Riemannian geometry - staff.science.uu.nlban00101/riemgeom2008/riemgeom.pdf · Riemannian geometry...
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Riemannian geometry Lecture Notes by E.P. van den Ban Fall 2008 1 Vector bundles In these notes smooth will always mean C ∞ , i.e., infinitely many times differ- entiable. In this section we will review the important notion of vector bundle. We will also define the notion of pull-back of a vector bundle. Definition 1.1 A vector bundle is a smooth map p : E → M of smooth manifolds E and M with the following properties: (a) p is surjective; (b) for every x ∈ M the fiber E x := p -1 (x) has the structure of a linear space; (c) for every a ∈ M there exists an open neighborhood U of a in M, a constant k ∈ N and a diffeomorphism τ from E U := p -1 (U ) onto U × R k , such that τ maps each fiber E x , for x ∈ U, linearly isomorphically onto {x}× R k R k . The space E is called the total space of the vector bundle, M is called the base manifold. The map τ is said to be a local trivialization of the vector bundle E over U. The constant k is called the rank of the bundle over U. Let p : E → M be a vector bundle. A smooth section of p is defined to be smooth map s : M → E such that p ◦ s = id M . Equivalently, this means that s(x) ∈ E x for all x ∈ M. We note that for U ⊂ M open the map p| E U : E U → U defines a smooth vector bundle over U. The space of smooth sections of this bundle is denoted by Γ ∞ (U, E). Note that this is the space of smooth maps s : U → E such that p ◦ s = id U . Let τ : E U → U × R k be a trivialization of E over U. For 1 ≤ j ≤ k let e j denote the j -th standard basis vector of R k . Then the map s j : U → E defined by τ (s j (x)) = (x, e j ), is a section. Clearly, the sections s 1 ,...,s k have the property that s 1 (x),...,s k (x) is a basis of E x , for every x ∈ U. Such an ordered k-tuple of sections is said to be a frame of E over U. Conversely, given a frame (s 1 ,...,s k ) of E over U, there exists a unique trivialization τ of E over U such that τ (s j (x)) = (x, e j ), for 1 ≤ j ≤ k. Example 1.2 Let M be a manifold, then the tangent bundle π : TM → M is a vector bundle. This bundle has trivializations induced by coordinate charts. As a set, TM is the disjoint union of the sets T x M. We will denote the elements 1
Transcript of Riemannian geometry - staff.science.uu.nlban00101/riemgeom2008/riemgeom.pdf · Riemannian geometry...
Riemannian geometry
Lecture Notes by E.P. van den Ban
Fall 2008
1 Vector bundles
In these notes smooth will always mean C∞, i.e., infinitely many times differ-entiable. In this section we will review the important notion of vector bundle.We will also define the notion of pull-back of a vector bundle.
Definition 1.1 A vector bundle is a smooth map p : E → M of smoothmanifolds E and M with the following properties:
(a) p is surjective;
(b) for every x ∈ M the fiber Ex := p−1(x) has the structure of a linear space;
(c) for every a ∈ M there exists an open neighborhood U of a in M, aconstant k ∈ N and a diffeomorphism τ from EU := p−1(U) onto U ×Rk,such that τ maps each fiber Ex, for x ∈ U, linearly isomorphically ontox × Rk ' Rk.
The space E is called the total space of the vector bundle, M is called the basemanifold. The map τ is said to be a local trivialization of the vector bundle Eover U. The constant k is called the rank of the bundle over U.
Let p : E → M be a vector bundle. A smooth section of p is defined to besmooth map s : M → E such that p s = idM . Equivalently, this means thats(x) ∈ Ex for all x ∈ M. We note that for U ⊂ M open the map p|EU
: EU → Udefines a smooth vector bundle over U. The space of smooth sections of thisbundle is denoted by Γ∞(U,E). Note that this is the space of smooth mapss : U → E such that p s = idU .
Let τ : EU → U × Rk be a trivialization of E over U. For 1 ≤ j ≤ k letej denote the j-th standard basis vector of Rk. Then the map sj : U → Edefined by τ(sj(x)) = (x, ej), is a section. Clearly, the sections s1, . . . , sk havethe property that s1(x), . . . , sk(x) is a basis of Ex, for every x ∈ U. Such anordered k-tuple of sections is said to be a frame of E over U. Conversely, givena frame (s1, . . . , sk) of E over U, there exists a unique trivialization τ of E overU such that τ(sj(x)) = (x, ej), for 1 ≤ j ≤ k.
Example 1.2 Let M be a manifold, then the tangent bundle π : TM → M isa vector bundle. This bundle has trivializations induced by coordinate charts.As a set, TM is the disjoint union of the sets TxM. We will denote the elements
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of TM as pairs (x, ξ), with x ∈ M and ξ ∈ TxM. Thus, π(x, ξ) = x. Let U ⊂ Mbe open and let χ : U → O be a diffeomorphism onto an open subset O of Rn.Then the map dχ : (x, ξ) 7→ (x, dχ(x)(ξ)). defines a trivialization of TM overU. Note that Γ∞(U, TM) equals the space X(U) of smooth vector fields on U.Let x1, . . . , xn be the local coordinates associated with χ, then the local frameassociated with the above trivialization is given by the vector fields ∂j = ∂
∂xj .
Example 1.3 Let M be a manifold, then the cotangent bundle T ∗M is avector bundle over M. Sections of T ∗M are just exterior 1-forms on M. A localcoordinatization (x1, . . . , xn) of a coordinate patch determines the local framedx1, . . . , dxn of T ∗M. We note that this local frame is dual to ∂1, . . . , ∂n in thesense that dxj(∂k) = δj
k for all 1 ≤ j, k ≤ n.
Example 1.4 More generally, for p ∈ N, the bundle ∧pT ∗M of exterior p-forms is an example of a vector bundle. The fiber above x equals ∧pT ∗
xM, thep-th exterior power of T ∗
xM, which may be naturally identified with the spaceof alternating p-forms on TxM. Its sections are the exterior p-forms on M. If(x1, . . . , xn) is a local coordinate system on U, then a local frame of ∧pT ∗U overU is given by the forms
dxI := dxi1 ∧ · · · ∧ dxip ,
with I running over the set of p-tuples (i1, . . . , ip) with 1 ≤ i1 < · · · < ip ≤ n.
Example 1.5 If p is a natural number and V is a finite dimensional real linearspace, we write
⊗pV =
p︷ ︸︸ ︷V ⊗ · · · ⊗ V .
If q is a second natural number, we write
T p,qV := ⊗pV ⊗⊗qV ∗
for the space of tensors of contravariance degree p and covariance degree q onV. If M is a manifold then the disjoint union
T p,qM :=
∐x∈M
T p,qTxM
carries a natural structure of smooth vector bundle over M whose sections arethe tensorfields of type (p, q).
Given a system x1, . . . , xn on an open subset U of M, a local frame of T p,qM
over U is given by
T JI = ∂i1 . . . ∂in ⊗ dxj1 ⊗ · · · ⊗ dxjn .
where I and J run over 1, . . . , np and 1, . . . , nq.
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Let p : E → M be a vector bundle. Then the zero section defines anembedding of M onto a closed submanifold of E. We will sometimes use thiszero section to view M as a submanifold of the bundle E.
A morphism from a vector bundles p : E → M to a vector bundle p′ : E′ →M ′ is a smooth map f : E → E′ such that f maps each fiber of E linearly intoa fiber of E′. In particular, this implies that f maps the zero section of E tothe zero section of E′. Hence, there exists a unique smooth map f0 : M → M ′
such that p′ f = f0 p. If M = M ′ and f0 = idM , we will say that f is theidentity on M.
The notion of isomorphism of vector bundles is now clear. We note that atrivialization of a vector bundle p : E → M over an open subset U of M is justan isomorphism of EU onto U × Rk, which is the identity on U.
Exercise 1.6 Let E,F be vector bundles over M. We denote by Hom(E,F )the space of morphisms from E to F which are identical on M. Show thatHom(E,F ) may be seen as a vector bundle whose fiber over a point x equalsthe space Hom(Ex, Fx) of linear maps Ex → Fx.
Let p : E → M be a vector bundle. Then by a subbundle of p we mean amorphism j : F → E of vector bundles over M, which is identical on M andwhich is injective on each fiber of F. This implies that j(Fx) is a linear subspaceof Ex for each x ∈ M.
Exercise 1.7 Let a ∈ M. Show that there exists an open neighborhood U ofa and trivializations σ : FU → U × Rm and τ : EU → U × Rk and a smoothmap L from U to Hom(Rm, Rk) such that
τ j(x, ξ) = Lx σ(x, ξ)
for all x ∈ U and ξ ∈ Fx.Conversely, show that if j admits a representation of this form everywhere
locally, then j : F → E is a subbundle of E.
We now come to the important notion of pull-back of a vector bundle. Letp : E → M be a vector bundle and f : N → M a smooth map. Then thepull-back f∗p : f∗E → N is a vector bundle over N, defined as follows.
As a set, f∗E is defined to be the collection of points (y, ξ) ∈ N × Esuch that ξ ∈ Ef(y). The projection map f∗p is defined to be the restrictionof prN : N × E → N. To see that this defines a vector bundle, let a ∈ N.Let U be a trivializing neighborhood of f(a) and let V := f−1(U). Under thistrivialization, EU corresponds to the trivial bundle U ×Rk. Accordingly, f∗EU
corresponds to graph(f |V ) × Rk hence is a smooth submanifold of N × EU . Itfollows that f∗E is a smooth submanifold of N×E. Moreover, f∗p|V correspondsto the projection map (y, f(y), v) 7→ y, showing that f∗p|V is a vector bundleisomorphic to V × Rk.
Restriction of the projection map N×E → E to f∗E defines a vector bundlemorphism f : f∗E → E. This vector bundle over N has the following universalproperty. Let q : F → N be a vector bundle and let ϕ : F → E be a vector
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bundle morphism which equals f : N → M on N. Thus, we have a commutativediagram
F −→ E↓ ↓N
f−→ M.
Then there exists a unique vector bundle morphism ϕ : F → f∗E which equalsthe identity on N, such that f ϕ = ϕ. This universal property has a nicerepresentation by arrows in the above diagram.
Exercise 1.8 Prove the above assertion. Show that the universal propertyproperty determines the bundle f∗E up to isomorphism.
2 Connections on a vector bundle
In this section, p : E → M will be a fixed vector bundle of rank k. We willdefine the notion of a connection on p. This notion will allow us to connectfibers of the vector bundle along curves on the base manifold. The connectingmaps will be called parallel transport or holonomies, and will play a crucial rolein the rest of this course.
At first we will define a connection in a geometric fashion. Later we willgive a different definition in terms of a differential operator.
Let x ∈ M, then the inclusion map ix : Ex → E induces the injectivetangent map dix(ξ) : TξEx → TξE, for each ξ ∈ Ex. As Ex has the structureof a linear space, TξEx ' Ex. Accordingly, we use dix(ξ) to identify Ex with alinear subspace of TξE.
On the other hand, the projection map p : E → M induces a surjectivelinear map dp(ξ) : TξE → TxM for each x ∈ M and ξ ∈ Ex. The kernel ofthis map, denoted Vξ, is called the space of vertical tangent vectors to E in thepoint ξ. Note that dim Vx = dim E − dim M = k.
Lemma 2.1 With notation as above, let x ∈ M and ξ ∈ Ex. Then dix(ξ) is alinear isomorphism from Ex onto Vξ.
Proof: From p ix(ξ) = x for all ξ ∈ Ex, it follows that dp(ξ) dix(ξ) = idEx .Hence dix(ξ) is a linear embedding of Ex into Vξ. The surjectivity of this mapfollows for dimensional reasons.
In the sequel we shall use the map dix(ξ) to identify Ex with the subspaceVξ of TξE.
Let Z be a smooth manifold. By an n-dimensional tangent system (alsocalled a distribution) of Z we mean a family of n-dimensional linear subspacesHz ⊂ TzZ such that z 7→ Hz is smooth in the following sense. For eachpoint of Z there should exist an open neighborhood U and smooth vector fieldsv1, . . . , vn ∈ X(U) such that Hz = span(v1(z), . . . , vn(z)), for all z ∈ U.
Definition 2.2 A connection on the vector bundle p : E → M is a smoothn-dimensional tangent system H = (Hξ)ξ∈E (n = dim M) such that TξE =Ep(ξ) ⊕Hξ for all ξ ∈ E.
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Thus, a connection may be viewed as a smooth family of spaces of ‘horizon-tal’ tangent vectors.
Example 2.3 If E = M ×Rk is the trivial bundle, then TξE ' Tp(ξ)M ×Rk,for all ξ ∈ E. The tangent system defined by
Hξ = Tp(ξ)M × 0
is a connection, which is called the trivial connection.
Since dp(ξ) : TξE → Tp(ξ)M is a surjective linear map, with kernel equal toVξ = Ep(ξ), its restriction to Hξ is a linear isomorphism onto Tp(ξ)M. It followsthat there exists a unique injective linear map αξ : Tp(ξ)M → TξE with
dp(ξ) αξ = idTp(ξ)M and im αξ = Hξ. (1)
Conversely, given a family of injective linear maps αξ : Tp(ξ)M → TξE depend-ing smoothly on ξ ∈ E and satisfying (1), the tangent system Hξ = image(αξ)defines a connection.
Given a connection as above, we use the direct sum decomposition to definethe projection map Pξ : TξE → Ep(ξ), for ξ ∈ E. Note that
Pξ = idTξE − αξ dp(ξ).
Example 2.4 In the example of the trivial bundle, αξ : Tp(ξ)M → TξE 'Tp(ξ)M × Rk is given by X 7→ (X, LξX), with Lξ : Tp(ξ)M → Rk a linear mapdepending smoothly on ξ ∈ M×Rk. Accordingly, Pξ is given by (X, v) 7→ (0, v−Lξ(X)). Note that the linear map Lξ : Tp(ξ)M → Rk is uniquely determined bythe requirement that
(X, LξX) ∈ Hξ, (X ∈ Tp(ξ)M). (2)
We will now describe how to transport connections under diffeomorphisms.Let q : F → N be a second vector bundle, and let ϕ : E → F be an isomorphismof vector bundles. Let x ∈ M and ξ ∈ Ex. Then the map ϕx : Ex → Fϕ(x) is alinear isomorphism. Moreover, dϕ(ξ) : TξE → Tϕ(ξ)F is a linear isomorphismwhich restricts to ϕx on Ex → TξE. Let H be a connection on p : E → M.Then it follows that for each η ∈ F the subspace
(ϕ∗H)η := dϕ(ϕ−1(η)) [ Hϕ−1(η) ].
is complementary to Fq(η) in TηF. We thus see that ϕ∗H defines a connectionon F.
Before proceeding, we note that connections behave well with respect topull-back. Indeed, let H be a connection on p : E → M as above and letf : N → M be a smooth map. Let f∗E be the pull-back of E under f andlet f : f∗E → E be the associated natural vector bundle morphism. For eachη ∈ f∗E we define
(f∗H)η := df(η)−1H ef(η).
Lemma 2.5 f∗H is a connection on f∗E.
We call this connection the pull-back of H under f.
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Proof: By localization we see that it suffices to prove this in case E is trivial.Thus, we may as well assume that E = M ×Rk, so that f∗E = graph(f)×Rk,with f∗p : f∗E → N given by (y, f(y), v) 7→ y. Let η ∈ f∗E. Then η = (y, x, v)with y ∈ N, x = f(y) and v ∈ Rk.
We note that ξ := (x, v) is the image of η under f . There exists a linearmap Lξ : TxM → Rk such that Hξ ⊂ TxM × Rk equals the space consisting ofthe vectors (X, LξX), for X ∈ TxM. The map df(η) : Tηf
∗E → TξE is givenby the restriction of prTxM × idRk to graph(df(y))× Rk. It follows that
(f∗H)η = (Y, df(y)Y, v) ∈ Tηf∗E | (df(y)Y, v) ∈ Hξ
= (Y, df(y)Y, v) ∈ Tηf∗E | v = Lξ df(y)Y .
Let Bη : TyN → Tηf∗E be defined by Bη(N) = (Y, df(y)Y, Lξdf(y)Y ). Then
Bη is injective linear with image (f∗H)η and depends smoothly on η. Moreover,dp(η) Bη = idTyN . It follows that f∗H is a connection on the bundle f∗E.
Remark 2.6 If f : N → M is a diffeomorphism, then f : f∗E → E is anisomorphism of vector bundles. It is readily seen that (f)∗f∗H = H in thiscase.
By a curve in a manifold Z we mean a continuous map c : I → Z, withI ⊂ R an interval.
Definition 2.7 Let p : E → M be a vector bundle equipped with a connectionH. A differentiable curve c : I → E is said to be horizontal if
c′(t) ∈ Hc(t)
for all t ∈ I.
Definition 2.8 Let p, H be as in the previous definition. Let c : I → M be adifferentiable curve. Then by a horizontal lift of c to E we mean a differentiablecurve c : I → E such that
(a) p c = c;(b) c is horizontal.
We shall now investigate the question whether a curve in M has horizontallifts. Since this is essentially a local question we first assume that the bundleis trivial, i.e., E = M × Rk. Then the connection is given by a family of linearmaps Lξ : Tp(ξ)M → Rk, depending smoothly on ξ ∈ M × Rk. Let c : I → Mbe a curve. Then a differentiable lift of c has the form c(t) = (c(t), d(t)), withd : I → Rk differentiable. The lift is horizontal if and only if (c′(t), d′(t)) ishorizontal for all t ∈ I, which is equivalent to
d′(t) = L(c(t),d(t))c′(t), (t ∈ I). (3)
If c is continuously differentiable, the above equation has always local solutions.More precisely, let t0 ∈ I be a fixed point, and let ξ0 ∈ Ec(t0). Then the equation
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(3) has solution with d(t0) = ξ0 defined on an interval J open in I and containingt0. Moreover, the uniqueness theorem for the initial value problem in ordinarydifferential equations guarantees that the solution is unique in the followingsense. If J1, J2 are two intervals as above, and ds : Js → Rk are solutions to (3)with ds(t0) = ξ0 for s = 1, 2, then d1 = d2 on J1 ∩ J2.
In the present generality there is no guarantee that the solution d extendsto the full interval I. If we require the map ξ 7→ Lx,ξ to be linear for everyx ∈ M, then the equation (3) becomes linear, and global existence of the solutionfollows.
Definition 2.9 Let E = M × Rk be the trivial bundle, and let H be a con-nection on E. For each ξ ∈ E, let Lξ : Tp(ξ)M → Rk be the unique linear mapdetermined by (2). The connection H is said to be affine if ξ 7→ Lξ is linear onthe fibers of E.
The following result implies that the notion of being affine can be extendedto arbitrary connections.
Lemma 2.10 Let H be a connection on the trivial bundle E = M × Rk. Letϕ : E → E be an isomorphism of vector bundles which is identical on M. ThenH is affine if and only if ϕ∗(H) is affine.
In the proof we shall use the notation Mk(R) for the space of k×k-matriceswith real entries, and GL(k, R) for the subset consisting of matrices a ∈ Mk(R)with det a 6= 0. Then Mk(R) may be viewed as a real linear space of dimensionk2. The matrix entries induce linear coordinates on this space. Accordingly,the function det : Mk(R) → R is polynomial hence continuous, and we see thatGL(k, R) is an open subset of Mk(R). In particular, GL(k, R) has the structureof a smooth manifold.
Proof: Assume that H is affine, and let L be associated to H as in Example 2.4.The isomorphism ϕ has the form ϕ(x, v) = (x, g(x)v), with g : M → GL(k, R)a smooth map. The tangent map of ϕ at a fixed point (x, v) ∈ M ×Rk is givenby
dϕ(x, v)(X, V ) = (X, (dg(x)X)v + g(x)V ).
Here we note that dg(x) is Mk(R)-valued, since GL(k, R) is open in Mk(R). Itfollows that ϕ∗(H)(x,g(x)v) consists of all elements of the form
(X, (dg(x)X)v + g(x)L(x,v)X), (X ∈ TxM).
This shows that the linear map Lϕ(x,g(x)v) associated to ϕ∗H is given by
Lϕ(x,g(x)v)X = (dg(x)X)v + g(x)L(x,v)X.
This expression depends linearly on v ∈ Rk if and only L(x,v)X does. The resultfollows.
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The above result guarantees correctness of the following definition.
Definition 2.11 Let H be a connection on a vector bundle p : E → M.The connection H is said to be affine if for each a ∈ M there exist an openneighborhood U and a trivialization τ : EU → U × Rk such that τ∗H is affine.
Theorem 2.12 Let H be an affine connection on the vector bundle p : E → M.Let c : I → M be a C1-curve, t0 ∈ I and ξ ∈ Ec(t0). Then
(a) the curve c has a unique horizontal lift cξ : I → E with cξ(t0) = ξ;
(b) the map (t, ξ) 7→ cξ(t), I × Ec(t0) → E is C1 and linear in the secondvariable.
Proof: For a trivial vector bundle, assertion (a) follows by our earlier discus-sion, motivating the definition of affine connection. The C1-part of assertion(b) follows from the known regularity and parameter dependence results forordinary differential equations. The statement about linearity follows from lin-earity of the ordinary differential equation (3) combined with uniqueness of thesolution.
We will establish the general result by reduction to the case of a trivialbundle.
Let J be the subset of I consisting of points t1 ∈ I such that the assertion ofthe theorem is valid for I replaced by the interval [t0, t1]. (This notation standsfor the interval of t contained between t0, t1, including the boundary points,and also makes sense for t1 ≤ t0). Clearly, if t1 ∈ J then [t0, t1] ⊂ J, so thatJ is an interval. We will show that this interval is both open and closed in I.From this the result will follow by connectedness of I.
Let J ′ be any interval which is open in I and such that c(J ′) is contained ina neighborhood on which E trivializes. We claim that if J and J ′ have a pointin common, then J ′ ⊂ J. To see this, fix t1 ∈ J ∩ J ′. Let γξ be the horizontallift of c|J with γ(t0) = ξ. For η ∈ Ec(t1), let γ′η be the horizontal lift of c|J withγ(t1) = η. By hypothesis the map ε : ξ 7→ γξ(t1) is linear from Ec(t0) to Ec(t1).By uniqueness of horizontal lift in the case of trivial bundles it follows thatγξ = γ′ε(ξ) on J ∩ J ′, for all ξ ∈ Ec(t0). This implies that γξ and γ′ε(ξ) are therestrictions of a C1-curve dξ : J∪J ′ → E which is a horizontal lift of c|J∪J ′ withinitial value dξ(t0) = ξ. It follows from the similar statements for the intervalsJ and J ′ that dξ is uniquely determined, and that the map (t, ξ) 7→ dξ(t) is C1
and linear in the second variable. Hence, J ′ ⊂ J.We can now show that J is open in I. Indeed, let t1 ∈ J. Fix an open
neighborhood U of c(t1) on which E trivializes. There exists an interval J ′ 3 t1,open in I, and such that c(J ′) ⊂ U. It follows from the above assertion thatJ ′ ⊂ J. Hence, t1 is an interior point of J and we see that J is open.
Similarly, we can show that J is closed in I. Let t2 ∈ I be an accumulationpoint of J. Then there exists a trivializing open neighborhood U of c(t2). Fix aninterval J ′ open in I, containing t2 and such that c(J ′) ⊂ U. Then J ∩ J ′ 6= ∅,hence J ′ ⊂ J by what we proved above. In particular it follows that J containsall of its accumulation points in I, hence is closed in I.
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Let p : E → M be a vector bundle equipped with an affine connection H.Let c : I → M be a C1-curve. Let t0, t1 ∈ I. We define the parallel transportTc,t1,t0 = Tt1,t0 from t0 to t1 along the curve c to be the linear map Ec(t0) → Ec(t1)
given byTt1,t0ξ = cξ(t1),
where cξ denotes the unique horizontal lift of the curve c with initial value ξ att = t0.
Lemma 2.13 With notation above, we have, for all s, s′, s′′ ∈ I,
(a) Ts′′,s′ Ts′,s = Ts′′,s ;(b) Ts,s = idEc(s)
;(c) Ts′,s is invertible with inverse Ts,s′ .
Proof: Easy and left to the reader.
Exercise 2.14 The purpose of this exercise is to explain the name paralleltransport.
First, let E = M×Rk be a trivial bundle, and let H be the trivial connectionon it. Show that for any C1-curve c : I → M the associated parallel transportTc,s′,s is given by
(c(s), v) 7→ (c(s′), v).
We now take M = U an open subset of Rn. Then for each x ∈ U themap v 7→ vx := d/dt(x + tv)|t=0 defines a linear isomorphism from Rn to TxU.Accordingly, ϕ : (x, v) 7→ vx defines a vector bundle isomorphism from U × Rn
onto TU. Let H be the connection on TU obtained by transference of the trivialconnection under ϕ. Show that the parallel transport in TU along a C1 curvec in U is given by
Tc,s′,s(vc(s)) = vc(s′)
and explain the name parallel transport.
The following lemma asserts that parallel transport behaves well with re-spect to reparametrization.
Lemma 2.15 With notation as in the previous lemma, let J be an intervaland ϕ : J → I a C1-map. Put d = c ϕ. Then for all s, s′ ∈ J,
Td,s′,s = Tc,ϕ(s′),ϕ(s).
Proof: Let c be any horizontal lift of c. Put d = c ϕ. Then for all t ∈ J, wehave that
d′(t) = ϕ′(t)c′(ϕ(t)) ∈ ϕ′(s)Hc(ϕ(t)) ⊂ Hd(t).
This implies that d is a horizontal lift of d. Moreover, if c(ϕ(s)) = ξ, thend(s) = ξ, and we see that
Td,s′,s(ξ) = d(s′) = c(ϕ(s′)) = Tc,ϕ(s′),ϕ(s)(ξ).
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Given a point x ∈ M we denote by Ω(x) the space of C1-curves c : [0, 1] → Mwith c(0) = c(1) = x. Given c ∈ Ω(x) we define the holonomy of H along c by
holx(c) = Tc,1,0.
Note that this holonomy is a linear transformation of Ex which by Lemma 2.13(c) is invertible. Let GL(Ex) denote the group of invertible linear transforma-tions of Ex.
Proposition 2.16 The transformations holx(c), with c ∈ Ω(x) form a sub-group of GL(Ex).
This subgroup is called the holonomy group of H at the point x.
Proof: Put I = [0, 1] and let ϕ : I → I be a fixed homeomorphism which isC1 and satisfies ϕ′(0) = ϕ′(1) = 0. Then it follows from Lemma 2.15 that
holx(c) = holx(c ϕ).
Given c, d ∈ Ω(x) we define d ∗ c : [0, 1] → M by
d ∗ c(t) =
d(ϕ(2t)) if 0 ≤ t ≤ 1/2;c(ϕ(2t− 1)) if 1/2 ≤ t ≤ 1.
Then it is readily seen that d ∗ c ∈ Ω(x). The reparametrization is needed togive c and d velocity zero at the enpoints, so that the composed loop is C1.Moreover, by application of Lemmas 2.15 and 2.13 it follows that
holx(d ∗ c) = holx(d) holx(c).
Given c ∈ Ω we define c−1 ∈ Ω(x) by c−1(t) = c(1− t). Applying Lemma 2.15with ϕ(t) = 1− t we see that
holx(c−1) = Tc−1,1,0 = Tc,0,1 = T−1c,1,0 = holx(c)−1.
Finally, the constant loop x : t 7→ x has holonomy idEx . The result follows.
The rest of the course will focus on several aspects of holonomy, in particularin the context of a Riemannian manifold.
3 Covariant differentiation
Let p : E → M be a vector bundle equipped with an affine connection H. Letx ∈ M and ξ ∈ Ex. Then the connection determines a unique projection Pξ :TξE → Ex with kernel Hξ. The projection allows us to measure the deviationfrom horizontality of a section under infinitesimal displacement. Indeed, let sbe a differentiable section of E defined in a neighborhood of x. Then ds(x) is alinear map TxM → Es(x).
Definition 3.1 Let Xx ∈ TxM and s a section of E defined in a neighborhoodof x. Then the covariant derivative of s at x in the direction X is defined by
∇Xxs := Ps(x) ds(x)X ∈ Ex.
10
Given a smooth vector field X ∈ X(M) and a smooth section s ∈ Γ∞(E)we define the section ∇Xs of E by ∇Xs(x) = ∇Xxs for every x ∈ M. Usingtrivializations of the vector bundle E it is readily seen that ∇Xs is a smoothsection again.
Lemma 3.2 Let s, t ∈ Γ∞(E), X, Y ∈ X(M) and f ∈ C∞(M). Then
(a) ∇fXs = f∇Xs;(b) ∇X+Y s = ∇Xs +∇Y s;(c) ∇X(s + t) = ∇Xs +∇Xt;(d) ∇X(fs) = (Xf)s + f∇Xs.
Proof: (a) and (b) are straightforward from the definitions. We will derive(c) and (d) by using that H is affine. The assertions are of a local nature onM, so that we may well assume that E = M × Rk. Then above a ∈ M theconnection is given by linear maps La,v : TaM → Rk, depending linearly onv ∈ Rk. A section of E is given by s(x) = (x, ϕ(x)), with ϕ a smooth mapM → Rk. Accordingly,
ds(a) = (I, dϕ(a)) : TaM → TaM × Rk.
The vertical projection at s(a) = (a, ϕ(a)) is given by
Ps(a)(X, v) = (0, v − La,ϕ(a)X).
It follows that∇Xas = (0, dϕ(a)Xa − La,ϕ(a)Xa),
viewed as an element of TξE ' TxM ⊕ Rk. Hence
∇Xas = (a, dϕ(a)Xa − La,ϕ(a)Xa) (4)
as an element of Ea = a×Rk. Assertions (c) and (d) follow from this formula,in view of the linear dependence of La,v on v and the C∞(M)-linear dependenceof ϕs on s.
The above rules for covariant differentiation allow a convenient expressionof covariant differentiation in terms of a local frame.
Let U be an open subset of M, and let e1, . . . , ek be a frame for E on U.Thus, ej are smooth sections of E on U and for each x ∈ U, the elementse1(x), . . . , ek(x) form a basis of Ex. We define the associated connection 1-formA = Ae on U to be the Mk(R)-valued 1-form on U given by
∇Xei =∑
j
A(X)jiej .
Let s be a smooth section of E on U. Then with respect to the frame (ej), thesection may be represented by a smooth function se : U → Rk. More precisely,
s =k∑
j=1
sjej
11
for uniquely defined smooth functions sj = sje ∈ C∞(U), and we put
se := (s1, . . . , sk)T.
Here T denotes that the transposed should be taken, in order to get a columnvector. In terms of this representation of sections, covariant differentiation maynow be described as follows.
Lemma 3.3 Let s ∈ Γ∞(U,E). Then for all X ∈ X(U),
(∇Xs)e = dse ·X + A(X)se.
Proof: This follows by a straightforward application of Lemma 3.2. Indeed,
∇Xs =k∑
j=1
∇X(sjej)
=k∑
j=1
(Xsj) ej + sj∇Xej
=k∑
j=1
dsj(X)ej +k∑
i,j=1
sjA(X)ijei
=k∑
j=1
dsj(X)ej +k∑
i=1
(A(X)s)iei
and the result follows.
We will now relate the connection H to the matrix A. The local framee1, . . . , ek of E on U determines the trivialization τ = τe : EU → U × Rk givenby
τ(s1e1(x) + · · ·+ skek(x)) = (x, s),
where s denotes the column vector with entries sj . The push-forward τ∗H of Hunder τ is a connection on the trivial bundle U ×Rk which may retrieved fromA in a simple manner.
Lemma 3.4 Let p : E → M be a rank k vector bundle equipped with an affineconnection H. Let e = (ej) be a local frame of E over an open subset U ⊂ M.Let A = Ae be the associated Mk(R)-valued 1-form on U and let τ = τe be theassociated trivialization of EU . Then for all (x, v) ∈ U × Rk,
(τ∗H)(x,v) = (X,−Ax(X)v) | X ∈ TxM.
Proof: Let s be any section of E over U and put se = (s1, . . . , sk)T. Then thesection τ s of the trivial bundle U × Rk is given by
τ s(x) = (x, se(x)).
Then for each vector field X on U,
τ ∇Xs(x) = (x, (∇Xs)e) = (x, dse(x)X + Ax(Xx)se(x)).
12
Let L be associated with the connection τ∗H on U × Rk as in (2). Thencomparing with (4) we find that
Ax(Xx)se(x) = −Lx,se(x)(Xx),
for all X ∈ X(U) and s ∈ Γ∞(U). This implies that
Ax(X)v = −Lx,v(X)
for all x ∈ U, X ∈ TxX and v ∈ Rk. The result now follows by using (2).
Proposition 3.5 Let ∇ : X(M) × Γ∞(E) → Γ∞(E), (X, s) 7→ ∇Xs be abilinear map satisfying properties (a) - (d) of Lemma 3.2. Then there existsa unique affine connection H on E such that ∇ is the associated covariantdifferentiation. Moreover, let x ∈ M and ξ ∈ Ex. Then Hξ is given by
Hξ = ds(x)Xx −∇Xs(x) | s ∈ Γ∞(E), s(x) = ξ, X ∈ X(M). (5)
Proof: By using cut off functions we see that the statement is local on M. Thus,we may restrict ourselves to the case that the bundle is trivial, E = M × Rk.Write ξ = (x, v). Let Kξ denote the set on the right-hand side of (5). Lete1, . . . , ek be the frame of E given by the standard basis of Rk, and let A be the1-form on M with values in Mk(R) given by ∇Xei =
∑j A(X)j
iej . A sections ∈ Γ∞(E) is any function of the form s(x) = (x, ϕ(x)), with ϕ ∈ C∞(M, Rk).As an element of TξE = TxM × Rk, the covariant derivative of such a sectionis given by
∇Xs(x) = (0, dϕ(x)Xx + Ax(Xx)ϕ(x)).
On the other hand, ds(x)Xx = (Xx, dϕ(x)Xx). Let Kξ denote the set on theright-hand side of (5). Then it follows that
Kξ = (Xx,−Ax(Xx)v | Xx ∈ TxM.
Clearly, this defines an affine connection on E. It is readily checked that theassociated covariant derivative is ∇. If ∇ comes from a connection H, then fromPs(x)ds(x)X = ∇Xs(x) it follows that ds(x)X−∇Xs(x) ∈ Hs(x). It follows thatKξ ⊂ Hξ. Since both spaces have the same dimension dim M, it follows thatHξ = Kξ.
Remark 3.6 In view of the above proposition, an affine connection is oftendefined to be an operator ∇ : X(M)×Γ∞(E) → Γ∞(E) such that the conditionsof Lemma 3.2 are fulfilled.
We end this section with another useful characterization of covariant differ-entiation in terms of parallel transport.
Lemma 3.7 Let a ∈ M and let Xa ∈ TaM. Let c : (−1, 1) → M be any C1
curve with c(0) = a and c′(0) = Xa. Moreover, let Tt = Tt,0 denote the paralleltransport along c from 0 to t. Then for every smooth section s of E,
∇Xas =d
dt
∣∣∣∣t=0
T−1t s(c(t)).
13
Proof: We note that Tt is a linear map from Ea to Ec(t). Thus ϕ(t) = T−1t s(c(t))
belongs to the finite dimensional vector space Ea for all t ∈ (−1, 1). We willshow that ϕ is a C1-function.
Let U be an open neighborhood of a on which the bundle E allows atrivialization τ : EU → U × Rk. Let J be an open interval in (−1, 1) con-taining zero and such that c(J) ⊂ U. Given x ∈ U we write τx for the lin-ear isomorphism Ex → Rk determined by τ(ξ) = (x, τx(ξ)). Then the mapt 7→ St := t 7→ τc(t) Tt τ−1
a is a C1-map from J to GL(k, Rk). By Cramer’sformula for the inverse of a matrix one sees that t 7→ S−1
t is C1 as well. Nowt 7→ τc(t)s(c(t)) is C1. It follows that
τa ϕ(t) = S−1t τc(t)s(c(t))
is a C1-function of t ∈ J. Hence ϕ is C1 on J. Other points of the interval(−1, 1) may be treated similarly, but we shall not need this.
We now observe that s(c(t)) = Ttϕ(t). This implies that
∇Xas = Ps(a)ds(a)c′(0)
= Ps(a)d
dt
∣∣∣∣t=0
s(c(t))
= Ps(a)d
dt
∣∣∣∣t=0
Ttϕ(t)
= Ps(a)d
dt
∣∣∣∣t=0
Ttϕ(0) + Ps(a)T0ϕ′(0).
Now the curve t 7→ Ttϕ(0) is horizontal by definition, so that its derivative att = 0 is horizontal. Hence, the first term of the above sum is zero. Moreover,T0 = idEa and ϕ′(0) belongs to Ea, on which Ps(a) equals the identity. It followsthat
∇Xas = ϕ′(0)
as was to be shown.
Exercise 3.8 Let p : E → M be a vector bundle equipped with a connectionH. Let c : I → M be C1 curve, and let s be a smooth section of E such that
∇c′(t)s = 0.
Show that s is parallel along c, i.e.,
Tc,′t′,ts(c(t)) = s(c(t′))
for all t, t′ ∈ I.
Exercise 3.9 Let p : E → M be a vector bundle equipped with a connectionH. We assume that F ⊂ E is a subbundle which has a complementary subbundleF ′ ⊂ E. By this we mean that Ea = Fa ⊕ F ′
a for all a ∈ M. Let π : E → Fdenote the projection map along F ′.
(a) Show that the map (X, s) 7→ π ∇Xs from X(M) × Γ∞(F ) to Γ∞(F )defines a covariant differential operator on F.
(b) Give a description of the connection HF on F associated to the operatorin (a) in terms of the connection H.
14
4 Curvature
Let p : E → M be a vector bundle equipped with a connection ∇. (From nowon we will always assume connections to be affine.) We define the associatedcurvature map R : X(M)× X(M)× Γ∞(M,E) → Γ∞(M,E) by
R(X, Y )s = ∇X∇Y s−∇Y∇Xs−∇[X,Y ]s,
for X, Y ∈ X(M), s ∈ Γ∞(M,E).
Lemma 4.1 The map R is C∞(M)-trilinear.
By this we mean of course that R is C∞(M)-linear in each of its variables.
Exercise 4.2 Prove the above lemma.
The following lemma will imply that R is essentially a vector bundle homo-morphism ⊗2TM → End(E). The first of these bundles is the vector bundle onM with fiber TxM ⊗TxM and the second of these bundles is the vector bundleon M with fiber End(Ex), for x ∈ M.
Lemma 4.3 Let p : E → M and q : F → M be vector bundles on M and letL : Γ∞(M,E) → Γ∞(M,F ) be a C∞-linear map. Then there exists a uniquevector bundle homomorphism λ : E → F, identical on M, such that
Ls(x) = λx(s(x)), (∀x ∈ M). (6)
Proof: Fix a ∈ M and consider the map La : Γ∞(E) → Fa given by Las =(Ls)(a). Select a local frame e1, . . . , ek of E over an open neighborhood U ofa in M. There exists a smooth function χ ∈ C∞(M) with compact supportcontained in U, such that χ(a) = 1. For a section s of E, let sj ∈ C∞(U) denotethe associated component functions with respect to the selected frame. Thus,s =
∑j sjej on U. The functions χsj may be viewed as smooth functions on
M , through extension by zero outside U. Similarly, the local sections χej maybe extended to smooth sections of E on U, vanishing outside the set U.
We now have, for any section s : M → E,
La(s) = χ(a)2La(s) = (χ2Ls)(a)
= L(χ2s)(a) = L(∑
j
(χsj)(χej))(a)
=∑
j
sj(a)La(χej).
Thus, if we define the linear map λa : Ea → Fa by λa(ej(a)) := La(χej), then
Las = λa(s(a)).
The evaluation map s 7→ s(a) is surjective from Γ∞(M,E) onto Ea. Hence, λa
is uniquely determined by the last displayed equality. Using local trivializationswe see that x 7→ λx depends smoothly on x. Hence, λ defines a smooth section ofHom(E,F ), or, equivalently, a vector bundle homomorphism E → F, identicalon M.
15
Corollary 4.4 There exist unique bilinear maps Ra : TaM×TaM → End(Ea),depending smoothly on a ∈ M, such that
R(X, Y )s = Ra(Xa, Ya)s(a),
for all smooth vector fields X, Y on M and smooth sections s of E.
Proof: This follows from Lemma 4.1, by application of Lemma 4.3 to each ofthe variables separately.
Let p : E → M be a rank k vector bundle equipped with a connection H.The connection is said to be flat, or locally trivial, if for each point a ∈ M thereexists a trivialization τ of E in a neighborhood U of a in which H becomesthe trivial connection, i.e., τ∗H is trivial. Equivalently, this means that the oneform A ∈ Ω1(U)⊗Mk(R) determined by τ vanishes.
Curvature measures the extent to which a connection is flat. More precisely,we have the following fundamental result.
Theorem 4.5 Let p : E → M be a vector bundle equipped with a connection∇. Then R = 0 if and only if the connection ∇ is flat.
For the proof we need the notion of restriction of (E,H) to a submanifoldN of M. Let N be such a submanifold, and let j : N → M denote the inclusionmap. Then the restriction of E to N is defined to be the pull-back bundlej∗(E). The restriction of H to EN is defined to be the pull-back f∗(H). Theseobjects were previously defined in the more general context of pull-back undersmooth maps between manifolds. The situation of a submanifold is particularlyimportant, and easier to handle. Therefore, we shall give the definitions for thissituation again and independently.
In the following p : E → M will be a vector bundle of rank k, H a connec-tion on it and ∇ the associated covariant differentiation. Let N be a smoothsubmanifold of M. Then the restriction bundle pN : EN → N is defined asfollows. The set EN is defined by EN := p−1(N). As p is a submersion, this isa smooth submanifold of E. The restriction pN := p|EN
: EN → N is a smoothsurjective map. Its fibers are p−1
N (x) = Ex for x ∈ N. Thus each of the fiberscomes equipped with the structure of a linear space. Moreover, each trivializa-tion of E in a neighborhood U of a point a ∈ N restricts to a trivialization τN
of EN on U ∩N. Hence, EN is a vector bundle. We note that a smooth sectionof EN is given by a map s : N → E such that p s = idN . For this reason, thespace of sections of EN is also denoted by Γ∞(N,E), and we speak of sectionsof E over the submanifold N.
If ξ ∈ EN , then the inclusion map EN → E induces a linear embeddingTξEN → TξE. Obviously, this embedding is compatible with the natural iden-tifications EpN (ξ) ' Ep(ξ) and with the natural embeddings EpN (ξ) → TξEN
and Ep(ξ) → TξE. Identifying elements of all these spaces accordingly, we haveEp(ξ) ⊂ TξEN ⊂ TξE. It follows that the linear projection map Pξ : TξE → Ep(ξ)
restricts to a linear projection map PNξ : TξEN → Ep(ξ). Put
HNξ := Hξ ∩ TξEN .
16
Then HNξ equals the kernel of PN
ξ so that TξEN = Ep(ξ) ⊕HNξ . It follows that
HNξ defines a connection on EN , which is readily checked to be affine (use local
trivializations). Let ∇N denote the associated covariant differentiation.
Lemma 4.6 Let s ∈ Γ∞(N,E) and let Xa ∈ TaN. Then
∇NXa
s = Ps(a) ds(a)Xa.
Proof: The equality is true with PN in place of P. Now use that ds(a) mapsTaN into Ts(a)EN and that Ps(a) restricts to PN
s(a) on Ts(a)EN .
Corollary 4.7 Let X be a vector field on M which is tangent to N everywhere,so that X|N is a vector field on N. Then for all sections s ∈ Γ∞(M,E),
(∇Xs)|N = ∇NX|N (s|N ).
Corollary 4.8 Let p : E → M be a vector bundle, equipped with a connection∇. Let N ⊂ M be smooth manifold, p|N : EN → N the restricted bundle and∇N the restricted connection. Then
RNa (Xa, Ya) = Ra(Xa, Ya)
for all a ∈ N and Xa, Ya ∈ TaN.
Exercise 4.9 With notation as above, let e1, . . . , ek be a local frame of Eon an open neighborhood U of a point a ∈ M. Assume that a ∈ N and putV := U ∩N.
(a) Show that e1|V , . . . , ek|V is a local frame of EN over U.
(b) Let Ae be the matrix valued 1-form on U associated with ∇ and the framee1, . . . , ek. Show that the matrix valued 1-form AN
e on V associated with∇N and the local frame ej |N is given by
ANe (x) = Ae(x)|TxN ,
for all x ∈ V.
We are now prepared for the proof of Theorem 4.5.
Proof of Thm. 4.5: If the connection is flat, then it is readily verified thatR = 0. To prove the converse, assume R = 0. In view of the local nature ofthe result, we may as well assume that M is the n-fold Cartesian product In
of the open interval I = (−1, 1). In this situation it suffices to establish theexistence of a frame e1, . . . , en of E over M that is flat. By this we mean that∇Xej = 0 for all X ∈ X(M). Let x1, . . . , xn denote the standard coordinatefunctions on M and ∂i = ∂/∂xi the associated standard vector fields. We willwrite ∇i := ∇∂i
. Then by the rules of calculation with connections it sufficesto show the existence of a frame e1, . . . , ek such that
∇iej = 0
17
for all 1 ≤ i ≤ n and 1 ≤ j ≤ k. We will achieve this by induction on n. Forn = 0 the manifold M consists of a single point, and there is nothing to prove.Thus, let n > 0 and assume the result has been established for strictly smallervalues of n.
Define the submanifold N of M by N = In−1 × 0. Then the connection∇N on the restricted bundle EN has curvature RN = 0 by Corollary 4.8. Bythe inductive hypothesis, there exists a frame eN
1 , . . . , eNk of E over N such that
∇Ni eN
j = 0, (1 ≤ i ≤ n− 1, 1 ≤ j ≤ k).
We now define the frame e1, . . . , ek of E over M as follows. For each x ∈ N weconsider the curve cx : I → M given by cx(t) = (x, t). Let Tx,t : Ex → E(x,t)
denote the parallel transport along this curve for the connection ∇ and define
ej = Tx,t(eNj (x)).
Then e1, . . . , ek defines a smooth frame of E over M. Moreover, in view ofLemma 3.7 it follows that
∇nej = 0 (1 ≤ j ≤ k).
As the curvature tensor vanishes it therefore also follows that for 1 ≤ i ≤ n− 1we have
∇n∇iej = ∇i∇nej = 0, (1 ≤ j ≤ k).
This implies that∇iej(x, t) = Tx,t(∇iej(x, 0)) = 0,
for 1 ≤ i ≤ n− 1 and 1 ≤ j ≤ k. The proof is complete.
Exercise 4.10 Let p : E → M be a vector bundle with a connection ∇. Lete1, . . . , ek be a local frame for E over an open subset U, and let A = Ae be theassociated k × k-matrix of 1-forms.
(a) Given X, Y ∈ X(U) we define R(X, Y )ij = Re(X, Y )i
j by
R(X, Y )ej =∑
i
R(X, Y )ijei.
Show that the R( · , · )ij define differential 2-forms on U. We write Re for
the associated Mk(R)-valued 2-form.
(b) Show thatRe = dAe + Ae ∧Ae.
It is customary to omit the subscript e in this notation. The action of don A is defined componentwise. Moreover,
(A ∧A)ij :=
∑r
Air ∧Ar
j .
18
5 Curvature and holonomy
In this section we will interpret curvature as infinitesimal holonomy. As prepa-ration we need a result relating the flows of two vector fields to their Lie bracket.We start by recalling some basic definitions and results concerning this bracket.
Let X be a smooth vector field on the manifold M, and let α : (t, x) 7→ αt(x)be its flow. We will write DX for the domain of α; it is an open subset of R×M.For each x ∈ M the set IX,x of t ∈ R with (t, x) ∈ DX is an interval containing0. In fact, it is the interval of definition of the maximal integral curve of X withstarting point x. The integral curve is given by t 7→ αt(x), IX,x → M. Thus,α0(x) = x and
∂
∂t
∣∣∣∣t=0
αt(x) = X(αt(x)), (t ∈ IX,x).
We recall that α0 = I and that for each compact set C ⊂ M we have αs αt =αs+t on C, for s, t sufficiently close to zero.
Likewise, let Y be a smooth vector field on M and let β denote its flow,with domain DY . For each x ∈ M, let IY,x denote the interval of t ∈ R with(t, x) ∈ DY . We recall that the Lie derivative of Y with respect to X is thevector field defined by
LXY :=∂
∂t
∣∣∣∣t=0
α∗t Y.
Clearly this notion is local and coordinate invariant. If M is an open subsetof Rn, then X and Y may be indentified with smooth functions M → Rn andthen the Lie derivative can be computed as follows:
LXY (x) =∂
∂t
∣∣∣∣t=0
d(α−t)(αt(x))Y (αt(x))
=∂
∂t
∣∣∣∣t=0
dα−t(x)Y (x) +∂
∂t
∣∣∣∣t=0
Y (αt(x))
= −dX(x) Y (x) + dY (x)X(x). (7)
Since the notion of Lie derivative is local and coordinate invariant, it followsfrom (7) that LXY = −LY X. Hence, the Lie bracket is anti-symmetric. Fromthe local expression (7) it is also readily verified that the Lie bracket satisfiesthe Jacobi identity:
[X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0,
for all smooth vector fields X, Y, Z on the manifold M. Accordingly, the reallinear space X(M) equipped with the Lie bracket is a Lie algebra.
We recall that a linear endomorphism D ∈ End(C∞(M)) which satisfiesthe Leibniz rule D(fg) = (Df)g + f(Dg) is called a derivation on C∞(M).The space of such derivations is denoted by Der(M). It is readily verified thatDer(M), equipped with the commutator bracket [D1, D2] = D1 D2 −D2 D1
is a Lie algebra. If X is a smooth vector field on M, we define
∂Xf(x) :=∂
∂t
∣∣∣∣t=0
f(αt(x)) = df(x)X(x).
19
It is readily verified that ∂X is a derivation on C∞(M).
Exercise 5.1 The map X 7→ ∂X defines an isomorphism of Lie algebas fromX(M) onto Der(M). Hint: use local coordinates and the local expression (7).
It is customary to abbreviate
Xf := ∂Xf, (f ∈ C∞(M), X ∈ X(M)).
With this notation the assertion about the map X 7→ ∂X being a Lie algebrahomomorphism may be rewritten as
X(Y f)− Y (Xf) = [X, Y ]f, (f ∈ C∞(M), X, Y ∈ X(M)).
The following result expresses the Lie bracket [X, Y ] in terms of the flows ofthe vector fields X and Y.
Lemma 5.2 Let X and Y be smooth vector fields on M with flows α and β,respectively. Then for each x ∈ M,
[X, Y ](x) =∂2
∂s∂t
∣∣∣∣s=t=0
β−tα−sβtαs(x).
The formula of the lemma should be interpreted as follows. First, there exists aδ > 0 such that (s, t) 7→ β−tα−sβtαs(x) is well-defined and smooth on (−δ, δ)×(−δ, δ). Moreover, for each s ∈ (−δ, δ), the element
∂
∂t
∣∣∣∣t=0
β−tα−sβtαs(x)
belongs to TxM and depends smoothly on s. Thus, the second differentiationis differentiation of a function with values in the fixed linear space TxM. Itsoutcome is an element of TxM.
Proof: By coordinate invariance and the local nature of the result, it sufficesto prove the identity in case M is an open subset of Rn. In this case the vectorfields X and Y may be viewed as functions on M with values in a single spaceRn; this simplifies the computations. We have
∂
∂t
∣∣∣∣t=0
β−tα−sβtαs(x) =∂
∂t
∣∣∣∣t=0
β−t(x) +∂
∂t
∣∣∣∣t=0
α−sβtαs(x)
= −Y (x) + dα−s(αs(x))Y (αs(x))= −Y (x) + (α∗sY )(x).
Differentiation with respect to s leads to
∂2
∂s∂t
∣∣∣∣s=t=0
β−tα−sβtαs(x) =∂
∂s
∣∣∣∣s=0
(α∗sY )(x) = [X, Y ].
20
We will now apply the above to the setting of a rank k vector bundle p : E →M equipped with a connection ∇. Let R be the associated curvature form. LetX, Y be two commuting vector fields on M, with flows α and β respectively. Leta ∈ M. It is a standard result that the flows of commuting vector fields locallycommute. Hence, there exists a constant δ > 0 such that for all s, t ∈ (−δ, δ)we have
β−tα−sβtαs(a) = a. (8)
We will define the so-called horizontal lifts of the vector fields X and Y. Itwill turn out that these vector fields on E need not commute.
For each x ∈ M and ξ ∈ Ex there exists a unique vector X(ξ) ∈ TξE suchthat
X(ξ) ∈ Hξ and dp(ξ)X(ξ) = X(x).
By using local trivializations it is readily seen that ξ 7→ X(ξ) defines a smoothvector field on E, called the horizontal lift of X. Let α be the flow of X.
Lemma 5.3 For every ξ ∈ E the integral curve s 7→ αs(ξ) is the horizontallift of the integral curve s 7→ αs(p(ξ)).
Proof: The tangent vector equals X(αs(ξ)) hence is horizontal, so that s 7→αs(ξ) is a horizontal curve in E. Write c(s) := p(αs(ξ)) for its projection to M.Then
c′(s) = dp(αs(ξ))X(αs(ξ)) = X(p(αs(ξ)) = X(c(s)).
Thus, c is an integral curve for X. Since c(0) = p(ξ) it follows that c(s) =αs(p(ξ)).
It follows from the above lemma that
αs(ξ) = Tc,s,0(ξ),
where Tc,s,0 denotes parallel transport from 0 to s along the integral curveσ 7→ ασ(x) of the vector field X.
Likewise, let Y be the horizontal lift of Y, and let β denote its flow. Thenβ is the horizontal lift of the flow β.
In view of (8) it follows from the above that β−tα−sβtαs(ξ) belongs to Ea
for all s, t ∈ (−δ, δ) and ξ ∈ Ea. In fact, the element may be viewed as theapplication of parallel transport along a loop based at a, composed of parts ofintegral curves of the vector fields X and Y. The following result expresses thatcurvature may be interpreted as infinitesimal holonomy.
Proposition 5.4 With notation as above, let a ∈ M and ξ ∈ Ea. Then
Ra(Xa, Ya)ξ =∂
∂s
∂
∂t
∣∣∣∣s=t=0
β−tα−sβtαs(ξ).
Proof: In view of the previous lemma, it suffices to show that
Ra(Xa, Ya)ξ = [X, Y ](ξ).
21
Let s be a smooth section of E with s(a) = ξ. We will show that
R(X, Y )s = [X, Y ] s
in a neighborhood of a. By the local nature of this assertion, and by invarianceunder diffeomorphism of manifolds and under isomorphism of vector bundles,we may assume that M is an open subset of Rn and that E is the trivialbundle M ×Rk. Let A be the k×k-matrix of 1-forms on M associated with theconnection and the standard frame e1, . . . , ek. Through the standard frame, asection s of E is identified with a map se : M → Rk. This being understood,covariant differentiation is given by
(∇Zs)e(x) = dse(x)Zx + Ax(Zx)se(x) = (Zse)(x) + A(Z)(x) se(x).
This implies that
(∇X∇Y s)e
= XY se + X[A(Y )se] + A(X)[Y se] + A(X)A(Y )se
= XY se + X[A(Y )]se + A(Y )[Xse] + A(X)[Y se] + A(X)A(Y )se.
Since X and Y commute, it follows from the definition of R that
(R(X, Y )s)e = (X[A(Y )]− Y [A(X)])se + [A(X)A(Y )−A(Y )A(X)]se.
We now observe that the horizontal lift of the vector field X at the pointξ = (x, v) is given by
X(ξ) = (Xx, Ax(Xx)v).
Similarly, the horizontal lift of Y is given by
Y (ξ) = (Yx, Ax(Yx)v).
It follows that
dY (ξ)(Z, V ) = (dY (x)Z , Z[A(Y )]x v + A(Y )xV )
so that
dY (ξ)X(ξ) = (dY (x)X(x) , X[A(Y )]xv + A(Y )xA(X)xv).
This implies that the Lie bracket of these horizontal lifts is given by
[X, Y ]ξ = d(Y )(ξ)X(ξ)− d(X)(ξ)Y (ξ)= (dY (x)X(x)− dX(x)Y (x) , V (x)),
where
V = (X[A(Y )]− Y [A(X)])v + [A(Y )A(X)v −A(X)A(Y )]v.
It follows that
[X, Y ]ξ = ([X, Y ]x, Rx(Xx, Yx)v))= (0, Rx(Xx, Yx)v).
In view of the natural identification 0×Rk = Tξ(Ex) ' Ex = Rk this impliesthe result.
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6 A useful derivative
In this section, we assume that p : E → M is a smooth vector bundle equippedwith a connection H. We recall that at every ξ ∈ E the tangent space decom-poses as TξE = Ep(ξ) ⊕Hξ. The corresponding projection map TξE → Ep(ξ) isdenoted by Pξ.
Let I be a real interval and s : I → E a smooth map. Then for each t ∈ Ithe tangent vector s′(t) = ds/dt belongs to Ts(t)E. We define
Ds
dt(t) := Ps(t)s
′(t).
We note that c := p s is a differentiable curve in M. Since s(t) ∈ Ec(t) for allt ∈ I we say that s is a section of E over the curve c. We note that Ds/dt isagain a section of E over the curve c.
Remark 6.1 Note that s : I → E is a section over the C1-curve c : I → M ifand only if s is a section of the pull-back bundle c∗(E). Moreover, let ∇c denotethe pull-back of ∇ under c, then Ds/dt = ∇c
1s.
Lemma 6.2 Let I be a real interval and s : I → E a differentiable curve.Then s is horizontal if and only if Ds/dt = 0.
Proof: The curve s in E is horizontal if and only if s′(t) ∈ Hs(t) for all t ∈ I.This is equivalent to Ps(t)s
′(t) = 0 for all t ∈ I, hence to Ds/dt = 0.
One would expect the derivative D/dt to be related to covariant differenti-ation. This is indeed the case.
Lemma 6.3 Let c : I → M be a C1-curve, and let s : M → E be a section.Then
D
dts(c(t)) = ∇c′(t)s, (t ∈ I).
Proof: By the chain rule,
D
dts(c(t)) = Ps(c(t))
d
dt[s(c(t))] = Ps(c(t)) ds(c(t)) c′(t) = ∇c′(t)s.
This result is probably the reason that in the literature one also sees thenotation ∇c′(t)s for Ds/dt in case s : I → E is a general section over c.
We will now express the derivative Ds/dt in terms of a local frame. Lete1, . . . , ek be a local frame, and let A = Ae the k × k matrix of one formsassociated with this frame. The local frame defines a local trivialization inwhich the vertical projection at a point ξ =
∑j vjej(x), x ∈ M, is given by
(X, V ) 7→ V + Ax(X)v, TaM × Rk → Rk.
23
Let se : I → Rk be the function defined by
s(t) =k∑
j=1
sje(t)ej(c(t)).
Then in the local trivialization s is given by t 7→ (c(t), se(t)), so that s′(t) isgiven by (c′(t), s′e(t)). Hence, (Ps(t)s
′(t))e = s′e(t) + Ax(c′(t))se(t) so that
(Ds
dt(t))e = s′e(t) + Ac(t)(c
′(t))se(t). (9)
From this expression for Ds/dt relative to a frame, one readily deduces thefollowing rules of calculation.
Lemma 6.4 Let c : I → E be a C1-curve, let σ, τ : I → E be differentiablesections over c and let f : I → R be a differentiable function. Then fσ andσ + τ are differentiable sections over c and
(a) Ddt [σ + τ ] = Dσ
dt + Dτdt (additivity);
(b) Ddt [fσ] = f ′σ + f Dσ
dt (Leibniz rule).
Proof: With respect to a local frame e1, . . . , ek one has [σ + τ ]e = σe + τe and[fσ]e = fσe. Now use (9). This proof is essentially the same as the proof ofLemma 3.2 (c),(d). The affineness of H again plays a crucial role.
Exercise 6.5 Let c : I → M be a C1-curve, and let s be a section of E overc. Show that for all t0 ∈ I,
Ds
dt(t0) =
d
dt
∣∣∣∣t=t0
T−1t (s(t)),
where Tt : Ec(t0) → Ec(t) denotes parallel transport along c. Hint: imitate theproof of Lemma 3.7.
7 Pseudo-Riemannian metrics
If V is a finite dimensional real linear space, then by a (not necessarily definite)inner product on V we mean a symmetric bilinear form β : V ×V → R which isnon-degenerate in the sense that β(v, w) = 0 for all w ∈ V implies that v = 0.The inner product is said to be positive definite if β(v, v) > 0 for all non-zerov ∈ V.
An inner product β on V induces a linear map V → V ∗, v 7→ β(v, ·) which isalso denoted by β. Non-degeneracy of the inner product means that β : V → V ∗
is injective. For dimensional reasons, β is also bijective. Thus we may use thelinear isomorphism β : V → V ∗ to transfer the inner product β to an innerproduct β∗ on V ∗. In formula,
β∗(β(v), β(w)) = β(v, w), ∀v, w ∈ V.
The inner product β∗ is called the dual inner product. If no confusion can arisefrom it, we will omit the star in the notation, and just write β for the dualinner product.
24
Exercise 7.1 Let e1, . . . , en be a basis for V and let e1, . . . , en be the dualbasis for V ∗, i.e., ei(ej) = δi
j for all i, j. Let βij = β(ei, ej) and βij = β(ei, ej).Show that the matrix βij is the inverse of the matrix βij . Hint: consider thematrix of the map β : V → V ∗ with respect to the basis and its dual.
Lemma 7.2 Let β be an inner product on a finite dimensional real linear spaceV of dimension n. Then there exists a direct sum decomposition V = V+ ⊕ V−,orthogonal with respect to β such that β is positive definite on V+ and negativedefinite on V−. The dimensions p = dim V+ and q = dim V− are independent ofthe particular direct sum decomposition.
The pair (p, q) in the above lemma is called the signature of the inner prod-uct β.
Proof: Let ei be a basis of V and let ei be the associated dual basis, as in theexercise preceding the lemma. Let ϕ : V → V ∗ be the linear isomorphism deter-mined by ϕ(ei) = ei for all 1 ≤ i ≤ n. Then the linear map h := ϕ−1β : V → Vhas matrix (β(ei, ej)) = (βij), which is symmetric. Hence h is symmetric withrespect to the unique inner product on V for which the basis (ei) is orthonor-mal. It follows that h diagonalizes with respect to an orthonormal basis. As h isinvertible, non of the eigenvalues is zero. Let V+ be the sum of the eigenspacesfor the positive eigenvalues, and V− the sum of the eigenspaces for the negativeeigenvalues. Then V = V+ ⊕ V−. Moreover, V+ and V− are orthogonal withrespect to β.
Let P be any subspace of V on which β is positive definite. Then β is bothpositive definite and negative definite on P ∩ V−. This implies that P ∩ V− = 0so that dim P ≤ n− q = p. It follows that p is the highest possible dimension ofa subspace on which β is positive definite. Therefore, p is independent of theparticular choice of decomposition.
In the following it will sometimes be used that the inner product β on Vis uniquely determined by the quadratic form f : v 7→ β(v, v) on V. Indeed, bysymmetry we have the identity
2β(v, w) = f(v + w)− f(v)− f(w).
By a quadratic form on V we mean a function f : V → R which is homogeneousof degree 2, i.e., f(λv) = λ2v for all v ∈ V and λ ∈ R. Conversely, each quadraticform defines a symmetric bilinear form β by the above formula. The quadraticform is said to be non-degenerate if and only if the associated symmetric bilinearform is. We note that
f(v) = β(v, v) =∑ij
βi,jvivj
which may also be expressed in terms of the dual basis as
f =∑i,j
βijeiej .
In this notation eiej stands for the product function V → R.
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Definition 7.3 Let M be a smooth manifold. A (pseudo-)Riemannian metricon M is a family of inner products gm on TmM, for m ∈ M, depending smoothlyon m in the sense that for all smooth vector fields X, Y ∈ X(M) the scalarfunction g(X, Y ) : m 7→ gm(Xm, Ym) is smooth. Equivalently, this means thatg defines a smooth section of the tensor bundle ⊗2T ∗M. The metric is saidto be Riemannian if gm is positive definite for all m. Otherwise, it is calledpseudo-Riemannian. A manifold equipped with a (pseudo-)Riemannian metricis said to be a (pseudo-)Riemmanian manifold.
Let x1, . . . , xn be a local coordinate system, defined on an open subset U ofM. Let ∂j = ∂/∂xj be the associated frame of vector fields over U and dxj theassociated frame of 1-forms. Let gij = g(∂i, ∂j), then viewing g as a section ofthe tensor bundle, we may write
g =∑ij
gij dxi ⊗ dxj on U.
In the literature one often encounters the local expression
g =∑i,j
gij dxidxj .
This expression makes sense if one interprets the symbol on the left-hand sideas the quadratic form associated with the metric g.
Proposition 7.4 Let M be a smooth manifold. Then there exists a Rieman-nian metric g on M.
For the proof we will need the existence of a partition of unity. A partitionof unity is a family of functions ϕα ∈ C∞
c (M), for α in some index set A, suchthat
(a) ϕα ≥ 0 for each α ∈ A;
(b) the supports suppϕα form a locally finite family in the sense that for eachcompact C ⊂ M the collection α ∈ A | suppϕα ∩ C 6= ∅ is finite;
(c)∑
α∈A ϕα = 1 (note that the sum is locally finite by (b)).
Let U be an open cover of M, i.e., a collection of open subsets of M suchthat the union
⋃U = ∪U∈UU equals M. A partition of unity ϕα is said to be
subordinate to the covering U if for each index α there exists an open subsetUα ∈ U containing the support suppϕα.
We need the following result for which we refer to the literature. For instanceone may consult the book ‘Differentiable manifolds’ by S. Lang.
Proposition 7.5 Let U be an open cover of M. Then there exists a partitionof unity subordinate to U .
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Proof of Proposition 7.4: Let U be a cover of M by coordinate charts. ForU ∈ U select a system of coordinates x1, . . . , xn and write gU for the associatedstandard metric
∑i dxidxi. Then gU m is positive definite at each m ∈ U.
Let ϕα, α ∈ A be a partition of unity subordinate to the cover U . For eachα ∈ A we select an open subset Uα ∈ U such that suppϕα ⊂ Uα and writegα := gUα . We note that ϕαgα may be extended by zero outside Uα and thusbecomes a smooth section of the tensor bundle ⊗2T ∗M. Define the smoothsection g of the same bundle by
g =∑α∈A
ϕαgα.
This definition makes sense, as the sum is locally finite. Note that gm is asymmetric bilinear form at every point m ∈ M. We will finish the proof byshowing that gm is positive definite. Indeed, let v ∈ TmM be a non-zero vector,then
g(v, v) =∑α∈A
ϕα(m)gαm(v, v).
By positive definiteness of the gαm, all terms of this sum are non-negative.Moreover, from
∑α ϕα(m) = 1 it follows that for some α ∈ A, ϕα(m) > 0. For
this α the associated term is positive. Therefore, g(v, v) > 0.
Remark 7.6 Note that the above proof uses positive definiteness in an essen-tial way, and cannot be extended to give a similar result for pseudo-Riemannianmetrics.
Let (M, g) be a Riemannian manifold, and let N be a smooth submanifold.For each a ∈ N the restriction gN,a of ga to TaN ⊂ TaM is a positive definiteinner product. Accordingly, gN defines a Riemannian metric on the submanifoldN, which is called the restriction of g to N. In general, for a pseudo-Riemannianmetric g it may happen that for some a ∈ N the restriction of ga may bedegenerate, hence does not define an inner product. The following exampledeals with a situation where the pseudo-Riemannian metric does restrict to ametric which turns out to be Riemannian.
Exercise 7.7 Let g be the standard pseudo-Riemannian metric of signature(n, 1) on Rn+1 given by
g =n∑
i=1
dxidxi − dxn+1dxn+1
Show that g restricts to a Riemannian metric gN on the submanifold N ⊂ Rn+1
given by the equationn∑
i=1
(xi)2 − (xn+1)2 = −1, xn+1 > 0.
Show that the coordinate functions x1, . . . , xn restrict to a global coordinatesystem on N and express gN in terms of the (restricted) one forms dx1, . . . , dxn
27
on N. The manifold N equipped with this Riemannian metric is called n-dimensional hyperbolic space. Advice: first do the case n = 2.
Exercise 7.8 Let g be the standard pseudo-Riemannian metric of signature(p, q) on Rn, n = p + q, p ≥ 2, given by
g =p∑
i=1
dxidxi −n∑
i=p+1
dxidxi.
Let N be the submanifold of Rn determined by the equation
p∑i=1
(xi)2 −n∑
i=p+1
(xi)2 = 1
Show that g restricts to a pseudo-Riemannian metric of signature (p − 1, q)on N. The resulting pseudo-Riemannian manifold is called pseudo-hyperbolicspace of signature (p− 1, q).
Definition 7.9 Let g be a (pseudo-)Riemannian metric on M and let ∇ be aconnection on the tangent bundle TM. The metric g is said to be covariantlyconstant, or flat, with respect to ∇ if
Zg(X, Y ) = g(∇ZX, Y ) + g(X,∇ZY )
for all X, Y, Z ∈ X(M).
Exercise 7.10 Let ∇ a connection (affine) on TM, and let g be a pseudo-Riemannian metric on M. Show that the following assertions are equivalent.
(a) The metric g is covariantly constant with respect to ∇;
(b) for any smooth curve c : [0, 1] → M the parallel transport Tt = Tc,t,0 :Tc(0)M → Tc(t)M is isometric (relative to gc(0) and gc(t)).
The following instructions should guide you through the exercise. We advisethe reader to recall the results of Section 6.
(1) First assume (b). Let X, Y, Z be smooth vector fields on M and leta ∈ M. Let c be any smooth curve with c′(0) = Za and use Lemma 3.7 todifferentiate
gc(0)(T−1t X(c(t)), T−1
t Y (c(t))− gc(t)(X(c(t)), Y (c(t))
at t = 0.
Now assume (a). Let c : [0, 1] → M be any smooth curve. Our goal is to prove
d
dtgc(t)(Xt, Yt) = gc(t)(
D
dtXt, Yt) + gc(t)(Xt,
D
dtYt)
for all smooth vector fields t 7→ Xt, Yt over c, i.e., Xt, Yt ∈ Tc(t)M.
28
(2) First prove the identity for Xt = X(c(t)) and Yt = Y (c(t)), with X, Y ∈X(M).
(3) Next prove the identity for Xt = f(t)X(c(t)), Yt = Y (c(t)).
(4) By using a frame to decompose Xt, prove the identity without conditionon t 7→ Xt, and with Yt = Y (c(t)).
(5) Prove the identity in general.
(6) Establish (b) by selecting X0, Y0 ∈ Tc(0)M and applying the identity tothe vector fields Xt = TtX0 and Yt = TtY0.
We will show that given a metric g, there always exists a connection forwhich the metric is flat. The connection will turn out to be unique if we requireits so-called torsion to vanish.
Definition 7.11 Let ∇ be a connection on TM. Its torsion form is the mapT = T∇ : X(M)× X(M) → X(M) defined by
T (X, Y ) = ∇XY −∇Y X − [X, Y ],
for all X, Y ∈ X(M). The connection ∇ is said to be torsion free (or symmetric)if the associated torsion form T∇ equals zero.
Exercise 7.12 Show that the torsion form T defines a tensor (called the tor-sion tensor). By this we mean that for each a ∈ TaM, the vector T (X, Y )a ∈TaM only depends on the vector fields X, Y through their values Xa, Ya. Thus,T determines a bilinear map Ta : TaM × TaM → TaM depending smoothly ona ∈ M and may be viewed as a section of the tensor bundle T 1,2
M .
Theorem 7.13 Let g be a (pseudo-)Riemannian metric on the manifold M.There exists a unique torsion free tensor ∇ for which g is covariantly constant.
This connection is called the Levi-Civita connection associated with themetric g.
Proof: Let ∇ be a connection with the asserted properties. Then by flatnessof the metric, it follows that, for smooth vector fields X, Y, Z ∈ X(M),
Xg(Y, Z) + Y g(Z,X)− Zg(X, Y )= g(∇XY +∇Y X, Z) + g(∇XZ −∇ZX, Y ) + g(∇Y Z −∇ZY, X).
By symmetry of the connection it follows that the expression on the right-handside equals
2g(∇XY, Z) + g([Y, X], Z) + g([X, Z], Y ) + g([Y, Z], X),
so that
2g(∇XY, Z) = Xg(Y, Z) + Y g(Z,X)− Zg(X, Y ) (10)−g([Y, X], Z)− g([X, Z], Y )− g([Y, Z], X).
It follows that ∇ is uniquely determined by the metric. Conversely, the latterformula may be used to define ∇ as a map X(M) × X(M) → X(M). It is astraightforward but somewhat tedious exercise to show that ∇ thus defined isa torsion free connection for which the metric is flat.
29
Let ∇ be any connection on the tangent bundle TM, and let x1, . . . , xn bea system of local coordinates, defined on an open subset U ⊂ M. Then thesmooth functions Γk
ij ∈ C∞(U) determined by
∇∂i∂j =
n∑k=1
Γkij ∂k
are called the Christoffel symbols of the connection with respect to the localcoordinate system. Let X, Y be two smooth vector fields on U. Let Xi and Y i
be the component functions defined by
X =∑
i
Xi∂i, and Y =∑
i
Y i∂i.
Then in terms of the Christoffel symbols the components of ∇XY are given by
(∇XY )k =∑
i
Xi∂iYk +
∑i,j
Γkij XiY j .
Exercise 7.14 Verify this.
Let us return to the situation of a (pseudo-)Riemannian metric g on M andthe associated Levi-Civita connection∇. It follows from (10) that the Christoffelsymbols with respect to a local coordinate system x1, . . . , xn may be expressedin terms of the components of the metric. Indeed, writing
g =∑ij
gij dxi ⊗ dxj
and applying (10) with X = ∂i, Y = ∂j Z = ∂l, it follows that
2∑
k
gklΓkij = ∂igjl + ∂jgil − ∂lgij
(note that the vector fields ∂i, ∂j , ∂l commute). Writing (glk) for the inverse ofthe matrix (gkl) (this is the matrix of the dual inner product), we find
Γkij =
12
∑l
glk[∂igjl + ∂jgil − ∂lgij ].
Note that the expression on the right-hand side, and hence the Christoffel sym-bol on the left-hand side, is symmetric in i, j. This reflects the symmetry of theconnection.
Exercise 7.15 Let g the standard metric of signature (p, q) on Rn, n = p+ q.Thus,
g =p∑
i=1
dxidxi −n∑
i=p+1
dxidxi.
Show that the Levi-Civita connection associated with g is given by
∇XY =∑i,j
Xi∂iYj∂j =
∑j
X(Y j)∂j .
30
The curvature form R associated with the Levi-Civita connection of the(pseudo-)Riemannian manifold (M, g) is also called the Riemannian curvaturetensor of (M, g). The following For each a ∈ M and all Xa, Ya ∈ TaM, the eval-uation Ra(Xa, Ya) defines an element of End(TaM), which depends on Xa, Ya
in an alternating fashion.
Lemma 7.16 Let R be the curvature tensor for a (pseudo-)Riemannian man-ifold (M, g). Then for all smooth vector fields X, Y, Z,W on M,
(a) R(X, Y )Z + R(Y, Z)X + R(Z,X)Y = 0 (Bianchi identity);(b) g(R(X, Y )Z,W ) = −g(Z,R(X, Y )W ) (anti-symmetry);(c) g(R(X, Y )Z,W ) = g(R(Z,W )X, Y ).
Proof: Assertion (a) is a straightforward consequence of the symmetry of theLevi-Civita connection∇. Assertion (b) may also be expressed as anti-symmetryof the form (Z,W ) 7→ g(R(X, Y )Z,W ). For this it suffices to establish theidentity g(R(X, Y )Z,Z) = 0 for all Z. By flatness of the metric it follows that
XY g(Z,Z) = 2X(g(∇Y Z,Z)) = 2g(∇X∇Y Z,Z) + 2g(∇Y Z,∇XZ)
and
Y Xg(Z,Z) = 2X(g(∇XZ,Z)) = 2g(∇Y∇XZ,Z) + 2g(∇XZ,∇Y Z).
Subtracting these identities we obtain
[X, Y ]g(Z,Z) = 2g(∇X∇Y Z −∇Y∇XZ).
By flatness of the metric the expression on the left-hand side may be rewrittenas 2g(∇[X,Y ]Z,Z) and (b) follows.
Assertion (c) can be obtained from (a) and (b) in the following way. Itfollows from (a) that
g(R(X, Y )Z,W ) = −g(R(Y, Z)X, W )− g(R(Z,X)Y, W ).
Adding the similar equations with (X, Y, Z, W ) replaced by (−Z,X, Y,W ),(W,Z,X, Y ) and (−X, W,Z, Y ) and using the two anti-symmetry propertiesof R, we obtain
2g(R(X, Y )Z,W ) + 2g(R(W,Z)X, Y ) = 0.
Using R(W,Z) = −R(Z,W ) we obtain (c).
8 Isometries and curvature
Definition 8.1 Let (M, g) and (N,h) be (pseudo-)Riemannnian manifolds.An isometry from M onto N is defined to be diffeomorphism f from M ontoN such that
hf(x)(df(x)v, df(x)w) = gx(v, w), (11)
for all x ∈ M and v, w ∈ TxM.
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Let now (M, g) be a pseudo-Riemannian manifold of signature (p, q), andlet ∇ be the associated Levi-Civita connection on the tangent bundle TM. Theassociated curvature form now is the tensor R : X(M)×X(M)×X(M) → X(M)given by
(X, Y, Z) 7→ R(X, Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z.
For p, q nonnegative integers, let Ep,q be the space Rp+q equipped with thestandard metric of signature (p, q) :
g =p∑
i=1
dxidxi −p+q∑
i=p+1
dxidxi.
In this case it follows from Exercise 7.15 that the associated curvature ten-sor vanishes identically. We will use Theorem 4.5 to show that R is the soleobstruction to M being locally isometric to the flat space Ep,q.
Theorem 8.2 Let (M, g) be a pseudo-Riemannian manifold of signature (p, q).Then the following assertions are equivalent.
(a) (M, g) is locally isometric to Ep,q.
(b) R = 0.
Proof: It remains to show that (b) implies (a). Assume (b) and let a ∈ M.The tangent space Ta admits a direct sum decomposition TaM = V+⊕V− suchthat V+ and V− are orthogonal, and such that ga is positive definite on V+
and negative definite on V−. It follows from Lemma 7.2 that dim V+ = p anddim V− = q. As ga is positive definite on V+ there exists a basis v1, . . . , vp ofV+ such that ga(vi, vj) = δij for all 1 ≤ i, j ≤ p. Similarly, there exists a basisvp+1, . . . , vn, n = p + q, such that ga(vi, vj) = −δij for all q + 1 ≤ i, j ≤ n. Putεi = 1 for 1 ≤ i ≤ p and εi = −1 for all p + 1 ≤ i ≤ n. Then it follows that
ga(vi, vj) = εiδij , (1 ≤ i, j ≤ n).
It follows from the proof of Theorem 4.5 that there exists a connected openneighborhood U of a in M and smooth vector fields X1, . . . , Xn on U such thatfor each 1 ≤ i ≤ n
(a) the vector field Xi is flat, i.e., ∇ZXi = 0 for all Z ∈ X(U);
(b) Xi(a) = vi.
In fact, in view of parallel transport, the vector fields Xi are uniquely deter-mined by these properties. By flatness of the metric, it follows that for allZ ∈ X(U),
Zg(Xi, Xj) = g(∇ZXi, Xj) + g(Xi,∇ZXj) = 0.
Hence, the scalar function g(Xi, Xj) is constant on U. By evaluation in (a), wesee that
g(Xi, Xj) = εiδij , (1 ≤ i, j ≤ n).
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By symmetry of the connection, it follows that
[Xi, Xj ] = ∇XiXj −∇XjXi = 0, (1 ≤ i, j ≤ n).
Thus, the vector fields Xi commute with each other. Let t 7→ etXi denote theflow of the vector field Xi. Then it is well known that there exists a δ > 0 suchthat the map
ϕ : (t1, . . . , tn) 7→ et1X1 · · · etnXn(a)
defines a diffeomorphism from (−δ, δ)n onto an open neighborhood of a inM. Adapting choices we may assume this neighborhood to be U. Obviously,ϕ∗(Xj) = ∂j . Hence ϕ∗g is the standard metric of signature (p, q) on (−δ, δ)n.This that ϕ is an isometric isomorphism from U onto the open subset (−δ, δ)n
of Ep,q. Assertion (a) follows.
9 Geodesics and the exponential map
We assume that M is a manifold whose tangent bundle p : TM → M isequipped with a torsion free (or symmetric) connection ∇. Of course the Levi-Civita connection associated to a pseudo-Riemannian metric on M is a moti-vating example of such a connection.
Let now c : I → M be a C2-curve. Then c′ : I → TM may be viewed as aC1-vector field over c. We define D/dt as in Section 6.
Definition 9.1 A geodesic relative to ∇ is defined to be a C2-curve γ : I → Msuch that
Dγ′
dt= 0. (12)
Remark 9.2 Equivalently, this means that γ′ is the horizontal lift of γ. Thus,if Tt,t0 denotes the parallel transport along γ from t0 to t then it follows thatTt,t0γ
′(t0) = γ′(t). Since parallel transport is an isometry for the metrics gγ(t0)
and gγ(t), it follows that
gγ(t)(γ′(t), γ′(t)) = constant, (t ∈ I).
Exercise 9.3 Let γ : I → M be a geodesic and let τ ∈ R. Show that the curvec : τ + I → M given by c(t) = γ(t− τ) is a geodesic again.
Let x1, . . . , xn be a local coordinate system on a coordinate patch U ofM. The associated vector fields ∂1, . . . , ∂n form a frame of TM over U. Theassociated one-form A on U with values in Mn(R) is given by
∇X(∂j) =n∑
k=1
A(X)kj ek.
It follows that the Christoffel symbols may be expressed in terms of A by
Γkij = A(∂i)k
j .
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Using that dx1, . . . , dxn is dual to ∂1, . . . , ∂n, we see that conversely
Akj =
∑i
Γkij dxi.
Put γk := xk γ. Then the k-th component of γ′ with respect to the local frame∂1, . . . , ∂n is given by the derivative γk = dγk/γk. Accordingly, relative to thelocal frame, the components of Dγ′/dt are given by
Dγ′
dt(t)k = γk(t) +
∑j
Aγ(t)(γ′(t))k
j γj(t).
Accordingly, the geodesic equation (12) becomes
γk(t) +∑
i
Γkij(γ(t))γi(t)γj(t) = 0, (1 ≤ k ≤ n).
This is a system of n non-linear second order differential equations, which maybe rewritten in the form
v′(t) = F (v(t)),
where v(t) is the column vector with components γk and γk, and where F isa smooth vector field on x(U) × Rn ⊂ R2n. It follows from the theory of ordi-nary differential equations that this system may be solved locally and uniquely,provided the initial data γ(0) and γ(0) are given. Moreover, the local solu-tion is smooth, and depends smoothly on the initial data. Applying this resulteverywhere locally, one obtains the following result.
Lemma 9.4 Let a ∈ M and v ∈ TaM. Then there exists a unique smoothcurve γv in M with domain an open interval Iv containing zero such that
(a) γv is a geodesic with γv(0) = a and γ′v(0) = v;
(b) if γ : I → M is any geodesic with γ(0) = a and γ′(0) = v, then I ⊂ Iv
and γ = γv|I .
Any linear reparametrization of a geodesic is a geodesic again. More pre-cisely, the following result is valid.
Lemma 9.5 Let v ∈ TM and s ∈ R. Then Isv = s−1Iv. Moreover,
γsv(t) = γv(st), (t ∈ s−1Iv).
Proof: The result is trivial for s = 0, provided we agree that s−1Iv = R inthat case. Thus, assume s 6= 0.
Let a = p(v). Then v, sv ∈ TaM. The curve γ(t) = γv(st) is well defined ons−1Iv. Moreover, γ′(t) = sγv(st) and γ′′(t) = s2γv(st). From the local expressionof the geodesic equation, which is linear in γ′′ and quadratic in γ′, one sees thatγ is a geodesic. Moreover, γ(0) = a and γ′(0) = sγ′v(0) = sv. It follows thats−1Iv ⊂ Isv and that γ = γsv on s−1Iv. The inclusion s−1Iv ⊃ Isv is obtainedby a similar reasoning, with sv, s−1 in place of v, s.
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In view of the above, the smooth dependence on initial values leads to thefollowing result.
Proposition 9.6 The set Ω = v ∈ TM | 1 ∈ Iv is an open neighborhood ofthe zero section in TM. Moreover, the map [0, 1] × Ω → M, (t, v) 7→ γv(t) issmooth.
Definition 9.7 With Ω defined as in the preceding proposition, we define theexponential map exp : Ω → M by exp(v) = γv(1). Given a ∈ M we define expa
to be the restriction of exp to Ωa = Ω ∩ TaM.
The preceding discussion leads to the following result.
Lemma 9.8 Let a ∈ M and v ∈ TaM. Then Iv = t ∈ R | tv ∈ Ω. Moreover,
γv(t) = exp(tv), (t ∈ Iv).
Proof: We note that tv ∈ Ω if and only if 1 ∈ Itv, which in turn is equivalentto t ∈ Iv, by Lemma 9.5. Moreover, by the same lemma, if the latter conditionis fulfilled then exp(tv) = γtv(1) = γv(t).
Proposition 9.9 Let a ∈ M. Then the exponential map expa : Ωa → Mrestricts to a diffeomorphism from an open neighborhood of the origin in TaMonto an open neighborhood of a in M. Viewed as a map from TaM ' T0TaMto a map TaM the derivative d expa(0) equals the identity.
Proof: The domain of expa is an open neighborhood of the origing in TaM.Let v ∈ TaM. Then γv(t) = expa(tv) for all t ∈ R sufficiently close to zero. Bydifferentiation with respect to t at t = 0 it follows that
d expa(0)v =d
dt
∣∣∣∣t=0
expa(tv) =d
dt
∣∣∣∣t=0
γv(t) = v.
The result now follows by application of the inverse function theorem.
We end this section with a result on the possibility to locally connect pointsby geodesics.
Proposition 9.10 Let (M, g) be a pseudo-Riemannian manifold and let a ∈M. Let V be an open neighborhood of a equipped with a Riemannian metric h.There exist a constant δ > 0 and an open neighborhood U of a in V such that:
(a) for every p ∈ U the exponentional map expp maps the open hp-ball ofradius δ diffeomorphically onto an open neighborhood of p containing U ;
(b) for each pair of distinct points p, q ∈ U there exists a unique geodesicγ : [0, b] → M with γ(0) = p, γ(b) = q and h(γ′(0), γ′(0)) = 1, 0 ≤ b < δ.
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Proof: From the implicit function theorem it follows that there exist an openneighborhood U of a and an open neighborhood ω of V in TM such that for allp ∈ ω∩M the exponential map exp maps Tp∩ω diffeomorphically onto an opensubset of M containing U. Let V0 be a compact neighborhood of a in V. Thenthere exists a constant δ > 0 such that for each p ∈ V0 the hp-ball Bp(0; δ) ofradius δ in TpM is contained in ω. Replacing U by a smaller neighborhood wemay assume that U ⊂ V0. Now (a) follows.
For (b), assume that p, q ∈ U are distinct points. Then q = expp(v) for aunique vector v ∈ Bp(0; δ). There exists a unique vector w ∈ TpM of hp-length1 such that bw = v for some 0 < b < δ. Then q = expp(v) = γv(1) = γw(b).Thus γ = γw satisfies the requirement.
Conversely, let γ : [0, r] → M be a geodesic as stated. Then γ(t) = expp(tw),for all 0 ≤ t < δ, where w = γ′(0). By injectivity of expp on Bp(0; δ), theequality expp(bw) = γ(b) = q = expp(v) implies that bw = v. Hence, γ = γv
and uniqueness of γ follows.
10 Riemannian distance
We now assume that (M, g) is a Riemannian manifold. Then gx is a positivedefinite inner product on TxM, for every x ∈ M. The associated norm is denotedby
‖v‖x :=√
gx(v, v), (v ∈ TxM).
A curve c : [a, b] → M is said to be piecewise C1 if c is continuous and thereexists a partition a = t0 < t1 < · · · < tk = b of the interval such that for each1 ≤ j ≤ k the restriction c|(tj−1,tj) extends to a C1-curve [tj−1, tj ] → M.
If c : [a, b] → M is a piecewise C1-curve, we define its length L(c) by
L(c) =∫ b
a‖c′(t)‖c(t) dt.
If ϕ : [α, β] → [a, b] is a surjective piecewise C1-map with ϕ′ ≥ 0 (or with ϕ ≤ 0)in all but finitely many points, then by the substitution rule for integration, itfollows that
L(c) = L(c ϕ).
In particular, it follows that L is invariant under a piecewise C1-reparametriz-ation.
If ϕ : (M, g) → (N,h) is an isometric isomorphism of Riemannian manifolds,then for all a ∈ M and v ∈ TaM we have ‖dϕ(a)v‖ϕ(a) = ‖v‖a. From this itreadily follows that L(ϕ c) = L(c) for any C1-curve in M.
Given two points p, q ∈ M we define the distance d(p, q) from p to q to bethe infimum
d(p, q) = infc
L(c),
as c varies over all curves as above with c(a) = p and c(b) = q. If ϕ : (M, g) →(N,h) is an isometric isomorphism, it is readily verified that dN (ϕ(p), ϕ(q)) =dM (p, q), for all p, q ∈ M.
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Lemma 10.1 (M,d) is a metric space whose underlying topology coincideswith that of M.
Proof: First, since L(c) ≥ 0 for all curves, it follows that d ≥ 0. The functiond is clearly symmetric, and it is readily seen to verify the triangle inequality. Tosee that it is a metric in the sense of topology, we must show that d(p, q) > 0when p and q are distinct. For this we select a chart (κ, U) with U an openneighborhood of p in M and κ a diffeomorphism of U onto an open ball of center0 in Rn such that κ(p) = 0. Let h denote the Riemannian metric on κ(U) forwhich κ becomes an isometry. For x ∈ U, put
m(x) = min‖v‖=1
hκ(x)(v, v), M(x) = max‖v‖=1
hκ(x)(v, v).
Replacing U by a smaller neighborhood if necessary, we may assume that thereexists a constant C > 0 such that C2 < m(p) and C−2 > M(p) for all p ∈ U.By homogeneity it follows that
C‖v‖ ≤ ‖v‖hκ(x)≤ C−1‖v‖
for all x ∈ U and v ∈ Rn. Given a piecewise C1-curve c in Rn let l(c) denoteits Euclidean length. Then it follows from the above that for any piecewiseC1-curve c in U we have
Cl(κ c) ≤ Lh(κ c) = L(c) ≤ C−1l(κ c). (13)
Here Lh denotes the length relative to the metric h, and L denotes the lengthrelative to g. Let r > 0 be such that the closed ball B := x ∈ Rn | ‖x‖ ≤ r iscontained in κ(U). Let S be its boundary, the sphere of center 0 and radius r.Replacing r by a smaller constant if necessary, we may assume that q /∈ κ−1(U).If c : [a, b] → M is a piecewise C1-curve connecting p and q, let t0 be the infimumof t ∈ [a, b] such that c(t) /∈ κ−1(B). Then it follows that c(t0) ∈ κ−1(S). Sinceκ c|[a,t0] is a curve connecting 0 and a point of S we conclude that its Euclideanlength is at least r. It follows that
L(c) ≥ L(c|[a,t0]) = Lh(κ c|[a,t0]) ≥ Cr.
Since this is true for any curve connecting p and q, we conclude that d(p, q) ≥Cr > 0. It follows that d is a metric in the sense of topology.
For any δ > 0 we denote by B(p; δ) the set of x ∈ M with d(p, x) < δ. ByB′(p, δ) we denote the set of x ∈ U with ‖κ(x)‖ < δ. We will complete the proofby showing that the system of neighborhoods B(p, δ), δ > 0, is equivalent tothe system of neighborhoods B′(p, δ), δ > 0.
Let ε < r and let x ∈ B(p;Cε). Then there exists a curve c in M with initialpoint p and end point x, such that L(c) < Cε. By what we showed earlier, thecurve c must be contained entirely in B. It follows that
L(c) = Lh(κ c) ≥ Cl(κ c) ≥ C‖κ(x)‖,
so that x ∈ B(p; ε). This establishes the inclusion B(p;Cε) ⊂ B′(p; ε) for allε < r.
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Conversely, let ε > 0 and let x ∈ B′(p;Cε). Let c be the curve in U suchthat κ c is the straight line connecting κ(p) = 0 and κ(x). Then
‖κ(x)‖ = l(κ c) ≥ CLh(κ c) = CL(c) ≥ Cd(p, x).
It follows that x ∈ B(p; ε). This establishes the inclusion B′(p;Cε) ⊂ B(p; ε)for all ε > 0.
Lemma 10.2 (Gauß lemma) Let M be a Riemannian manifold and p ∈ M.Let δ > 0 be such that expp is a diffeomorphism from Bgp(0; δ) onto an openneighborhood of p in M. Let S be the unit sphere in TpM and let ϕ : [0, δ)×S →M be defined by ϕ(r, σ) = expp(rσ). Then
ϕ∗(g) = dr2 + r2hr,
where hr is a Riemannian metric on S and limr↓0 hr = gp|S .
Proof: We observe that the map ϕ restricts to a diffeomorphism from D :=(0, δ) × S onto a punctured neighborhood of p in M. Let ∂/∂r be the vectorfield on (0, δ) × S for which the associated derivation is partial differentiationwith respect to the first variable. Let v be a vector field on S and let v := (0, v)denote the associated vector field on (0, δ) × S. Then ∂/∂r and v commute,hence so do their images R and V under ϕ. By symmetry of the Levi-Civitaconnection associated to g it follows that ∇RV = ∇V R.
We note that for (r, σ) ∈ D,
R(r, σ) = dϕ(r, s)(
∂
∂r
)r,σ
=∂
∂rϕ(rσ) = γσ(r),
so that ∇RR = 0 on ϕ(D). We see that
ϕ∗(g)(∂
∂r,
∂
∂r)(r,σ) = g(R,R)ϕ(r,σ)
= gγσ(r)(γσ(r), γσ(r))= gp(σ, σ) = 1. (14)
Also, for (r, σ) ∈ D,
V (r, σ) = dϕ(r, s)(0, v) = rd(expp)(rσ)v,
so thatϕ∗(g)(v, v)(r,σ) = g(V, V )ϕ(r,σ) = r2 hr(v, v)σ, (15)
wherehr(v, v)σ = gϕ(r,σ)(d expp(rσ)v, d expp(rσ)v).
It remains to consider the cross term
(ϕ∗g)(∂/∂r, v) = ϕ∗(g(R, V )). (16)
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For this we note that
Rg(R, V ) = g(∇RR, V ) + g(R,∇RV )= 0 + g(R,∇V R)
=12V g(R,R) = 0.
This implies that g(R, V )ϕ(r,σ) is constant as a function of r. Now
g(R, V )ϕ(r,σ) = rgϕ(r,σ)(γσ(r), d expp(rσ)) → 0
as r ↓ 0. Hence,g(R, V ) = 0. (17)
The result follows from combining (14), (15), (16) and (17).
Corollary 10.3 Let M,p, δ, S be as above. Then for every (r, σ) ∈ [0, δ)× S,
d(p, expp(rσ)) = r. (18)
Let c be any piecewise C1-curve of length r with endpoints p and expp(rσ). Thenc is a piecewise C1-reparametrization of the geodesic γσ : [0, r] → M.
Proof: The geodesic γσ : [0, r] → M has tangent vector of length
gp(γσ(0), γσ(0)) = gp(σ, σ) = 1
(see Remark 9.2). It follows that L(γσ) = r, so that d(p, expp(rσ)) ≤ r.On the other hand, let c : [τ1, τ2] → M be any piecewise C1-curve connecting
p and q := expp(rσ). Let a be the supremum of τ ∈ [τ1, τ2] such that c(τ) = p.Then c(a) = p and c(τ) 6= p for τ > a. Let b be the supremum of τ ∈ [a, τ2]such that c(τ) ∈ expp(B(0, r)). Then c(b) = expp(rσ′) for a σ′ ∈ S. There existunique piecewise C1-functions r : (a, b] → (0, δ) and σ : (a, b] → S, such thatc(t) = expp(r(t)σ(t)) for t ∈ (a, b]. Hence
L(c) =∫ τ1
τ0
‖c′(t)‖c(t) dt
≥∫ b
a‖c′(t)‖c(t) dt
=∫ b
a
√|r′(t)|2 + r(t)2hr(σ′(t), σ′(t)) dt
≥∫ b
a|r′(t)| dt ≥
∫ b
ar′(t) dt
= r(b)− r(a).
The latter expression equals r − 0 = r. Hence, d(p, q) ≥ L(c) ≥ r. This estab-lishes (18). Now assume that the length of c above equals r. Then all inequalitiesmust be equalities. Hence (except for possibly finitely many points) c′ = 0 out-side [a, b], σ′ = 0, and r′(t) ≥ 0, so that c(t) = expp(r(t)σ) = γσ(r(t)) is apiecewise C1-reparametrization of γσ : [0, r] → M.
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A piecewise C1-curve c : I → M is said to be parametrized by arc length ifI = [0, r] for some r ≥ 0 and if L(c|[0,t]) = t for all t ∈ [0, r]. Note that the lastcondition is equivalent to ‖c′(t)‖c(t) = 1 for all t ∈ [0, r].
Let γ : [a, b] → M be a (non-constant) geodesic. Then ‖γ′(t)‖γ(t) equals aconstant r > 0 (see Remark 9.2). The reparametrization γ : [0, r(b− a)] → Mdefined by
γ(t) = γ(a + r−1t)
is again a geodesic. As this geodesic is parametrized by arc length, it is said tobe the reparametrization of γ by arc length.
Corollary 10.4 Let (M, g) be a Riemannian manifold and let a ∈ M. Thereexist a constant δ > 0 and an open neighborhood U of a in M such that:
(a) for every p ∈ U the exponentional map expp maps the open ball of radiusδ in TpM diffeomorphically onto an open neighborhood of p containing U ;
(b) for each pair of distinct points p, q ∈ U there exists a unique geodesic γp,q,parametrized by arc length and of length less than δ, with initial point pand end point q;
(c) the length of γp,q equals d(p, q).
Proof: Applying Proposition 9.10 with V = M and h = g, we obtain U suchthat (a) and (b) are valid. Finally, (c) follows from Corollary 10.3.
Corollary 10.5 Let c : [a, b] → M be piecewise C1-curve such that L(c) =d(c(a), c(b)). Then the reparametrization of c by arc length is a geodesic.
Proof: Let a < α < β < b. Then by the triangle inequality for d,
L(c) = d(c(a), c(b)) ≤ d(c(a), c(α)) + d(c(α), c(β)) + d(c(β), c(b))= L(c|[a,α]) + L(c|[α,β]) + L(c|[β,b])= L(c).
It follows that the length of c|[α,β] equals d(c(α), c(β)). In view of the previouscorollary, it now follows that for each t ∈ [a, b] there exists a ε > 0 such thatthe restriction of c to [a, b]∩ (t− ε, t + ε) has a reparametrization by arc lengthwhich is a geodesic. By compactness of [a, b], the result follows.
A Riemannian manifold (M, g) is said to be geodesically complete if eachgeodesic extends to a geodesic with domain R. Equivalently, this means thatthe exponential map exp has domain TM. Not every Riemannian manifold isgeodesically complete. Indeed, let γ : R → M be a geodesic in M, let b > 0be such that γ is injective on [0, b]. Then M ′ = M \ γ(b) is a Riemannianmanifold and γ[0,b) is a geodesic in M ′, whose maximal extension to a geodesicin M ′ equals γ|I , where I is the connected component of R \ γ−1(b) whichcontains 0.
It is easily seen that the metric of M ′ is the restriction of the metric of M.Let (tn) be an increasing sequence in [0, b) with limit b. Then the sequence γ(tn)
40
is Cauchy in M ′ but has no limit in M ′. Therefore, M ′ is not complete as ametric space.
We end this section with the important Hopf-Rinow theorem, which givesthe relation between geodesic completeness and metric completeness.
Theorem 10.6 (Hopf-Rinow) The following assertions are equivalent:(c) M is complete as a metric space.
(a) there exists a p ∈ M such that expp has domain TpM ;(b) exp has domain TM ;
We will not give the proof of this theorem here. The interested reader isreferred to the lecture notes by E.J.N. Looijenga, which can be found on thewebsite for the course.
11 Gauss curvature of surfaces
In this section we assume that (M, g) is a Riemannian manifold, and that N isa smooth submanifold of M. The induced Riemannian metric on N is denotedby h. Thus, for all x ∈ N the positive definite inner product hx is the restrictionof the positive definite inner product gx to the subspace TxN of TxM. We willdenote the Levi-Civita connection associated to g by ∇, and the one associatedto h by ∇h.
In Section 4, text below Theorem 4.5, we defined the restriction of a vectorbundle to a smooth submanifold. The restriction TM |N of the tangent bundleof M to N is a particular example. It is a vector bundle of rank equal to thedimension of M ; its fiber above a point x ∈ N equals TxM. The inclusion mapN → M induces a natural injective morphism of vector bundles TN → TM |N ,over N. Fiberwise this morphism gives the natural injection TxN → TxM, forx ∈ N.
We define TxN⊥ to be the gx-orthocomplement of TxN in TxM, for x ∈N. The union TN⊥ of the spaces TxN⊥ form a subbundle of TM |N , calledthe normal bundle of N relative to M. The restricted tangent bundle TM |Ndecomposes as the following direct sum of vector bundles:
TM |N = TN ⊕ TN⊥.
The associate projection map π : TM |N → TN is a morphism of vector bundles.For each x ∈ N the map πx : TxM → TxN is the orthogonal projection relativeto the inner product gx.
In Section 4 we also defined the restriction of a connection to a submanifold.In particular, the Levi-Civita connection ∇ of (M, g) may be restricted to theto the restricted bundle TM |N . The restricted connection is denoted ∇N . IfX, Y are smooth vector fields on N which extend to smooth vector fields Xand Y on M, then by Corollary 4.7 it follows that
∇NXY = ∇X Y |N ;
note however that the restriction on the right is a section of TM |N , but notnecessarily a section of TN.
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Lemma 11.1 The Levi-Civita connection ∇h is given by
∇hXY = π ∇N
XY,
for X, Y ∈ X(N).
Proof: Since the assertion is of a local nature, we may assume that M,N aresuch that every vector field X on N extends to a smooth vector field X on M.We must show that π ∇N is a torsion free affine connection, for which h iscovariantly constant. That π ∇N is an affine connection is readily verified.
Let X, Y ∈ X(N) be smooth vector fields on N with smooth extensionsX, Y to M. Then
π ∇NXY − π ∇h
Y X − [X, Y ] = π (∇NXY −∇N
Y X − [X, Y ])= π (∇X Y −∇Y X − [X, Y ])|N= π T∇(X, Y )|N = 0.
This shows that π ∇N is torsion free.To see that h is covariantly constant for π ∇N , let X, Y,X be smooth vector
fields on N, and let X, Y , Z be extensions to smooth vector fields on M. Then
Zh(X, Y ) = Zg(X, Y )|N = g(∇ZX, Y )|N + g(X,∇Z Y )|N= h(∇N
Z X, Y ) + h(X,∇NZ Y )
= h(π ∇NZ X, Y ) + h(X, π ∇N
Z Y ).
We will now describe the relation between the curvature tensor R of (M, g)and the curvature tensor Rh of (N,h). The key ingredient for the comparisonis the so-called second fundamental form.
Given X, Y ∈ X(N) we define
H(X, Y ) := ∇NXY −∇h
XY = ∇NXY − π ∇N
XY.
Then H(X, Y ) is a section of the normal bundle TN⊥. The map H : X(N) ×X(N) → Γ(TN⊥) is called the second fundamental form. It is readily verifiedto be bilinear over the ring C∞(N), so that in fact H(X, Y )x only dependson the values Xx, Yx, for x ∈ N. Thus, H defines vector bundle morphismTN ⊗ TN → TN⊥.
Lemma 11.2 H is symmetric.
Proof: Since the assertion is of a local nature, we may assume that everyX ∈ X(N) extends to a vector field X ∈ X(M). Then
H(X, Y )−H(Y, X) = ∇NXY −∇N
Y X − (∇hXY −∇h
XY )= (∇X Y −∇Y X)|N − π (∇X Y −∇Y X)|N= [X, Y ]|N − π [X, Y ]|N= [X, Y ]− π [X, Y ]= [X, Y ]− [X, Y ] = 0.
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In the following we will denote the curvature associated to (M, g) by R andthe curvature associated to (N,h) by Rh. The relation between the two may beexpressed by means of the second fundamental form.
Lemma 11.3 Let X, Y, V, W ∈ X(N). Then
h(Rh(X, Y )V,W )= g(R(X, Y )V,W )− g(H(X, V ),H(Y, W )) + g(H(Y, V ),H(X, W )).
Proof: From ∇hY V = ∇N
Y V −H(Y, V ) it follows that
∇hX∇h
Y V = π ∇NX(∇N
Y V −H(Y, V )),
hence
h(∇hX∇h
Y V,W ) = g(∇NX∇N
Y V,W )− g(∇NXH(Y, V )),W )
= g(∇NX∇N
Y V,W ) + g(H(Y, V )),∇NXW )
= g(∇NX∇N
Y V,W ) + g(H(Y, V )),H(X, W )).
Here the second equality follows from the fact that g(H(Y, V ),W ) = 0 andthat the restriction of g to TM |N is covariantly constant with respect to ∇N .From the equality obtained we may subtract the similar equality with X andY interchanged. This gives
h(Rh(X, Y )V,W ) + h(∇h[X,Y ]V,W ) =
= g(R(X, Y )V,W ) + g(∇N[X,Y ]V,W ) +
+ g(H(Y, V )),H(X, W ))− g(H(X, V )),H(Y, W )).
We now use that
h(∇h[X,Y ]V,W ) = g(∇h
[X,Y ]V,W ) = g(∇N[X,Y ]V,W )
to complete the proof.
We will now apply the above to the special case that N is a smooth twodimensional submanifold (surface) of three dimensional Euclidean space E3, andthat h is the restriction of the Euclidean metric to N.
Corollary 11.4 Let N be a smooth submanifold of Euclidean space E3, equippedwith the restricted metric h. Then for all X, Y, V, W ∈ X(N),
−h(Rh(X, Y )V,W ) = 〈H(X, V ) , H(Y, W )〉 − 〈H(Y, V ) , H(X, W )〉.
Proof: This is an immediate consequence of the fact that R = 0 and the factg is given by the standard inner product 〈 · , · 〉.
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Let a ∈ N be given and let U be an open neighborhood of a in N diffeo-morphic to a disc. Then there exists a smooth local section ν : U → TU⊥ with‖ν‖ = 1. Of course, ν is uniquely determined up to sign. We agree to define
Hx(Xx, Yx) := 〈Hx(Xx, Yx) , ν(x)〉,
for x ∈ U and Xx, Yx ∈ TxN.The unit normal ν may be viewed as a smooth map from U to the unit
sphere S in E3. Accordingly, for every x ∈ U, the derivative dν(x) is a linearmap TxN → Tν(x)S. Now TxN ' ν(x)⊥ ' Tν(x)S, so that dν(x) may be viewedas a linear map TxN → TxN. The determinant of this map is called the Gaußcurvature of N at the point x, and denoted by K(x). Here we note that K(x)is independent of the sign of ν(x). This implies that the local definitions of Kare compatible and the restrictions of a uniquely defined smooth map N → R,called the Gauß curvature of N.
Lemma 11.5 Let x ∈ N and v, w ∈ TxN. Then
Hx(v, w) = −〈dν(x)v , w〉.
Proof: Let V,W be vector fields on N such that Vx = v and Wx = w. Then itfollows from the fact that 〈W , ν〉 = 0 that
0 = 〈∇NV W , ν〉+ 〈W , ∇N
V ν〉 = 〈H(V,W ) , ν〉+ 〈W , dν(V )〉.
Evaluation in x yields the desired identity.
We note that it follows from this lemma that dν(x) : TxN → TxN is sym-metric with respect to the restriction hx of the inner product of E3. Hence dν(x)diagonalizes with real eigenvalues. If these are different and non-zero, they arecalled the principal curvatures of N at x; the corresponding eigenspaces arecalled the principal axes. Note that these are perpendicular.
The Gauß curvature may be expressed in terms of the curvature tensor asfollows.
Corollary 11.6 Let x ∈ N and let v, w be an orthonormal basis of TxN. Then
K(x) = −hx(Rh(v, w)v, w). (19)
Proof: The determinant of dν(x) equals
K(x) = 〈dν(x)v , v〉〈dν(x)w , w〉 − 〈dν(v) , w〉〈dν(w) , v〉= Hx(v, v)Hx(w,w)−Hx(v, w)Hx(w, v)= 〈Hx(v, v) , Hx(w,w)〉 − 〈Hx(v, w) , Hx(w, v)〉= −hx(Rh
x(v, w)v, w).
It follows from the above result that the Gauß curvature, although definedby using the isometric embedding of (N,h) in E3, is really an intrinsic notionof the Riemannian manifold (M,h). In particular, it does not depend on theembedding used. This was discovered by Gauß who found the result so strikinghe named it the Theorema Egregium.
In view of the Theorema Egregium, we may extend the definition of Gaußcurvature to any two dimensional manifold by using the formula (19.
44
12 Several notions of curvature
In this section we assume that (M, g) is a pseudo-Riemannian manifold. Letx ∈ M, then Rx : TxM ⊗TxM → End(TxM) is anti-symmetric in its argument,hence induces a map Rx : ∧2Tx → End(TxM). It follows from Lemma 7.16 (b)that
(X ∧ Y, V ∧W ) 7→ gx(Rx(X, Y )V,W )
extends to a well-defined bilinear map ∧2TxM × ∧2TxM → R. Moreover, from(c) of the mentioned lemma it follows that this bilinear map is symmetric.Finally, in view of the Bianchi identity (a), it satisfies
β(v1 ∧ v2, v3 ∧ v4) + β(v2 ∧ v3, v1 ∧ v4) = β(v1 ∧ v3, v2 ∧ v4) (20)
for all v1, v2, v3, v4 ∈ TxM.
Lemma 12.1 Let V be a finite dimensional real linear space, and let β :∧2V × ∧2V → R be a symmetric bilinear form such that (20) holds for allv1, . . . , v4 ∈ V. Then β is completely determined by the values β(v ∧ w, v ∧ w),for v, w ∈ V.
Exercise 12.2 The goal of this exercise is to prove Lemma 12.1. Let β be aform as above. Assume that β(v ∧ w, v ∧ w) is known for all v, w ∈ V. Then itsuffices to show that β is known.
(a) Let v1, v2, v3 ∈ V. Show that β(v1 ∧ v2, v2 ∧ v3) can be written as a linearcombination of β((v1 + v3) ∧ v2, (v1 + v3) ∧ v2), β(v1 ∧ v2, v1 ∧ v2) andβ(v3 ∧ v2, v3 ∧ v2), with rational coefficients independent of the particularchoice of v1, v2, v3.
(b) Given a vector λ ∈ Z4 we define the linear map λ : V 4 → V by λ(v1, . . . , v4) =∑i λvi. Show that there exists a finite sequence of integers m1, . . . ,mr and
for every 1 ≤ j ≤ r vectors λj , µj ∈ Z4 such that for all v1, v2, v3, v4 ∈ Vwe have
β(v1 ∧ v2, v3 ∧ v4) =14
∑j
mj β(λj(v) ∧ µj(v), λj(v) ∧ µj(v)).
Hint: expand β((v1 + v2) ∧ (v3 + v4)).
(c) Complete the proof of the lemma.
It follows from the lemma above that the curvature tensor at x ∈ M is com-pletely determined by the values gx(Rx(v, w)v, w), for v, w ∈ TxM.
Sectional curvature We now assume that (M, g) is Riemannian, i.e., g ispositive definite. If σ is a two-dimensional linear subspace of TxM, for x ∈ M,then ∧2σ is 1-dimensional, hence has two elements of length 1 for the metricinduced by gx. If v, w form an orthonormal basis of σ, then v ∧ w has length 1in ∧2σ, so that v ∧w and w ∧ v form the elements of length 1 in ∧2σ. It follows
45
from this that the endomorphism Rx(v, w) ∈ End(TxM) is uniquely determinedby σ, up to sign. Moreover, the scalar
K(σ) = −gx(Rx(v, w)v, w)
does not depend on the particular choice of orthonormal basis, and is uniquelydetermined by σ. In case M is two-dimensional, the scalar K(TxM) equals theGauß curvature of M at x. In general, K(σ) is called the sectional curvature ofM along σ.
Remark 12.3 It follows from Lemma 12.1 that the curvature tensor Rx :∧2TxM → End(TxM) is completely determined by the sectional curvaturesK(σ), with σ a two-dimensional linear subspace of TxM.
Exercise 12.4 Let (M, g) be a Riemannian manifold, let x ∈ M and let σ bea two dimensional subspace of TxM. Show that for any basis v, w of σ we have
K(σ) = − gx(Rx(v, w)v, w)gx(v, v)gx(w,w)− gx(v, w)2
.
Components Let x1, . . . , xn be a system of local coordinates on an openset U ⊂ M. Let ∂1, . . . , ∂n be the associated vector fields, and dx1, . . . , dxn theassociated one forms over U. Then ∂1, . . . , ∂n is a frame for TU and dx1, . . . , dxn
is the dual frame of T ∗U. We agree to write
Rlijk = dxl(R(∂i, ∂j)∂k),
so thatR = Rl
ijkdxi ⊗ dxj ⊗ dxk ⊗ ∂l.
Here we agree to use the Einstein summation convention. Each time a symboloccurs both as a superscript and as a subscript, we sum over its full range.Thus, with summation symbols the above may be written as
R =∑
1≤i,j,k,l≤n
Rlijkdxi ⊗ dxj ⊗ dxk ⊗ ∂l.
In terms of components, the anti-symmetry of the curvature tensor becomes
Rlijk = −Rl
jik.
The Bianchi identity (see Lemma 7.16 (a)) may be formulated as
Rlijk + Rl
jki + Rlkij = 0.
The tensor R : (X, Y, V, W ) 7→ g(R(X, Y )V,W ) has components
Rijkm = gmlRlijk.
Its symmetry and anti-symmetry properties are now described by
Rijkm = −Rjikm = −Rijmk and Rijkm = Rkmij .
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In terms of the component notation, the symmetry and anti-symmetry prop-erties of the curvature tensor may be reformulated as
Rlijk = −Rl
jik
Ricci curvature We return to the situation that (M, g) is a pseudo-Riemannianmanifold. The Ricci curvature of M at a point x ∈ M is defined to be the bi-linear form Ricx : TxM × TxM → R given by
Ricx(X, Y ) = tr [ Rx(X, · )Y ].
In terms of components of the curvature tensor, the Ricci tensor is given by
Ricik = Rjijk,
where again the Einstein convention has been used to suppress the summationsymbol. We note that the Ricci tensor is symmetric. Indeed, if X, Y ∈ TxM,and if e1, . . . , en is a basis of TxM such that gx(ei, ej) = εiδij , with εi = ±1,then
Ric(X, Y ) =∑
j
[R(X, ej)Y ]j =∑
j
εjg(R(X, ej)Y, ej)
=∑
j
εj(R(Y, ej)X, ej) = Ric(Y, X).
in view of Lemma 7.16 (c).If g is positive definite, and e ∈ TxM a vector of unit length, let e =
e1, . . . , en be an extension to an orthonormal basis of TxM. Then
Ricx(e, e) =∑
i
g(R(e, ei)e, ei) =∑
i
K(σ1i),
where σij = spanei, ej. Thus, Ricx(e, e) is the sum of the sectional curvaturesalong the coordinate planes which contain e.
Scalar curvature In general, if (M, g) is pseudo-Riemannian, we define thelinear map Ricx : TxM → TxM by Ricx = g−1
x Ricx, or,
gx(Ricx(X), Y ) = Ricx(X, Y ),
for X, Y ∈ TxM. The components of this tensor are given by
Ricil = glkRicik.
The scalar curvature K(x) at the point x ∈ M is defined by contraction:
K(x) = tr (Ricx).
In components:K = gikRicik = gikRj
ijk.
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If g is positive definite, and x ∈ M, let e1, . . . , en be an orthonormal basis forTxM. Then it follows that
K(x) =∑ij
gx(Rx(ei, ej)ei, ej) = 2∑i<j
Kx(σij),
so that the scalar curvature equals twice the sum of the sectional curvaturesalong the coordinate hyperplanes in TxM.
Exercise 12.5 Let (M, g) be a 3-dimensional Riemannian manifold. Thepurpose of this exercise is to show that the sectional curvature at a point x ∈ Mcan be retrieved from the Ricci form Ricx : TxM × TxM → R (and the metricgx). (Thus, also the full curvature tensor can be retrieved from the Ricci form.)The three dimensionality of M is a crucial assumption.
(a) The metric gx on TxM induces a linear isomorphism TxM → TxM∗ .The dual metric g∗x on TxM∗ is defined to be the unique inner product forwhich this isomorphism becomes an isometry. Let f1, f2, f3 be a basis forTxM and let f1, f2, f3 be the dual basis for T ∗
xM. Show that the followingassertions are equivalent:
• f1, f2, f3 is orthonormal for g;
• g(fj) = f j for alle j;
• f1, f2, f3 is orthonormal for g∗.
(b) Equip TxM with an orientation, and let ω ∈ ∧3TxM be the associatedunit element. This means that ω = f1∧f2∧f3 for any positively orientedorthonormal basis. We define the pairing γ : ∧2TxM × TxM → R byλ ∧X = γ(λ, X)ω. Show that γ is non-degenerate, hence induces an iso-morphism ∧2TxM → TxM∗. Show that for f1, f2, f3 a positively orientedorthonormal basis of TxM∗ we have
γ(f1 ∧ f2) = f3.
Determine γ(f2 ∧ f3), γ(f3 ∧ f1).
(c) Via the isomorphism γ the form β may be transfered to a bilinear form
β∗ : T ∗xM × T ∗
xM → R.
Show that β∗ diagonalizes with respect to a g∗-orthonormal basis e1, e2, e3
of TxM∗. Let e1, e2, e3 be the associated dual basis of TxM. Show thatβ diagonalizes with respect to the basis e1 ∧ e2, e2 ∧ e3, and e3 ∧ e1 of∧2TxM.
(d) Show that for each i 6= j the value β(ei∧ej , ei∧ej) is a linear combinationof Ricx(ek, ek), k = 1, 2, 3. Hint: solve a system of three linear equationswith three unknowns.
(e) Show that all sectional curvature at x can be retrieved from the Ricciform.
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Exercise 12.6 We assume that M is a three dimensional connected Rieman-nian manifold, whose Ricci form is proportional to the metric g, i.e., there existsa smooth function f : M → R such that Ricx = f(x) · g for all x ∈ M. Thepurpose of this exercise is to show that f is constant.
(a) Let e1, e2, e3 be an orthonormal basis of TxM. Show that for every X ∈TxM,
Rx(X, e1)e1 + Rx(X, e2)e2 + Rx(X, e3)e3 = f(x)X.
(b) Write A = Rx(e1, e2), B = Rx(e2, e3), C = Rx(e3, e1). Argue that thematrices of A,B, C with respect to the basis e1, e2, e3 are anti-symmetric.Use this in combination with the previous item to conclude that A13 =A23 = B12 = B13 = C12 = C23 = 0 and that A12 = B23 = C31 = 1
2f(x).
(c) Show that for all U, V, X ∈ TxM,
Rx(U, V )X =12f(x) [gx(V,X)U − gx(U,X)V ].
Hint: use that both sides are bilinear and anti-symmetric with respect toU, V and test the identity at basis elements.
(d) As an intermezzo, conclude that for each plane σ ∈ TxM the sectionalcurvature K(σ) equals −1
2f(x).
The rest of the exercise is more complicated, and relies on the so-called secondBianchi identity, which we shall not prove here. It says that for all vector fieldsU, V,W, one has
∇UR(V,W ) +∇V R(W,U) +∇W R(U, V ) = 0.
(e) Let R be the tensor given by R(U, V ) = g(V, · )U − g(U, · )V, for vectorfields U, V. Then the conclusion of item (e) may be rephrased as
R(U, V ) =12f ·R(U, V ).
Show that the tensor R satisfies the analogue of the second Bianchiidentity for all commuting vector fields U, V,W. Conclude that
Uf ·R(V,W ) + V f ·R(W,U) + Wf ·R(U, V ) = 0
for all commuting vector fields U, V,W. Conclude that the same identityholds at a fixed point x for any choice of vectors U, V, W ∈ TxM.
(f) Conclude that df(x) = 0. Hint: choose an orthonormal basis U, V,Wfor TxM and apply the above identity to U. Finally, conclude that f isconstant.
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13 Exercises
Exercise 13.1 Let p : E → M be a rank k vector bundle over a smoothmanifold M. Let q : E∗ → M be the dual bundle. Thus, for each x ∈ M thefiber q−1(x) equals the dual E∗
x of Ex. The vector bundle structure of E∗ isdetermined by the requirement that for each local frame e1, . . . , ek of E over anopen set U ⊂ M, the maps ej : U → E∗ determined by ej(x)(ei(x)) = δj
i forma local frame of E∗ over U.
(a) Given s ∈ Γ(E) and s∗ ∈ Γ(E∗) we define the function 〈s , s∗〉 : M → Rby 〈s , s∗〉(x) = s∗(x)(s(x)). Show that 〈s , s∗〉 is smooth.
(b) Let ∇ be a connection on E. Show that there exists a unique connection∇∗ on E∗ such that
〈∇Xs , s∗〉+ 〈s , ∇∗Xs∗〉 = X〈s , s∗〉
for all s ∈ Γ(E), s∗ ∈ Γ(E∗) and X ∈ X(M).
Let p′ : E′ → M be a rank l-vector bundle over M. The tensor product E ⊗E′
is a rank kl-vector bundle whose fiber over x ∈ M equals Ex ⊗ E′x. If U is an
open subset of M and if e1, . . . , ek and f1, . . . , fl are local frames of E and E′
over U, then ei ⊗ fj , (1 ≤ i ≤ k, 1 ≤ j ≤ l) is a local frame of E ⊗ E′ over U.
(c) Let ∇ be a connection on E and let ∇′ be a connection on E′. Show thatthere exists a unique connection ∇′′ on E′′ = E ⊗ E′ such that
∇′′X(s⊗ s′) = ∇Xs⊗ s′ + s⊗∇Xs′
for all s ∈ Γ(E), s′ ∈ Γ(E′) and X ∈ X(M).(d) Let M be a manifold whose tangent bundle is equipped with a connection
∇. Let ∇∗ be the induced connection on T ∗M and let ∇′′ be the inducedconnection on T ∗M ⊗T ∗M. Let g be a pseudo-Riemannian metric on M.Then g may be viewed as a section of T ∗M ⊗ T ∗M. Show that g is flat ifand only if
∇′′Xg = 0, (∀X ∈ X(M)).
Exercise 13.2 Let M be a smooth surface in Euclidean space E3 (R3 equippedwith the standard inner product). Let g be the restricted Riemannian metricon M. Thus, gx is the restriction of 〈· , ·〉 to TxM ⊂ E3, for every x ∈ M. Foreach x ∈ M, let πx : E3 → TxM denote the orthogonal projection.
(a) Show that the Levi-Civita connection on TM is given by the formula
∇XY (x) = πxdY (x)X(x),
for X, Y ∈ X(M). Here dY (x) denotes the derivative of Y viewed as amap M → E3.
(b) Let c : I → M be a C1-curve. Show that for every vector field t 7→ X(t)along c,
DX
dt(t) = πc(t)(X
′(t)).
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(c) Show that a C2-curve γ : J → M is a geodesic if and only if
γ′′(t) ⊥ Tγ(t)M, (∀t ∈ J).
Exercise 13.3 The purpose of this exercise is to exhibit a different realisationof two dimensional hyperbolic space H2. Let D be the unit disk in R2, consistingof the points (x, y) with x2 + y2 < 1. Put r = r(x, y) =
√x2 + y2 and define
the map f : R2 → R3 by
f(x, y) =1
1− r(x, y)2(2x, 2y, 1 + r(x, y)2).
(a) Show that the pull-back of the metric dx21 + dx2
2 − dx23 on R3 under f
equals
f∗(dx21 + dx2
2 − dx23) = 4
dx2 + dy2
(1− r(x, y)2)2(∗)
To keep the amount of calculation limited, it is a good idea to introducethe function s = 1 − r2, and to keep working with s and the form ds aslong as possible.
(b) Show that f is an isometry from D, equipped with the Riemannian metricon the right-hand side of (*), onto H2.
The disk D, equipped with the given Riemannian metric, is called the Poincaredisk.
Exercise 13.4 Let (M, g) be a compact Riemannian manifold.
(a) Assume that r > 0 and that γ : [0, r) → M is a geodesic. Show that thesequence xn := γ(r− 1
n) is a Cauchy sequence for the metric d determinedby g.
(b) Show that limt↑r γ(t) exists.
(c) Show that every geodesic γ : I → M extends to a geodesic R → M.
Exercise 13.5 In this exercise we assume that (M, g) and (N,h) are Rieman-nian manifolds, and that ϕ : M → N is an isometry.
(a) Let a ∈ M and v ∈ TaM. Let γv; Iv → M be the associated maximalgeodesic with initial point a and initial tangent vector v. Show that ϕ γv
is a geodesic in N. In fact, let w = dϕ(a)v. Show that ϕ γv = γw.
(b) Let a ∈ M and put b = ϕ(a). Show that dϕ(a) maps the domain Ωa
of expa onto the domain Ωb of expb . Moreover, show that the followingdiagram commutes:
Mϕ−→ N
expa ↑ ↑expb
Ωadϕ(a)−→ TbN.
Let now σ be an isometry of M onto itself. Let S := x ∈ M | σ(x) = x bethe set of fixed points for σ.
51
(c) For a ∈ S, let da = dim ker(dσ(a) − I). Show that at the point a the setS is a submanifold of M of dimension da.
(d) Let a ∈ S and v ∈ TaS. Show that the geodesic γv : Iv → M is containedin S. A submanifold of M with this property is called totally geodesic inM.
(e) Let now M be the unit sphere in E3. Show that the geodesics of M areprecisely the curves traversing big circles with constant velocity.
Exercise 13.6 We assume that p : E → M is a rank k vector bundle overthe manifold M, equipped with a connection. Given an open subset U ⊂ Mwe agree to write Ωp(U,E) for the space of E-valued p-forms on U. By this wemean the space of smooth sections of the bundle E⊗∧pT ∗M. In particular, wewrite Ω0(U,E) for the space of smooth sections of E over U. We agree to writeΩp(E) for Ωp(M,E).
(a) Show that the connection may be viewed as a map D : Ω0(E) → Ω1(E)which (1) is locally defined, (2) is R-linear, and (3) satisfies the Leibnizrule
D(fs) = fDs + s⊗ df
for all s ∈ Γ(E) and f ∈ C∞(M).
We agree to write Ω(E) for the direct sum of the spaces Ωp(E), for p ≥ 0. ThenD may be viewed as a map Ω0(E) → Ω(E).
(b) Show that D has a unique extension to a map Ω(E) → Ω(E) which (1)is locally defined, (2) is R-linear, and (3) satisfies the generalized Leibnizrule
D(s⊗ α) = Ds ∧ α + s⊗ dα,
for s ∈ E(E) and α ∈ Ωp(M). Hint: use a local frame for E.
(c) Show that there exists a unique bilinear map ∧ : Ω(End(E)) × Ω(E) →Ω(E) such that
(A⊗ α) ∧ (b⊗ β) = Ab⊗ (α ∧ β)
for all A ∈ Γ(End(E)), b ∈ Γ(E), and α, β ∈ Ω(M).
(d) We recall from Exercise 13.1 that the connection of E induces a connec-tion on End(E) ' E ⊗ E∗ such that the associated operator D satisfiesD(Aa) = DA ∧ a + A ∧Da, for all A ∈ Γ(End(E)) and a ∈ Γ(E). Showthat
D(Φ ∧ λ) = DΦ ∧ λ + (−1)pΦ ∧Dλ
for Φ ∈ Ωp(End(E)) and λ ∈ Ω(E).
(e) The curvature form R of the connection on E may be viewed as an elementof Ω2(End(E)). Show that for every ω ∈ Ω(E) we have
DD(ω) = R ∧ ω.
52
(d) Show that D(R) = 0. (This is called the Bianchi identity.)
Exercise 13.7 Let (M, g) be a Riemannian manifold. A submanifold N ⊂ Mis said to be totally geodesic if for any a ∈ N and v ∈ TaN the maximal geodesicγv is entirely contained in N. Let h be the restriction metric on N. Show thatthe following three assertions are equivalent.
(a) N is totally geodesic.(b) For every v ∈ TN, the maximal geodesic γN
v in N relative to h equals themaximal geodesic γv in M.
(c) Every geodesic in N relative to h is a geodesic in M.
Exercise 13.8 Let (M, g) be a Riemannian manifold, and let N be a subman-ifold of M. Let h be the restriction metric on N. Let c : I → N be a C1-curve,and let X : I → TN be a vector field over c. We denote by DX/dt the covariantderivative of X along c with respect to the Levi-Civita connection associatedwith g. Moreover, by DhX/dt we denote the covariant derivative of X along cwith respect to the Levi-Civita connection associated with h.
(a) Show thatDhX
dt(t) = πc(t)
DX
dt(t),
where πx denotes the orthogonal projection TxM → TxN, for x ∈ N.Hint: first prove this for a vector field of the form X(t) = Y (c(t)), withY ∈ X(N). Then prove it for vector fields of the form X(t) = f(t)Y (c(t)),with f ∈ C∞(N).
(b) Show that
H(c′(t), X(t)) =DX
dt(t)− DhX
dt(t).
(c) Show that c : I → N is a geodesic in N if and only if
Dc′(t)dt
⊥ Tc(t)N
for all t ∈ I.
(d) The submanifold N is said to be totally geodesic if and only if for allv ∈ TN the maximal geodesic γv : Iv → M is entirely contained in N.Show that the following assertions are equivalent
(1) N is totally geodesic;
(2) the second fundamental form H is identically zero.
Exercise 13.9 Let S be the sphere of center 0 and radius R > 0 in E3.Determine the Gauß curvature of S. Give an a priori reason why the Gaußcurvature is constant.
Exercise 13.10 Let R > 0. We consider the cilinder C in E3 given by theequation x2 + y2 = R2.
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(a) Show that the Gauß curvature of C is zero.
(b) Show that the curvature form of C is zero.
(c) Specify an isometry from R× (−π, π) onto an open subset of C and provethe correctness of your answer.
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