˙γ(a), ˙γ(b))
measured by transporting parallelly to any point along γ.
Let us consider now a curved triangle. We can use Gauss-Bonet formula after smoothing the
corners to obtain
C1
Kg ds + (π − α3) +
C2
Kg ds + (π − α1) +
C3
Kg ds + (π − α2) = 2π −
D
K dσ
the terms π − αi being exactly the angle differences (in limit). We obtain
Theorem 5.14. ∂∆
Kg ds = (α1 + α2 + α3 − π) − ∆
K dσ.
When all the sides Ci of the triangle are geodesic then Kg = 0 and we obtain
Theorem 5.15. The sum of the internal angles in a geodesic triangle is
α1 + α2 + α3 = π +
∆
K dσ.
When the curvature is constant the defect (α1 +α2 +α3 −π) is proportional to the area of the
triangle. For the Euclidean geometry K = 0 and α1 + α2 + α3 = π. For K = 1 we have triangles
with defect up to 4π.
Lemma 5.16. Let γ : [a, b] → M be a piecewise smooth path such that γ = d(γ(a), γ(b)).
Then γ is a reparametrization of a geodesic path.
Proof. We have proved this when γ(a) is sufficiently close to γ(b). For an arbitrary γ the
statement holds locally. But geodesics are described locally thus γ must be itself a reparametrization
of a geodesic.
4Thus it is convenient to assume that the derivative ϕ∗0 at zero is an isometry.
6. GEODESIC VARIATIONS 76
Theorem 5.17 (Hopf-Rinow). Let (M, g) be a connected geodesically complete Riemannian
space. Then arbitrary x, y ∈ M can be joined by a geodesic path γ satisfying (γ) = d(x, y). Such
paths are called minimal geodesics.
Proof. Let us define the “shell” Sh(x, r) = expx(S(x, r)) where S(x, r) is a sphere in TxM
cetred at 0x and of radius r. We choose r small enough so that expx is a diffeomorphism on the
closed ball of radius r. Since Sh(x, r) is compact there exists p ∈ Sh(x, r) such that d(p, y) is
minimal. Then p = expx(r · v) with |v| = 1. We will show that y = expx(d · v) where d = d(x, y).
This will prove the theorem. But first observe that d(p, y) equals exactly d(x, y) − r for it cannot
be smaller as that would give
d(x, y) ≤ d(x, p) + d(p, y) < r + (d(x, y) − r)
and it cannot be bigger either as that would contradict the minimality of d(x, p).
Now we will prove that the set
T = {t0 ∈ [0, d] | ∀0 ≤ t ≤ t0 : d(expx(t · v), y) = d − t}
equals [0, d]. Clearly T is closed in [0, d] and contains 0. It remains to show that it is open by
connectedness. Therefore let t0 ∈ T, p0 = expx(t0 · v) and again let p1 be the closest to y of
the points from Sh(p0, r0). We have shown in the first paragraph that d(p1, y) = d(p0, y) − r0 =
d − t0 − r0 and thus the concatenation of the geodesic from x to p0 and that from p0 to p1 is a
path having the minimal length t0 + r0 = (x, p1). By the previous lemma it must be a geodesic
and in particular p1 = expx((t0 + r0) · v). Since r0 was arbitrary (small) t0 + r0 ∈ T.
Remark. For a simply connected geodesically complete Riemannian space of non-positive
sectional curvature the minimal geodesic is unique and the exponential map expx : TxM → M is
a diffeomorphism.
Corollary 5.18. A geodesically complete Riemannian space is complete as a metric space.
Proof. Pick a point x ∈ M and let xn be a Cauchy sequence. The set d(x, xn) is necessarily
bounded by some r and hence xn lie in a compact subspace expx(B(x, r)) which implies the
convergence.
6. Geodesic variations
Let F be a vector field along f as in
f∗
TM
φ
GG

TM

N
f
GG
F
XX
M
and write F for the covariant derivative using the induced connection f∗
. We will now compute
the torsion T(f∗A, f∗B) in terms of the covariant derivative on f∗
TM. In terms of the equivariant
maps we have
Dθ(f∗A, f∗B) = dθ(f∗A, f∗B) = dθ(φ∗
˜A, φ∗
˜B) = d(φ∗
θ)( ˜A, ˜B)
= ˜A(φ∗
θ( ˜B)) − ˜B(φ∗
θ( ˜A)) − φ∗
θ[ ˜A, ˜B]
= ˜A(θ(φ∗
˜B)) − ˜B(θ(φ∗
˜A)) − φ∗
θ[A, B]
= ˜A(θ(f∗B)) − ˜B(θ(f∗A)) − θ(f∗[A, B])
which corresponds back to Af∗B − Bf∗A − f∗[A, B]. We conclude that
0 = T(f∗A, f∗B) = Af∗B − Bf∗A − f∗[A, B].
Analogously we obtain
R(f∗A, f∗B)F = A BF − B AF − [A,B]F
6. GEODESIC VARIATIONS 77
Definition 6.1. Consider a path γ : [a, b] → M and let I ⊆ R be an open interval containing
zero. By a variation of γ we understand a smooth map V : [a, b]×I → M satisfying V (t, 0) = γ(t).
Definition 6.2. A geodesic variation of a geodesic path γ is a variation V such that V (−, s)
is geodesic for each s ∈ I.
On [a, b] we use parameter t and on I parameter s. On the product [a, b] × I we have vector
fields ∂
∂t , ∂
∂s . We denote
V∗
∂
∂t = ∂tV V∗
∂
∂s = ∂sV
For a vector field F : [a, b] × I → RM along V we denote
∂
∂t
F = DtF ∂
∂s
F = DsF
Our formula for torsion for vactor fields ∂
∂t
∂
∂s can be written as
Dt∂sV − Ds∂tV = 0
since [ ∂
∂t , ∂
∂s ] = 0. For a geodesic variation we compute
D2
t ∂sV = DtDt∂sV = DtDs∂tV = DsDt∂tV + R(∂tV, ∂sV )∂tV
Writing ˙γt = ∂tV we see that Dt∂tV = ˙γt
˙γt = 0 and finally
D2
t ∂sV = R(˙γt, ∂sV )˙γt.
Definition 6.3. A vector field X along a geodesic path γ is called a Jacobi field if 2
˙γX =
R(˙γ, X)˙γ.
The condition on a Jacobi field is a second order linear differential equation. Thus a solution is
determined uniquely by X(a) and ˙γX(a). We have shown above that for every geodesic variation
V of γ the vector field ∂sV (t, 0) is a Jacobi field. In the opposite direction we have.
Theorem 6.4. For every Jacobi field X along γ there exists a geodesic variation V of γ such
that ∂sV (t, 0) = X(t).
Proof. We assume a = 0 for simplicity. Let β : I → M be any path with ˙β(0) = X(0). Put
γ(s) = Ptβ(˙γ(0) + s · ( ˙γX)(0), s)
and V (t, s) = expβ(s)(t · γ(s)). Since V is a geodesic variation of γ the derivative ∂sV (t, 0) is
a Jacobi field along γ and we will now show that it equals X(t). But the initial conditions for
∂sV (t, 0) are
∂sV (0, 0) = ∂
∂s s=0
β(s) = X(0)
(Dt∂sV )(0, 0) = (Ds∂tV )(0, 0) = (Dsγ)(0)
= ( ˙β Ptβ(˙γ(0), s))
0
(0) + ˙β(s · Ptβ(( ˙γX)(0), s))(0) = ( ˙γX)(0)
i.e. the same as that for X and thus the vector fields must also agree.
Example 6.5. Let γ : [a, b] → M be a geodesic path. Then both
γ(t + s) and γ((1 + s)t)
are geodesic variations (for each s they are affine reparametrizations of γ). The corresponding
Jacobi fields are
∂sγ(t + s)|s=0 = ˙γ(t)
∂sγ((1 + s)t)|s=0 = t · ˙γ(t) = ˆγ(t).
Lemma 6.6. For each Jacobi field X along γ it holds
d2
dt2 g(X, ˙γ) = 0.
6. GEODESIC VARIATIONS 78
Proof. We compute
d2
dt2 g(X, ˙γ) = d
dt (g( ˙γX, ˙γ) + g(X, ˙γ ˙γ
0
)) = g( 2
˙γX, ˙γ) + g( ˙γX, ˙γ ˙γ
0
)
= g(R(˙γ, X)˙γ, ˙γ) = 0
since the curvature tensor is antisymmetric in its last two variables.
From the lemma it follows that g(X, ˙γ) = α + βt. Assuming for simplicity that |˙γ| = 1 we
have g(ˆγ, ˙γ) = t. Therefore
g(X − α ˙γ − βˆγ, ˙γ) = 0
We have proved
Theorem 6.7. Every Jacobi field X along a geodesic γ can be uniquely decomposed as
X = α ˙γ + βˆγ + Y
where Y is a Jacobi field perpendicular to ˙γ.
We are now in the position to prove Gauss lemma asserting that
g(expx∗(v, v), expx∗(v, w)) = g0
(v, w)
for all v, w ∈ TxM.
Proof. Consider the geodesic variation expx(t(v + sw)) and its Jacobi field
X(t) = ∂s expx(t(v + sw))|s=0 = expx∗(tv, tw)
With γ(t) = expx(tv) the last lemma says that
g(X(t), ˙γ(t)) = g(expx∗(tv, tw), expx∗(tv, v)) = t · g(expx∗(tv, w), expx∗(tv, v))
should be linear in t. Therefore g(expx∗(tv, w), expx∗(tv, v)) must be constant and
g(expx∗(v, w), expx∗(v, v)) = g((0, w), (0, v)) = g0
(w, v)
Remark. The above Jacobi field is the only one with X(0) = 0.
Definition 6.8. We say that two points γ(α), γ(β) are conjugate if there exists a nonzero
Jacobi filed X satisfying X(α) = 0 = X(β).
Definition 6.9. For x ∈ M consider expx : Ux → M. A point y ∈ Ux (i.e. a small vector in
TxM) is said to be conjugate to x if the rank of expx∗ at y is less than dim M.
Theorem 6.10. A point y ∈ Ux is conjugate to x if and only if x = expx 0 and z = expx y
are conjugate points of the geodesic expx(ty), t ∈ [0, 1].
Proof. For the implication “⇒” let w ∈ ker expx∗y. Then the Jacobi field expx∗(ty, tw) of
the geodesic variation expx t(y + sw) has zeroes for t = 0, 1.
For the reverse implication let X be a nonzero Jacobi field along expx ty satisfying X(0) =
0 = X(1). There exists a geodesic variation of the form expx(t · y(s)), with y(0) = y, generating
X. Then
X(t) = ∂
∂s s=0
expx(t · y(s)) = expx∗(ty, t ˙y(0))
and 0 = X(1) = expx∗(y, ˙y(0)). Moreover ˙y(0) = 0 as that would imply X ≡ 0.
Theorem 6.11. If −g(R(˙γ, Y )˙γ, Y ) ≤ 0 for any vector field Y along γ then no points of γ
are conjugate. In particular if K(p) ≤ 0 then expx is a local diffeomorphism (on its domain).
6. GEODESIC VARIATIONS 79
Proof. We start with a computation
d
dt g( ˙γX, X) = g( ˙γX, ˙γX) + g( 2
˙γX, X) = | ˙γX|2
+ g(R(˙γ, X)˙γ, X) ≥ 0
Integrating from a to b we obtain
g( ˙γX(b), X(b)) − g( ˙γX(a), X(a)) ≥ 0
and the equality can occur only for a parallel vector field. But if X(a) = 0 = X(b) then both
terms are zero and thus necessarily X ≡ 0.
Theorem 6.12. If M is a connected complete Riemannian space with non-positive sectional
curvature then every expx : TxM → M is a covering. In particular when M is simply connected
then expx is a global diffeomorphism.
Proof. Let v, w ∈ TxM and consider the geodesic variation expx(t(v + sw)) and its Jacobi
field X(t) = expx∗(tv, tw). In particular X(1) = expx∗(v, w). We will now study the behaviour of
|X(t)| for t > 0.
d
dt g(X, X)1/2
=
g( ˙γX, X)
|X|
d2
dt2 g(X, X)1/2
=
| ˙γX|2
+ g(R(˙γ, X)˙γ, X)
|X|
−
g( ˙γX, X)2
|X|3
=
(|X|2
| ˙γX|2
− g( ˙γX, X)2
) − |X|2
R(˙γ, X, ˙γ, X)
|X|3
≥ 0
In the numerator the first bracket is non-negative by the Cauchy-Schwarz inequality while the
second is non-positive by our assumption on the sectional curvature. For t ≥ 0 let us denote
f(t) = |X(t)| − t|w| and study its Taylor expansion. In local coordinates we can write
X(t) = expx∗(tv, tw) = t · w(t)
where w is a curve with w(0) = w which we may assume to be non-zero. Thus
|X(t)| = t · |w(t)|
is smooth and hence so is f whose value and first derivative at zero are zero. By continuity the
second derivative on [0, ∞) must be non-negative and thus the same must be true for the first
derivative and finally also for the value. For t = 1 this means | expx∗(v, w)| = |X(1)| ≥ |w|. In
other words expx∗ is non-contracting.
We will now show that expx : TxM → M possesses the path-lifting property. Let γ : [a, b] → M
be a path with γ(a) = expx y0. Denote by
T = {t ∈ [a, b] | γ|[a,t] can be lifted to ˜γ with ˜γ(a) = y0}
We will show that T = [a, b] by connectedness. Clearly T is nonempty and open since expx is a
local diffeomorphism. Let tn → b0 ≤ b be a sequence with tn ∈ T and denote by ˜γ : [a, b0) → TxM
a lift with ˜γ(a) = y0. It exists by the uniqueness of the lifts (thanks to the local diffeomorphism
property). Then
|˜γ(tn) − ˜γ(tm)| ≤ (˜γ|[tn,tm]) ≤ (γ|[tn,tm]) < ε
for n, m 0 since expx is non-contracting and ˙γ is bounded. Thus ˜γ(tn) is a Cauchy sequence
and converges to some y. As expx is a local diffeomorphism at y the lift ˜γ can be prolonged.
It is a simple matter to deduce that a local diffeomorphism expx is a covering from the pathlifting
property. Namely a trivialization is produced from radial rays in a coordinate chart.
Remark. If M and N are two simply connected complete Riemannian manifolds of the same
dimension and the same constant non-positive sectional curvature then in the diagram
TxM ∼=
isometry
GG
∼=expx

TyN
∼= expy

M ∼=
GG N
7. PROBLEMS 80
the dotted arrow is an isometry. The same is true for positive curvature but the vertical arrows
are not isomorphisms. We will try to explain the situation by a computation. Let us denote the
constant value of the curvature by K > 0. We know that
R(X, Y )Z = K · (g(Y, Z)X − g(X, Z)Y )
If γ is a geodesic parametrized by the arc length and X is a Jacobi field perpendicular to ˙γ then
2
˙γX = R(˙γ, X)˙γ = −K · X
If we put K = ϕ2
then the solution of this equation is
X(t) = sin(ϕt) · Ptγ(w, t)
and we see that X(π/ϕ) = 0 for all w. Thus the whole sphere S(x, π/ϕ) is mapped to a single
point and expx induces a map
D(x, π/ϕ)/S(x, π/ϕ)
expx
−−−−→ M
which is a diffeomorphism on the interior of D(x, π/ϕ). Its metric properties are the following:
it preserves orthogonality of the radial rays to the spheres and preserves the metric on the radial
rays while on the sphere of radius r it multiplies it by sin(ϕr). The point is that this behaviour
only depends on the curvature K and thus for two manifolds Sm
and M in the diagram
Dm
(π/ϕ)/Sm−1
(π/ϕ)
∼= isometry

expy
GG Sm
(1/ϕ)

D(x, π/ϕ)/S(x, π/ϕ) expx
GG M
the dotted arrow, which is defined on the image of the interior of Dm
, preserves the metric. A
similar map can be defined using a different point on the sphere and together they provide a local
isometry from Sm
to M. It is a covering by the proof of the last theorem and thus an isometry.
7. Problems
Problem 7.1. Determine the Levi-Civita connection (or the corresponding covariant derivative)
for the Euclidean space Em by guessing what it might be and then proving that it indeed is
symmetric and metric.
Problem 7.2. For Em determine the Christoffel symbols, all curvatures and geodesics.
Problem 7.3. Determine the connection form of the Levi-Civita connection on Sm
.
Problem 7.4. Show that the sphere has constant sectional curvature by studying the effect
of an orthogonal transformation.
Problem 7.5. Determine the sectional curvature of the unit sphere.
Problem 7.6. In Rm,1
= Rm
× R we consider the scalar product
y, y = x2
1 + · · · + x2
m − x2
0
of signature (m, 1) where y = (x, x0) = ((x1, . . . , xm), x0). The hyperbolic space of dimension m is
the subset
Hm
= {y = (x, x0) ∈ Rm,1
| y, y = −1, x0 > 0}
Show that with the induced scalar product Hm
is a Riemannian manifold. Show that Hm ∼=
O+
(m, 1)/ O(m) and has a constant sectional curvature. Determine this sectional curvature.