Vector bundles and connections

Kim A. Frøyshov

September 6, 2018

1 Vector bundles

In the following, let K = R or C. Smooth manifolds are assumed to be second countable. For a smooth manifold M we denote by D(M, K) the ring of smooth functions M → K, and we set D(M) := D(M, R). By definition, a smooth K-vector bundle of rank r over a smooth manifold M consists of • a smooth manifold E called the total space of the bundle,

• a smooth map π : E → M called the projection,

−1 • for each p ∈ M a K-vector space structure on the fibre Ep := π (p), such that the axiom of local triviality holds: For each p ∈ M there should r exist an open neighbourhood U of p and a diffeomorphism Φ : U × K → π−1U such that

r • π ◦ Φ(q, v) = q for all (q, v) ∈ U × K , • for each q ∈ U the map

r Φq : K → Eq, v 7→ Φ(q, v) is a linear isomorphism. Such a diffeomorphism Φ is called a local trivialization of E. If r = 1 then E is called a (real or complex) line bundle over M.

Examples (i) The product bundle

M × V → M, (q, v) 7→ q, where V is any finite-dimensional K-vector space.

1 (ii) The tangent bundle TM → M. (iii) If π : E → M is a vector bundle and N ⊂ M a submanifold, then −1 E|N := π N is a vector bundle over N. (iv) More generally, if π : E → M is a vector bundle and f : N → M a smooth map, then

f ∗E := {(q, v) ∈ N × E | f(q) = π(v)} is a vector bundle over N called the pull-back of E by f. (v) Many operations on vector spaces can be applied fibrewise on a vector bundle to produce new vector bundles. For instance, to any vector space V we can associate its dual space V ∗. Applying this operation on each fibre of a vector bundle E → M yields the dual bundle E∗ → M whose fibre over ∗ ∗ a point p ∈ M is (Ep) . In the case E = TM, the dual bundle T M is called the cotangent bundle of M. More details about this type of construction will be given in Section 2.

Let E → M be a vector bundle of rank r. By a subbundle of E of rank s we mean a subset E0 ⊂ E such that for every point p ∈ M there exist a r −1 local trivialization Φ : U × K → π U of E around p and an s–dimensional r linear subspace V ⊂ K such that

Φ(U × V ) = π−1U ∩ E0.

Then E0 is in a natural way a vector bundle over M.

Example If N ⊂ M is a submanifold then the tangent bundle TN is a subbundle of TM|N . By a section of a vector bundle π : E → M we mean a smooth map s : M → E such that π ◦ s = IdM . The vector space Γ(E) of all sections of E is a module over the ring D(M, K). Addition in Γ(E) and multiplication by functions are defined pointwise by

(s + t)(p) := s(p) + t(p), (fs)(p) := f(p) s(p) for s, t ∈ Γ(E) and f ∈ D(M, K).

By definition, a vector field on M is a section of TM. We denote by X (M) = Γ(TM) the D(M)–module of all vector fields on M.

2 0 0 0 0 Let π : E → M and π : E → M be K-vector bundles and f : M → M a map. A smooth map F : E → E0 is called a bundle homomorphism 0 over f if F maps each fibre Ep linearly into Ef(p). (This implies that f is smooth.) If in addition M = M 0 and f is the identity map then F is called a bundle homomorphism. A bundle homomorphism F : E → E0 which is also a diffeomorphism is called a bundle isomorphism. A vector bundle is called trivial if it is isomorphic to a product bundle. An isomorphism E → E is called a bundle automorphism or gauge transformation.

Examples (i) If f : M → M 0 is a smooth map between manifolds then the tangent map T f : TM → TM 0 is a bundle homomorphism over f. (ii) If π : E → M is a vector bundle and f : N → M a smooth map then there is a canonical bundle homomorphism f ∗E → E obtained by restricting the projection N × E → E to f ∗E. r s (iii) Let E = M × K and E = M × K be product bundles over M. Then a bundle homomorphism E → E0 has the form r s M × K → M × K , (p, v) 7→ (p, α(p)v) for some smooth map α from M into the space M(s × r, K) of s × r matrices with entries in K.

Let E → M be a K–bundle of rank r and let U ⊂ M be an open subset. By a (local) frame for E over U we mean an r–tuple (s1, . . . , sr) of sections of E|U such that (s1(p), . . . , sr(p)) is a basis for Ep for all p ∈ U. If U = M then (s1, . . . , sr) is called a global frame. In that case the map r r X j M × K → E, (p, v) 7→ v sj(p) j=1 is a bijective bundle homomorphism and therefore an isomorphism. Con- versely, such an isomorphism clearly gives rise to a global frame.

2 Operations on vector bundles

Let C be the category of finite-dimensional K–vector spaces and linear iso- morphisms, and let F : C → C be a covariant functor. In other words, F is a rule that associates to any vector space V a vector space F(V ) and to any isomorphism A : V → W between vector spaces an isomorphism F(A): F(V ) → F(W ), such that the following hold:

3 • If V →A W →B X are isomorphisms then F(B ◦ A) = F(B) ◦ F(A).

•F(IdV ) = IdF(V ). Such a functor is called smooth if for any vector space V the map

GL(V ) → GL(F(V )),A 7→ F(A) is smooth (and therefore a homomorphism of Lie groups).

Example Given a non-negative integer k, let F(V ) := ΛkV ∗ be the vector space of alternating k–forms on V . If A : V → W is an isomorphism let A∗ : W ∗ → V ∗ be the dual isomorphism and set

F(A) := (A∗)−1 = (A−1)∗.

Given a smooth functor F : C → C and a vector bundle π : E → M we construct a new vector bundleπ ˜ : E˜ → M as follows. (We will sometimes write F(E) := E˜.) As a set, [ E˜ := {p} × F(Ep), p∈M andπ ˜(p, w) := p. For any local trivialization

r −1 Φ: U × K → π U the map ˜ r −1 Φ: U × F(K ) → π˜ U, (p, w) 7→ (p, F(Φp)w) is a bijection. Moreover, if Ψ is another local trivialization of E defined over an open subset V ⊂ M then

˜ −1 ˜ −1 Ψ Φ(p, w) = (p, F(Ψp Φp)w)

r is an automorphism of the product bundle (U ∩ V ) × F(K ) over U ∩ V . Hence, E˜ has a unique structure of vector bundle over M with projectionπ ˜ such that each map Φ˜ defines a local trivialization of E˜ (after fixing a basis r for F(K )). Examples (i) Taking F(V ) := ΛkV ∗ we obtain a vector bundle ΛkE∗. Sim- P k ∗ ilarly, taking F(V ) to be the whole exterior algebra Λ(V ) := k≥0 Λ V yields a vector bundle (in fact, bundle of real algebras) ΛE∗ which contains each ΛkE∗ as a subbundle.

4 (i) For E = TM, the bundles ΛkE∗ and ΛE∗ will be denoted by ΛkM and ΛM, respectively. Note that the sections of ΛkM are by definition the differential k–forms on M.

The above construction can also be applied to functors of several vari- ables. For instance, to any pair V,W of vector spaces we can associate the direct sum V ⊕ W , the tensor product V ⊗ W , and the space of homomor- phisms Hom(V,W ) (the latter being isomorphic to V ∗ ⊗W ). Applying these functors fibrewise to a pair E,E0 of vector bundles over M we obtain new vector bundles E ⊕E0, E ⊗E0, and Hom(E,E0) over M. The fibre of E ⊕E0 0 over a point p ∈ M is Ep ⊕ Ep, and similarly for the other examples. In the case E0 = E we obtain the endomorphism bundle End(E) := Hom(E,E). Sections of the bundle ΛkM ⊗ E are called differential k–forms on M with values in E. The space of all such forms is denoted by Ωk(M; E), or by Ωk(M; V ) in the case of a product bundle E = M × V .

3 Connections

Given a section s of a vector bundle E → M and a vector field X on M we would like to have some kind of derivative ∇X s of s with respect to X. This derivative should be a new section of E. Because there is no canonical isomorphism between the fibres of E at two different points, we are unable to define a canonical derivative of this kind. Instead, we will formulate some properties that we would like such a derivative to have. By a connection (or covariant derivative) in a K-vector bundle E → M we mean a map

∇ : X (M) × Γ(E) → Γ(E), (X, s) 7→ ∇X s such that for all X,Y ∈ X (M), s, t ∈ Γ(E), and f ∈ D(M, K) one has

(i) ∇X+Y (s) = ∇X s + ∇Y s,

(ii) ∇fX (s) = f∇X s,

(iii) ∇X (s + t) = ∇X s + ∇X t,

(iv) ∇X (fs) = (Xf)s + f∇X s. Note that if f is constant, say f ≡ α, then Xf = 0, so (iv) gives

∇X (αs) = α∇X s.

5 A section s is called covariantly constant, or parallel, if ∇X s = 0 for all vector fields X.

Example A product bundle E = M ×V → M has a canonical connection ∇ called the product connection. To define this, note that to any function h : M → V we can associate a section h˜ of E given by

h˜(p) = (p, h(p)), ˜ and any section of E has this form. Now define ∇X h to be the section corresponding to the function Xh, i.e. ˜ ∇X h := Xh.g

Theorem 3.1 Any vector bundle E → M admits a connection ∇.

Proof. Choose an open cover {Uα}α∈I of M such that E|Uα is trivial for each α. Choose a partition of unity {gα} subordinate to this cover. By the α example above, each bundle E|Uα admits a connection ∇ . Now define, for any section s of E and vector field X on M,

X α ∇X s := gα∇X (s), α

α where ∇X (s), defined initially on Uα, is extended trivially to all of M. Then ∇ clearly satisfies the first three axioms for a connection. To verify the fourth one, let f ∈ D(M, K). Then

X α ∇X (fs) = gα(Xf · s + f∇X (s)) α X α = Xf · s + f gα∇X (s) α

= Xf · s + f∇X (s).

For an arbitrary connection ∇ we will now investigate the dependence of ∇X s on the two variables X and s, beginning with s.

Proposition 3.1 Let ∇ be a connection in E → M and X a vector field on M. Then ∇X is a local operator, i.e. if a section s of E vanishes on an open subset U ⊂ M then ∇X s vanishes on U, too.

6 Proof. Suppose s|U = 0 and let p ∈ U. Choose a smooth real function f on M which is supported in U and satisfies f(p) = 1. Then fs = 0, so

0 = ∇X (fs) = Xf · s + f∇X s.

Evaluating this equation at p gives (∇X s)(p) = 0.

0 Proposition 3.2 Let E,E be K-vector bundles over M and

A : Γ(E) → Γ(E0) a D(M, K)–linear map. Then there exists a unique bundle homomorphism a : E → E0 such that As = as for all s ∈ Γ(E).

Explicitly, the assumption on A is that

A(s + t) = As + At, A(fs) = fAs for all s, t ∈ Γ(E) and f ∈ D(M, K). The main point in the proposition is that (As)(p) depends only on the value of s at p. Proof of proposition. Uniqueness follows from the fact that for any p ∈ M and v ∈ Ep there exists a section s of E with s(p) = v. To prove existence of a, let p ∈ M and v ∈ Ep. Choose a section s of E with s(p) = v and define av := (As)(p). To show that av is independent of the choice of s it suffices to verify that if t is any section of E with t(p) = 0 then (At)(p) = 0. Let r be the rank of E. Choose sections s1, . . . , sr of E which are linearly independent at every point in some neighbourhood U of p. Choose a smooth real function f on M which is supported in U and satisfies f(p) = 1. Then

X j ft = g sj j

1 r for some uniquely determined K–valued functions g , . . . , g on M that van- ish outside U. Clearly, gj(p) = 0. Applying A to both sides of the above equation we obtain X j fAt = g Asj, j and evaluating both sides of this equation at p gives (At)(p) = 0 as required.

7 Corollary 3.1 Let ∇ be a connection in E → M and s ∈ Γ(E), p ∈ M. If X,Y are vector fields on M satisfying Xp = Yp then (∇X s)(p) = (∇Y s)(p).

Proof. This follows from the proposition because for given s the map

X 7→ ∇X s is D(M)–linear. The corollary allows us to define ∇vs for any tangent vector v ∈ TpM. Namely, choose any vector field X with Xp = v and set

∇vs := (∇X s)(p).

Proposition 3.3 Connections in E are in 1-1 correspondence with R–linear maps ∇¯ :Ω0(M; E) → Ω1(M; E) satisfying ∇¯ (fs) = df ⊗ s + f∇¯ s for all f ∈ D(M) and s ∈ Γ(E). If ∇¯ is such a map then the corresponding connection ∇ is given by

∇X s = (∇¯ s)(X). (1)

Proof. It is clear that (1) defines a connection in E. To reconstruct ∇¯ from ∇, let X1,...,Xn be a frame of vector fields on an open subset U ⊂ M, and let α1, . . . , αn the dual frame of 1–forms. Then

¯ X i (∇s)|U = α ⊗ ∇Xi (s). i

Henceforth we will not distinguish between ∇ and ∇¯ . For any vector bundle E let A(E) denote the set of all connections in E. The following proposition says that A(E) is an affine space.

Theorem 3.2 Given a connection ∇ in a vector bundle E → M, the map

Ω1(M; End(E)) → A(E), a 7→ ∇ + a is a bijection.

8 Proof. It is easy to see that ∇ + a is a connection. Conversely, if ∇0 0 is any other connection in E then ∇ − ∇ is D(M, K)–linear and therefore given by a bundle homomorphism E → T ∗M ⊗ E, or equivalently, a section of the bundle E∗ ⊗ T ∗M ⊗ E = T ∗M ⊗ End(E).

Example Let E → M be a trivial vector bundle and (s1, . . . , sr) a global frame for E. Then there is a 1-1 correspondence between connections in E i and r × r matrices ω = (ωj) of K–valued 1–forms on M specified by the formula r X i ∇X sj = ωj(X)si (2) i=1 for vector fields X on M. To deduce this from the theorem, let ∇0 be the product connection in E given by ∇X sj = 0. Then any other connection in E has the form ∇0 + ω, where

1 ω ∈ Ω (M; gl(r, K)).

We call ω the connection form of ∇ with respect to the given global frame.

0 Proposition 3.4 Let E1,...,Em,E be K–vector bundles over M and

0 B : Γ(E1) × · · · × Γ(Em) → Γ(E ) a D(M, K)–multilinear map. Then there exists for each p ∈ M a K– multilinear map 0 Bp :(E1)p × · · · × (Em)p → Ep such that if sj ∈ Γ(Ej), j = 1, . . . , m then

B(s1, . . . , sm)(p) = Bp(s1(p), . . . , sm(p)). (3)

All these maps Bp together define a bundle homomorphism E1 ⊗· · ·⊗Em → E0.

Proof. If at least one of the sections sj vanishes at p then, by Proposi- tion 3.2, B(s1, . . . , sm) also vanishes at p. By repeated application of this we see that if for each j we are given a pair of sections sj, tj of Ej with the same value at p then

B(s1, . . . , sm)(p) = B(t1, . . . , tm)(p).

We can now define Bp by (3).

9 If ∇ is a connection in TM then the map

X (M) × X (M) → X (M), (X,Y ) 7→ ∇X Y − ∇Y X − [X,Y ] is easily seen to be D(M)–linear. Since it is also skew-symmetric, it is given by a TM–valued 2–form T ∈ Ω2(M; TM) called the torsion of ∇. If T = 0 then ∇ is called torsion-free. For in- stance, the formula for the Lie-bracket in local coordinates says that the n n n product connection in T R = R × R is torsion-free. Any smooth mani- fold admits a torsion-free connection in TM, as follows from Theorems 8.1 and 8.4 below. (Alternatively, one can prove this by pathing together local torsion-free connections using a partition of unity.)

4 Pull-back connections

Let π : E → M be a vector bundle and f : M˜ → M a smooth map. By a section of E along f we mean a smooth map t : M˜ → E such that π ◦ t = f. We will usually identify such a map t with the corresponding sections ˜ of the pull-back bundle f ∗E given bys ˜(p) = (p, t(p)). Note that if s is a section of E then f ∗s := s ◦ f is a section of E along f.

Proposition 4.1 Let ∇ be a connection in E → M and f : M˜ → M a smooth map. Then there exists a unique connection ∇˜ = f ∗∇ in E˜ := f ∗E such that for all s ∈ Γ(E), p ∈ M˜ , and v ∈ TpM˜ we have ˜ ∗ ˜ ∇v(f s) = ∇f∗v(s) in Ep = Ef(p), (4) ˜ where f∗ : TpM → Tf(p)M is the tangent map of f.

Proof. Uniqueness follows from the fact that if {sj} is a frame for E over ∗ −1 an open set U ⊂ M then {f sj} is a frame for E˜ over f U. We prove existence in three steps. (i) First suppose M˜ is an open subset of M and f the inclusion map, so that E˜ is just the restriction of E to M˜ . In this case the existence of ∇˜ follows from Proposition 3.1 and Corollary 3.1. (ii) Next we consider the case when E is trivial. Let (s1, . . . , sr) be a i global frame for E and (ωj) the matrix of 1–forms on M with

X i ∇Y sj = ωj(Y )si i

10 ∗ i ∗ i ˜ for vector fields Y on M. Sets ˜j := f sj andω ˜j := f ωj, and let ∇ be the unique connection in E˜ such that

˜ X i ∇X s˜j = ω˜j(X)˜si i for vector fields X on M˜ . We now verify that ∇˜ satisfies (4). Let p ∈ M˜ , v ∈ ˜ P j j TpM, and s ∈ Γ(E). Then s = j h sj for some functions h ∈ D(M, K). j j ∗ P j Set q := f(p), w := f∗v, and g := h ◦ f. Then f s = j g s˜j, and

˜ X i X i ∇vs˜j = ω˜j(v)˜si(p) = ωj(w)si(q) = ∇wsj, i i so

∗ X j j ∇˜ v(f s) = [v(g ) · s˜j(p) + g (p) · ∇˜ vs˜j] j X j j = [w(h ) · sj(q) + h (q) · ∇wsj] j

= ∇ws.

(iii) In the general case, choose an open cover {Uα}α∈I of M such that ˜ −1 E|Uα is trivial for each α. Set Uα := f Uα. By case (ii), we have a pull- back connection ∇˜ α in E˜| . By uniqueness, ∇˜ α and ∇˜ β restrict to the U˜α α same connection on U˜α ∩ U˜β for each α, β ∈ I. Hence, the connections ∇˜ patch together to give the desired connection in E˜.

Example Let E → M be a K–vector bundle with connection ∇ and γ : I → M, t 7→ γ(t) a smooth path in M, where I ⊂ R is an interval. Proposition 4.1 provides D ∗ a K–linear operator := (γ ∇) d acting on sections of E along γ with the dt dt follwing properties.

• For all smooth functions f : I → K and sections σ of E along γ one has D df Dσ (fσ) = · σ + f · . dt dt dt • If σ = f ∗s is the pull-back of a section s of E then Dσ = ∇ dγ (s). dt dt

11 5 Holonomy

Proposition 5.1 Let E → M be a K–vector bundle with connection ∇, and let γ : I → M be a smooth curve. Let t0 ∈ I and v ∈ Eγ(t0). Then there exists a unique parallel section σ of E along γ such that σ(t0) = v.

Dσ Here, σ is called parallel if dt = 0. Proof. It is a simple exercise to show that any vector bundle over an interval is trivial. Hence, there exists a global frame (σ1, . . . , σr) of the pull- ∗ D back bundle γ E. In terms of this frame, the operator dt is given by an r ×r i matrix (cj) of functions I → K such that

Dσj X = ci σ . dt j i i

1 r We are therefore seeking smooth K–valued functions f , . . . , f on the in- terval I with specified values at t0 such that ! D X X df j X 0 = f jσ = · σ + f j ci σ dt j dt j j i j j i   X df i X = + ci f j σ ,  dt j  i i j or, equivalently, df i X + ci f j = 0, i = 1, . . . , r. (5) dt j j Because this is a linear system of ordinary differential equations, it has a unique solution with given values at t0. The proposition allows us to define for any a, b ∈ I the holonomy map, or parallel transport,

b b ≈ Hola := Hola(∇, γ): Eγ(a) → Eγ(b) of ∇ along γ. This is the linear isomorphism characterized by the property that if σ is any parallel section of E along γ then

b Hola(σ(a)) = σ(b). Clearly, if a, b, c ∈ I then

c c b Hola = Holb ◦ Hola.

12 b We can exploit this property to define Hola even if γ is only piecewise smooth. Namely, if a = a0 < a1 < ··· < an = b b and γ[ai−1,ai] is smooth for i = 1, . . . , n we define Hola to be the composite of the holonomies along each subinterval, i.e.

b an a2 a1 Hola := Holan−1 ◦ · · · ◦ Hola1 ◦ Hola0 . (6) The following property expresses a connection in terms of its holonomy.

Proposition 5.2 Let ∇ be a connection in a vector bundle E → M and s a section of E. Let γ :(−, ) → M be a smooth curve, where  > 0. For − < t <  set 0 ht := Hol t (∇, γ): Eγ(t) → Eγ(0).

Then d ∇γ0(0)(s) = ht(s(γ(t))). dt 0

Note that t 7→ ht(s(γ(t))) is a curve in the finite-dimensional vector space Eγ(0), and we are differentiating this curve at 0 in the usual sense. ∗ Proof. Let (σ1, . . . , σr) be a global frame for γ E consisting of parallel ∗ P j j sections. Then γ s = j f σj for some functions f :(−, ) → K. Now,

X j ht(s(γ(t))) = f (t) · σj(0), j and ∗ Dγ s X j 0 d ∇γ0(0)(s) = (0) = (f ) (0) · σj(0) = ht(s(γ(t))). dt dt j 0

Proposition 5.3 Let ∇ be a connection in a vector bundle E over M, and F a smooth functor. Then there is a unique connection ∇˜ in F(E) such that for any path γ : I → M and a, b ∈ I one has

b ˜ b Hola(∇, γ) = F(Hola(∇, γ)) : F(Eγ(a)) → F(Eγ(b)).

Proof. The uniqueness of such a connection ∇˜ follows from Proposi- tion 5.2, which expresses a connection in terms of its holonomy. To prove ex- istence, it then suffices to consider the case of a product bundle E = M ×V . Let a ∈ Ω1(M; gl(V )) be the connection form of ∇, so that ∇ = d+a, where

13 d is the product connection. Recall that the functor F gives defines a Lie group homomorphism f : GL(V ) → GL(V˜ ), where V˜ := F(V ). Let L : gl(V ) → gl(V˜ ) be corresponding Lie algebra homomorphism, i.e. the derivative of f at the identity. We define ∇˜ to be the connection in the product bundle M × V˜ over M with connection form La ∈ Ω1(M; gl(V˜ )). To show that ∇˜ has the desired property, let γ : I → M be a curve. Define c : I → gl(V ) by

c(t) := a(γ0(t)).

If s is any section of E along γ, thought of as a map I → V , then Ds = s0 + cs, dt 0 where s is the usual derivative of s. Now fix t0 ∈ I and define u : I → GL(V ) by t u(t) := Holt0 (∇, γ). For any v ∈ V the map t 7→ u(t)v defines a parallel section of E along γ, hence D(uv) 0 = = u0v + cuv. dt Therefore, u is unique solution to the equations

0 u + cu = 0, u(t0) = IdV . We claim that the map

u˜ := f ◦ u : I → GL(V˜ ) satisfies a similar equation. Namely,

0 0 d u˜ (t) = f∗(u (t)) = −f∗(c(t)u(t)) = − f(exp(sc(t)) · u(t)) ds 0

d = − f(exp(sc(t)) · u˜(t) = −Lc(t)˜u(t). ds 0

Becauseu ˜(t0) = IdV˜ , we conclude that t ˜ u˜(t) = Holt0 (∇, γ). This proves the existence of the desired connection in F(E) in the case when E is trivial, hence also in general.

14 Proposition 5.4 Let ∇ be the connection in ΛM induced by a given con- nection in the tangent bundle TM. Then for any vector field X on M and differential forms α, β on M one has

∇X (α ∧ β) = ∇X α ∧ β + α ∧ ∇X β.

Here, one could replace the tangent bundle with any vector bundle over M, but we are mainly interested in TM. Proof. Let ∇0 be the given connection in TM and γ :(−, ) → M a smooth curve. Let p := γ(0) and v := γ0(0) and for − < t <  set

t 0 ut := Hol0(∇ , γ),

−1 so that ut = (ht) in the notation of Proposition 5.2. Then

d ∗ ∇v(α ∧ β) = ut (αγ(t) ∧ βγ(t)) dt 0

d ∗ ∗ = (ut αγ(t) ∧ ut βγ(t)) dt 0     d ∗ d ∗ = ut αγ(t) ∧ βp + αp ∧ ut βγ(t) dt 0 dt 0 = ∇vα ∧ βp + αp ∧ ∇vβ.

Proposition 5.3 can be generalized to smooth functors of several variables such as the tensor product. Given vector bundles E and E0 over M, equipped with connections ∇ and ∇0, respectively, we obtain a connection ∇˜ in E ⊗E0 which is uniquely characterized by the fact that for any vector field X on M and sections s, s0 of E,E0 one has

˜ 0 0 0 0 ∇X (s ⊗ s ) = ∇X s ⊗ s + s ⊗ ∇X s .

Such a connection can also be constructed using local frames for E and E0.

6 Curvature

∇ The curvature F = F of a connection ∇ in a K–vector bundle E → M associates to any pair X,Y of vector fields on M the map Γ(E) → Γ(E) given by F (X,Y )s := ∇X ∇Y s − ∇Y ∇X s − ∇[X,Y ]s.

15 If F (X,Y ) = 0 for all X,Y then ∇ is called flat. We will sometimes use the notation [∇X , ∇Y ] := ∇X ∇Y − ∇Y ∇X .

r Example Let ∇ be the product connection in E := M × K → M. Iden- r tifying sections of E with functions h : M → K we have ∇X h = Xh, so ∇X ∇Y h − ∇Y ∇X h = XY h − Y Xh = [X,Y ]h = ∇[X,Y ]h. Therefore, the product connection is flat.

Proposition 6.1 Let ∇ be a connection in a vector bundle E → M. Then the map

X (M) × X (M) × Γ(E) → Γ(E), (X, Y, s) 7→ F (X,Y )s is D(M, K)–multilinear. The curvature of ∇ can therefore be regarded as a bundle-valued 2–form F ∈ Ω2(M; End(E)).

Proof. We prove linearity in X (and hence in Y because of the skew- symmetry), leaving the linearity in s as an exercise for the reader. Let g ∈ D(M, K). Then

[gX, Y ] = g[X,Y ] − (Y g)X, so

F (gX, Y ) = ∇gX ∇Y s − ∇Y ∇gX s − ∇[gX,Y ]s   = g∇X ∇Y s − [(Y g)∇X s + g∇Y ∇X s] − g∇[X,Y ]s − (Y g)∇X s = gF (X,Y )s.

It follows from the proposition that if E is trivial and (s1, . . . , sr) is a i global frame for E then there is an r×r matrix (Ωj) of 2–forms on M, called the curvature form of ∇ with respect to the given frame, such that for all vector fields X,Y on M one has

X i F (X,Y )sj = Ωj(X,Y )si. i

16 Theorem 6.1 Let ∇ be a connection in a trivial vector bundle E → M, i i and let (s1, . . . , sr) be a global frame for E. Let ω = (ωj) and Ω = (Ωj) be the corresponding connection form and curvature form of ∇, respectively. Then for all i, k one has

i i X i j Ωk = dωk + ωj ∧ ωk. j

In matrix notation, Ω = dω + ω ∧ ω.

Proof. Recall that for any 1–forms α, β on M one has

(α ∧ β)(X,Y ) = α(X)β(Y ) − α(Y )β(X), (dα)(X,Y ) = X(α(Y )) − Y (α(X)) − α([X,Y ]).

Now

X j ∇X ∇Y sk = ∇X (ωk(Y )sj) j ! X j j X i = X(ωk(Y ) · sj + ωk(Y ) ωj(X)si j i   X i X i j = X(ωk(Y ) + ωj(X)ωk(Y ) si, i j and of course there is a similar formula with X,Y switched. Combining this with X i ∇[X,Y ]sk = ωk([X,Y ])si i we obtain   X i X i j F (X,Y )sk = dωk + ωj ∧ ωk (X,Y ) · si. i j

Corollary 6.1 In the situation of the theorem, let f : M˜ → M be a smooth ˜ i ∗ map and (Ωj) the curvature form of the pull-back connection f ∇ with re- ∗ ∗ spect to the global frame (f sj) for f E. Then ˜ i ∗ i Ωj = f Ωj.

17 A connection ∇ in a vector bundle E → M is called trivial if there exists a global frame (sj) for E such that ∇sj = 0 for each j. The connection is called locally trivial if every point in M has a neighbourhood over which ∇ is trivial. Theorem 6.2 A connection is flat if and only if it is locally trivial. Proof. A locally trivial connection can locally be represented by the zero connection form and is therefore flat by Theorem 6.1. To prove the converse, n let ∇ be a flat connection in a trivial vector bundle E → R . We will construct a parallel section of E with any prescribed value at the origin. 1 n n ∂ Let (x , . . . , x ) be coordinates on R and Xj := ∂xj . For k = 0, . . . , n k n let Vk := R × {0} ⊂ R and Ek := E|Vk . We first observe that for k > 0, any section σ of Ek−1 can be extended to a section τ of Ek by means of holonomy along lines parallel to the kth coordinate axis. Explicitly, τ is the unique section of Ek such that

∇Xk τ = 0, τ|Vk−1 = σ. (7)

To see that these equations have a unique smooth solution τ, let (s1, . . . , sr) be a global frame for E and

X i ∇Xk sj = cjsi, i i n P j where the coefficients cj are smooth functions on R . If σ = j f sj and P j j j τ = j g sj, where f and g are functions on Vk−1 and Vk, respectively, then the equations (7) are equivalent to ∂gi X + ci gj = 0, gi| = f i (i = 1, . . . , r). ∂xk j Vk−1 j The theory of ordinary differential equations guarantees that these equations have a unique smooth solution (gi), see for instance [John Lee: Introduction to Smooth Manifolds, Theorem D.6]. We now prove that if σ is parallel then so is τ. In fact, for 1 ≤ j < k we have

∇Xk ∇Xj τ = ∇Xj ∇Xk τ = 0. Because

(∇Xj τ)|Vk−1 = ∇Xj σ = 0 it follows that ∇Xj τ = 0, so τ is parallel as claimed. Applying the above procedure for k = 1, . . . , n we obtain a parallel sec- tion of E with any prescribed value at the origin.

18 7 The exterior covariant derivative

A connection ∇ in a K–vector bundle E → M gives rise to a K–linear operator d∇ :Ωk(M; E) → Ωk+1(M; E) for k ≥ 0, called the exterior covariant derivative. In degree zero, the operator d∇ is just the connection ∇ itself. To define it in arbitrary degrees, we use an auxiliary torsion-free connection ∇0 in TM. Together, ∇ and ∇0 induce a connection ∇00 in ΛM ⊗ E, and we define d∇ in degre k to be the composite map

00 Γ(ΛkM ⊗ E) ∇→ Γ(Λ1M ⊗ ΛkM ⊗ E) →∧ Γ(Λk+1M ⊗ E), where the second map is given by the wedge product on forms. The following proposition provides an alternative formula for d∇ which does not involve ∇0.

k Proposition 7.1 For any ω ∈ Ω (M; E) and vector fields X0,...,Xk on M one has

k ∇ X i   (d ω)(X0,...,Xk) = (−1) ∇Xi ω(X0,..., Xci,...,Xr) i=0 X i+j + (−1) ω([Xi,Xj],X0,..., Xci,..., Xcj,...,Xk). i

Here, as usual, a hat over a term indicates that the term is to be omitted. 1 n ∂ Proof. Let (x , . . . , x ) be local coordinates on M and set ∂j := ∂xj . In the coordinate domain we then have n X d∇ω = dxj ∧ ∇00 ω, ∂j j=1

19 which gives

n k X X (d∇ω)(X ,...,X ) = (−1)idxj(X )(∇00 ω)(X ,..., X ,...,X ) 0 k i ∂j 0 ci k j=1 i=0 k X i 00 = (−1) (∇Xi ω)(X0,..., Xci,...,Xk) i=0 k X i   = (−1) ∇Xi ω(X0,..., Xci,...,Xk) i=0 X i+1 0 + (−1) ω(X0,..., Xci,..., ∇Xi Xj,...,Xk). i6=j

Let S denote the sum on the last line. Then

X i+1 0 S = (−1) ω(X0,..., Xci,..., ∇Xi Xj,...,Xk) i

Interchanging i and j in the last sum and recalling that

0 0 ∇Xi Xj − ∇Xj Xi = [Xi,Xj] since ∇0 is torsion-free we obtain

X i+j S = (−1) ω([Xi,Xj],X0,..., Xci,..., Xcj,...,Xk). i

For α ∈ Ω1(M; E) the proposition says that

∇ (d α)(X,Y ) = ∇X (α(Y )) − ∇Y (α(X)) − α([X,Y ]). (8)

∇ k Proposition 7.2 If ∇ is the product connection in M×R then d :Ω (M) → Ωk+1(M) coincides with the exterior derivative d.

20 n Proof. This is a local statement, so we may assume M = R . Since d and ∇ I d are both R–linear it suffices to show that they agree on forms ω = f dx , n I i i where f ∈ D(R ) and dx = dx 1 ∧· · ·∧dx k for some increasing multi-index 0 n n n I = (i1, . . . , ik). Taking ∇ to be the product connection in T R = R × R 0 I ∂ we have ∇ (dx ) = 0. Setting Xj := ∂xj we obtain

∇ X j d ω = dx ∧ ∇Xj ω j X j I = dx ∧ (Xjf)dx j = df ∧ dxI = dω.

Proposition 7.3 Let ∇ be a connection in a vector bundle E → M. Then for any α ∈ Ωk(M) and φ ∈ Ω`(M; E) one has

d∇(α ∧ φ) = dα ∧ φ + (−1)kα ∧ d∇φ.

Note that this property, together with the fact that d∇s = ∇s for s ∈ Ω0(M; E) = Γ(E), determines d∇ completely. Proof. Let E1,...,En be a frame of vector fields on an open subset U ⊂ M and η1, . . . , ηn the dual frame of 1–forms. Then over U one has

∇ X j 00 d (α ∧ φ) = η ∧ ∇Ej (α ∧ φ) j X j  0 00  = η ∧ ∇Ej α ∧ φ + α ∧ ∇Ej φ j   X j 0 k X j 00 =  η ∧ ∇Ej α ∧ φ + (−1) α ∧ η ∧ ∇Ej φ j j = dα ∧ φ + (−1)kα ∧ d∇φ.

Theorem 7.1 Let ∇ be a connection in a vector bundle E → M, and let F be the curvature of ∇. Then for all ω ∈ Ωk(M; E) one has

d∇(d∇(ω)) = F ∧ ω.

Here, the pairing F ∧ω combines the wedge product Λ2M ⊗ΛkM → Λk+2M with the evaluation map End(E) ⊗ E → E.

21 Proof. Since the statement is local and both sides of the equation are k R–linear in ω we may assume ω = α ⊗ s for some α ∈ Ω (M) and s ∈ Γ(E). For vector fields X,Y on M, Equation 8 yields

∇ ∇ (d d s)(X,Y ) = ∇X ∇Y s − ∇Y ∇X s − ∇[X,Y ]s, i.e. d∇d∇s = F s. Applying Proposition 7.3 twice we get

d∇d∇ω = d∇(dα ⊗ s + (−1)kα ∧ ∇s) = α ∧ F s = F ∧ ω.

Theorem 7.2 (Bianchi) Let ∇ be a connection in E → M with curvature F . Let ∇¯ denote the induced connection in End(E). Then

¯ d∇F = 0.

Proof. For all sections s of E one has

¯ F ∧ ∇s = d∇d∇d∇s = d∇(F s) = (d∇F )s + F ∧ ∇s, hence (d∇¯ F )s = 0.

8 Orthogonal connections

By a Euclidean metric on a real vector bundle E → M we mean a choice of scalar product h·, ·i on each fibre Ep such that for every pair s, t of sections of E the function M → R, p 7→ hs(p), t(p)i is smooth. A real vector bundle equipped with a Euclidean metric is called a Euclidean vector bundle.

k Example The product bundle M × R → M where each fibre has the standard Euclidean metric.

Theorem 8.1 Every real vector bundle admits a Euclidean metric.

Proof. Glue together local Euclidean metrics using a partition of unity.

Let E → M be a Euclidean vector bundle of rank k. By an orthonormal frame for E over an open set U ⊂ M we mean a k–tuple (s1, . . . , sk) of sections of E|U such that (s1(p), . . . , sk(p)) is an orthonormal basis for Ep for every p ∈ U. Note that the Gram-Schmidt process applied pointwise to an arbitrary frame for E over U yields an orthonormal frame.

22 Let E → M be a real vector bundle equipped with a connection ∇ as well as a Euclidean metric. Then ∇ is called orthogonal, or compatible with the Euclidean metric, if for all sections s, t of E and vector fields X on M one has Xhs, ti = h∇X s, ti + hs, ∇X ti. k Example If each fibre of a product bundle E = M × R → M has the same scalar product, then the product connection in E is orthogonal.

Theorem 8.2 Every Euclidean vector bundle admits an orthogonal connec- tion.

Proof. This is proved just as Theorem 3.1.

For any finite-dimensional Euclidean space V let so(V ) denote the Lie algebra of skew-symmetric linear endomorphisms of V . If E → M is a Euclidean vector bundle then we can apply this construction fibrewise to obtain a subbundle [ so(E) := so(Ep) p∈M of the endomorphism bundle End(E).

Proposition 8.1 Let ∇ be an orthogonal connection in a Euclidean vector bundle E → M, and a ∈ Ω1(M; End(E)). Then the connection ∇ + a is orthogonal if and only if a takes values in so(E).

Proof. Set ∇0 := ∇ + a. Then for any vector field X on M and sections s, t of E one has

0 0 h∇X s, ti + hs, ∇X ti = Xhs, ti + ha(X)s, ti + hs, a(X)ti.

From this the proposition follows immediately.

Theorem 8.3 Let E → M be a real vector bundle equipped with both a connection ∇ and a Euclidean metric. Then the following are equivalent.

(i) ∇ is orthogonal.

(ii) For every piecewise smooth curve γ :[a, b] → M the holonomy Eγ(a) → Eγ(b) of ∇ along γ is an orthogonal linear map.

23 Proof. Suppose ∇ is orthogonal and let γ :[a, b] → M be a piecewise smooth curve. We will show that the holonomy of ∇ along γ is orthogonal. Because of the composition law (6) for the holonomy we may assume γ is smooth. We may in fact also assume E is trivial, since for any sufficiently fine partion of the interval [a, b] the image of each subinterval under γ will be contained in an open subset of M over which E is trivial. i Let (ωj) be the connection form of ∇ with respect to a global orthonor- mal frame (s1, . . . , sk) for E. By Proposition 8.1, the connection form is skew-symmetric. To see this explicitly, let X be a vector field on M. Then

0 = Xhsi, sji

= h∇X si, sji + hsi, ∇X sji * + * + X k X k = ωi (X)sk, sj + si, ωj (X)sk k k j i = ωi (X) + ωj(X),

i j so that ωj = −ωi . Now let σ be a parallel section of E along γ. We will show that the pointwise norm |σ| is a constant function on [a, b], which implies ∗ that the holonomy of ∇ along γ is orthogonal. Since (γ si) is a global ∗ P i ∗ orthonormal frame for γ E we have σ = i f γ si for some real-valued functions f i on the interval [a, b], and X |σ|2 = (f i)2. i

i ∗ i i Let cj be the functions given by γ ωj = cjdt. Equation (5) now yields

d X df i X |σ(t)|2 = 2 f i · = −2 f if jci = 0 dt dt j i ij

i by the skew-symmetry of the matrix (cj). Hence, σ has constant length as claimed. To prove the reverse implication in the proposition, suppose (ii) holds and let s1, s2 be a pair of sections of E and X a vector field on M. Given a point p ∈ M we can find  > 0 and a smooth curve

γ :(−, ) → M

0 such that γ(0) = p and γ (0) = Xp. For − < t <  let

0 ht = Holt (∇, γ): Eγ(t) → Ep

24 be the holonomy of ∇, which is orthogonal by assumption. Using Proposi- tion 5.2 we obtain

d Xphs1, s2i = hs1(γ(t)), s2(γ(t))i dt 0

d = hhts1(γ(t)), hts2(γ(t))i dt 0     d d = hts1(γ(t)), s2(p) + s1(p), hts2(γ(t)) dt 0 dt 0 = h∇X s1, s2ip + hs1, ∇X s2ip, which shows that ∇ is orthogonal.

Proposition 8.2 Let ∇ be an orthogonal connection in a Euclidean vector bundle E → M and F the curvature of ∇. Then F (X,Y ) is a section of so(E) for all vector fields X,Y on M, i.e.

F ∈ Ω2(M; so(E)).

Proof. For any sections s, t of E we have

XY hs, ti = h∇X ∇Y s, ti + h∇Y s, ∇X ti + h∇X s, ∇Y ti + hs, ∇X ∇Y ti.

Combining this with the corresponding equality with X and Y interchanged we obtain

h[∇X , ∇Y ]s, ti + hs, [∇X , ∇Y ]ti = [X,Y ]hs, ti

= h∇[X,Y ]s, ti + hs, ∇[X,Y ]ti.

Therefore, hF (X,Y )s, ti + hs, F (X,Y )ti = 0. A Riemannian manifold is a smooth manifold M equipped with a Eudclidean metric in TM.

Theorem 8.4 (Levi-Civita) If M is a Riemannian manifold then TM has a unique torsion-free orthogonal connection.

The proof can be found in any book on Riemannian geometry.

25