Lectures On Fibre Bundles and Differential Geometry

By J.L. Koszul

Notes by S. Ramanan

No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1

Tata Institute of Fundamental Research, Bombay 1960

Contents

1 Differential Calculus 1 1.1 ...... 1 1.2 Derivationlaws ...... 2 1.3 Derivation laws in associated modules ...... 4 1.4 TheLiederivative...... 6 1.5 Lie derivation and exterior product ...... 8 1.6 Exterior differentiation ...... 8 1.7 Explicit formula for exterior differentiation ...... 10 1.8 Exterior differentiation and exterior product ...... 11 1.9 The curvature form ...... 12 1.10 Relations between different derivation laws ...... 16 1.11 Derivation law in C ...... 17 1.12 Connections in pseudo-Riemannian manifolds ...... 18 1.13 Formulae in local coordinates ...... 19

2 Differentiable Bundles 21 2.1 ...... 21 2.2 Homomorphisms of bundles ...... 22 2.3 Trivial bundles ...... 23 2.4 Induced bundles ...... 24 2.5 Examples ...... 25 2.6 Associated bundles ...... 26 2.7 Vector fields on manifolds ...... 29 2.8 Vector fields on differentiable principal bundles . . . . . 30 2.9 Projections vector fields ...... 31

iii iv Contents

3 Connections on Principal Bundles 35 3.1 ...... 35 3.2 Horizontal vector fields ...... 37 3.3 Connection form ...... 39 3.4 Connection on Induced bundles ...... 39 3.5 Maurer-Cartan equations ...... 40 3.6 Curvatureforms...... 43 3.7 Examples ...... 46

4 Holonomy Groups 49 4.1 Integral paths ...... 49 4.2 Displacement along paths ...... 50 4.3 Holonomygroup ...... 51 4.4 Holonomy groups at points of the bundle ...... 52 4.5 Holonomy groups for induced connections ...... 53 4.6 Structure of holonomy groups ...... 54 4.7 Reduction of the structure group ...... 55 4.8 Curvature and the holonomy group ...... 57

5 Vector Bundles and Derivation Laws 61 5.1 ...... 61 5.2 Homomorphisms of vector bundles ...... 63 5.3 Induced vector bundles ...... 63 5.4 Locally free sheaves and vector bundles ...... 64 5.5 Sheaf of invariant vector fields ...... 67 5.6 Connections and derivation laws ...... 70 5.7 Parallelism in vector bundles ...... 74 5.8 Differential forms with values in vector bundles . . . . . 76 5.9 Examples ...... 77 5.10 Linear connections and geodesics ...... 78 5.11 Geodesic vector field in local coordinates ...... 79 5.12 Geodesic paths and geodesic vector field ...... 79

6 Holomorphic Connections 81 6.1 Complex vector bundles ...... 81 6.2 Almost complex manifolds ...... 82 Contents v

6.3 Derivation law in the complex case ...... 83 6.4 Connections and almost complex structures ...... 85 6.5 Connections in holomorphic bundles ...... 88 6.7 Atiyah obstruction ...... 91 6.8 Line bundles over compact Kahler manifolds ...... 93

Chapter 1 Differential Calculus

1.1

Let k be a commutative ring with unit and A a commutative and asso- 1 ciative algebra over k having 1 as its element. In Applications, k will usually be the real number field and A the algebra of differentiable func- tions on a manifold.

Definition 1. A derivation X is a map X : A A such that → i) X Hom (A, A), and ∈ k ii) X(ab) = (Xa)b + a(Xb) for every a, b A. ∈ If no non-zero element in k annihilates A, k can be identified with a subalgebra of A and with this identification we have Xx = 0 for every x k. In fact, we have only to take a = b = 1 in (ii) to get X = 0 and ∈ 1 consequently Xx = xX(1) = 0. We shall denote the set of derivations by C. Then C is obviously an A-module with the following operations:

(X + Y)(a) = Xa + Ya (aX)(b) = a(Xb) for a, b A and X, Y C. ∈ ∈ We have actually something more: If X, Y, C, then [X, Y] C. ∈ ∈ 1 2 1. Differential Calculus

This bracket product has the following properties:

[X1 + X2, Y] = [X1, Y] + [X2, Y] [X, Y] = [Y, X] − X, [Y, Z] + Y, [Z, X] + Z, [X, Y] = 0,       2 for X, Y, Z C. The bracket is not bilinear over A, but only over k. We ∈ have

[X, aY](b) = X(aY) (aY)(X) (b) { − } = (Xa)(Yb) + a[X, Y](b)

so that [X, aY] = (Xa)Y + a[X, Y] for X, Y C, a A. The skew com- ∈ ∈ mutativity of the bracket gives

[aX, Y] = (Ya)X + a[X, Y]. − When A is the algebra of differentiable functions on a manifold, C is the space of differentiable vector fields.

1.2 Derivation laws

Definition 2. A derivation law in a unitary A-module M is a map D : C Hom (M, M) such that, if D denotes the image of X C under → k X ∈ this map, we have

i) DX+Y = DX + DY D = aD for a A, X, Y C. aX X ∈ ∈ i.e., D Hom (C, Hom (M, M)). ∈ A k 3 ii) D (au) = (Xa)u + aD u for a A, u M. X X ∈ ∈ In practice, M will be the module of differentiable sections of a vec- tor bundle over a manifold V. A derivation law enables one thus to differentiate sections of the bundle in specified directions. 1.2. Derivation laws 3

If we consider A as an A-module, then D defined by DXa = Xa is a derivation law in A. This will hereafter be referred to as the canonical derivation in A. Moreover, if V is any module over k, we may define on the A-module A V, a derivation law by setting D (a v) = Xa X ⊗ ⊗ Nk v and extending by linearity. This shall also be termed the canonical derivation in A V. Nk There exist modules which do not admit any derivation law. For instance, let A be the algebra k[t] of polynomial in one variable t over k; then C is easily seen to be the free A-module generated by P = ∂/∂t. Let M be the A-module A/N where N is the ideal of polynomials without constant term. If there were a derivation law is this module, denoting by e the identity coset of A/N, we have

0 = Dp(te) = (P.t)e + t.Dpe = e, which is a contradiction. However, the situation becomes better if we confine overselves to free A-modules.

Theorem 1. Let M be a free A-module, (ei)i I being a basis. Given any 4 ∈ system (ωi)i I of elements in HomA(C, M), there exists one and only one derivation law∈ D in M such that D e = ω (X) for every i I. X i i ∈ Let u be an arbitrary element of M. Then u can be expressed in the term u = λiei. If the conditions of the theorem have to be satisfied, = + we have toP define DXu (X i)ei λiλi(X). It is easy to verify that ∧ this is a derivation law. P P We shall now see that the knowledge of one derivation law is enough to compute all the possible derivation laws. In fact, if D, D′ are two such laws, then (D D )(au) = a(D D )(u). Since D, D X − ′X X − ′X ′ ∈ Hom (C, Hom (M, M)), it follows that D D Hom (C, Hom A k − ′ ∈ A A (M, M)). Conversely, if D is a derivation law and h any element of HomA(C, HomA (M, M)), then D′ = D + h is a derivation law as can be easily verified. 4 1. Differential Calculus

1.3 Derivation laws in associated modules

i Given module Mi with derivation laws D , we proceed to assign in a canonical way derivation laws to module which are obtained from the Mi by the usual operations. Firstly, if M is the direct sum of the modules Mi, then DX(m) = i =Σ DX(mi) where m mi, gives a derivation law in M. i P Since D are k-linear, we may define D in M1 Mp by set- X ··· Nk Nk ting D (u , u ) = u Di u u . Now, it is easy to see X 1 ⊗···⊗ p 1 ⊗··· X i ⊗··· p that this leaves invariant theP ideal generated by elements of the form

u au u u au u with a A. 1 ⊗··· i ⊗···⊗ p − 1 ⊗··· j ⊗···⊗ p ∈

5 This therefore induces a k-linear map DX of M1 Mp into A ··· A itself, where N N

= i DX(u1 up) u1 DXui up ⊗···⊗ X ⊗··· ⊗···⊗ where u u M M . 1 ⊗···⊗ p ∈ 1 ··· p NA NA It is easily seen that D is a derivation law. We will be particularly interested in the case when M = M = = 1 2 ··· M = M. In this case, we denote M M by T p(M). Since we p 1 ⊗···⊗ p have such a law in each T p(M) (for T 0(M) = A, we take the canonical derivation law) we may define a derivation law in the tensor algebra T ∗(M) of M. If t, t′ are two tensors, we still have

D (t t′) = D t t′ + t D t′. X ⊗ X ⊗ ⊗ X

Now let N be the ideal generated in T ∗(M) by elements of the form u v v u with u, v M. It follows from the above equality that ⊗ − ⊗ ∈ D N N. Consequently D induces a derivation law in T (M)/N, which X ⊂ ∗ is the symmetric algebra over M. Again, if N′ is the ideal in T ∗(M) whose generators are of the form u u, u M, then it is immediate ⊗ ∈ that D N N . Thus we obtain a derivation law in exterior algebra X ′ ⊂ ′ T ∗(M)/N′ of M. 1.3. Derivation laws in associated modules 5

Let M, L be two A-modules with derivation laws DM, DL respec- tively. We define a derivation law D in HomA(L, M) by setting (DXh) = DMh hDL for every h Hom (L, M) and X C. X − X ∈ A ∈ In fact D (ah) = DM(ah) (ah)DL . X X − X D (ah)(l) = DM(ah)(l) (ah)DL (l) X X − X = DM(a.h(l)) a.h(DL (l)) X − X = (Xa)(h(l)) + aDM(h(l)) a.h(DL (l)) L − X = (Xa)h (l) + (aD h) for every I L. { } X ∈ In particular, if L = M with DL = DM, then we have 6 = L DXh DX, h for every h HomA(L, L). h i ∈ Moreover, this leads to a derivation law in the dual L∗ of L by taking M = A with the canonical derivation law. The corresponding law is

L (D f )(u) = X( f (u)) f (D (u)) for every f L∗ and u L. X − X ∈ ∈ Now let L, M be modules with derivation laws DL, DM respectively. We may then define a derivation law in the A-module F p(M, L) of mul- tilinear forms on M of degree p with values in L in the following way: p (D ω)(u ,..., u ) = DL ω(u ,..., u ) ω(u ,..., DMu ,... u ). X 1 p X 1 p − 1 X r p Xr=1 If U p(M, L) is the submodule of F p(M, L) consisting of alternate forms , then it is easy to see that D U p(M, L) U p(M, L). This leads ∧ ⊂ to a derivation law in the A-modules F p(M, A), U p(M, A), if we take 7 L = A with the canonical derivation law. Let M , M , M be three modules with a bilinear product M M 1 2 3 1× 2 → M , denoted (u, v) uv. If D1, D2, D3 are the respective derivation 3 → laws, we say that the product is compatible with the derivation laws if

D3 (uv) = (D1 u)v + u(D2 v) for X C, u M and v M . X X X ∈ ∈ 1 ∈ 2 This was the case the we took M1 = M2 = M3 = A with the canon- ical derivation law and the algebra-product. Again, we seen that the 6 1. Differential Calculus

above condition is satisfied by M1 = M2 = M3 = T ∗(M) with the asso- ciated derivation laws and the usual multiplication. Moreover, it will be noted that if M is an A-module with derivation law D, the condition D (au) = (Xa)u + aD u for every a A, u M X X ∈ ∈ expresses the fact that the map A M M defining the module struc- × → ture is compatible with the derivation laws in A and M. If we denote p p q U p(L, M ) by U (i = 1, 2, 3), we may define for α U β U ,α β i i ∈ 1 ∈ 2 ∧ by setting

(α β)(a1,..., ap+q) = σ α(aσ(1),..., aσ(p))β(aσ(p+1),..., aσ(p+q)) ∧ X ∈ where the summation extends over all permutations σ of (1, 2,..., p+q) such that σ(1) < σ(2) < < σ(p) and σ(p + 1) < σ(p + 2) < < ··· ··· σ(p + q) and σ is its signature. If the derivation laws are compatible ∈ with the product, we have D3 (α β) = (D1 α) β + α (D2 β) for every X C. X ∧ X ∧ ∧ X ∈ 1.4 The Lie derivative

8 Let V be a manifold, U the algebra of differentiable functions on V and C the module of derivations of U (viz. differentiable vector fields on V. Any one-parameter group of differentiable automorphisms on V gener- ates a differentiable vector field on V. Conversely, every differentiable vector field X gives rise to a local one-parameter group s(t) of local au- tomorphisms of V. If ω is a p-co variant tensor, i.e., if ω F p(C , U ), ∈ then we may define differentiation of ω with respect to X as follows:

s∗(t)ω ω θXω = lim − , t o t → where s∗(t) stands for the k-transpose of the differential map lifted to F p(C , U ). This is known as the Lie derivative of ω with respect to X and can be calculated to be p θ ω(u ,..., u ) = Xω(u ,..., u ) ω(u ,... [X, u ],... u ). X 1 p 1 p − 1 i p Xi=1 1.4. The Lie derivative 7

It will be noted if a U , then θ ω , a(θ ω) if p > 0. At a point ∈ aX X ξ V, the Lie derivative θ ω, unlike the derivation law, does not depend ∈ X only on the value of the vector field x at ξ. Lemma 1. Let D be a derivation law in the A-module M and α a multilinear form on C of degree p with values in M. Then the map β : CP M defined by → p β(Z ,..., Z ) = D α(Z ,..., Z ) α(Z ,... [X, Z ],..., Z ) 1 p X 1 p − 1 i p Xi=1 is multilinear. 9 + = + In fact, β(Z1,... Zi Zi′,... Zp) β(Z1,... Zi,... Zp) β(Z1,... Zi′,... Zp) and

β(Z1,... aZi,... Zp) = (Xa)α(Z1,... Zp) + a(DXα(Z1,... Zp)) p aα(Z ,... [X, Z ],... Z ) α(Z ,... (Xa)Z ,... Z ) − 1 j p − 1 i p Xj=1 = aβ(Z ,... Z ) for every Z ,... Z C, a A. 1 p 1 p ∈ ∈ p p Definition 3. The map X θX of C into Homk(F (C, M)F (C, M)) → = p defined by (θXα)(Z1,..., Zp) DXα(Z1,... Zp) i=1 α(Z1,... [X, Zi] − p Zp) is called the Lie derivation in the A-modulePF (C, M) and θXα ··· is defined to be the Lie derivative of α with respect to X.

The Lie derivation satisfies the following

θX(α + β) = θXα + θXβ

θX(aα) = (Xa)α + a(θXα)

θX+Y (α) = θXα + θY α

θλX(α) = λ(θX(α) for every X, Y C, αβ F p(C, M) and λ k. ∈ ∈ ∈ Thus θ looks very much like a derivation law, but θaX , aθX in gen- 10 eral. From the definition, it follows that if α is alternate (resp. symme- tric), so is θXα. 8 1. Differential Calculus

1.5 Lie derivation and exterior product

As in ch.1.3, let M1, M2, M3 be A-module with derivations D1, D2, D3 respectively. Let there be given a bilinear product M1 M2 M3 with × →p reference to which an exterior product (α, β) α β of U (C, M1) + → ∧ × U q(C, M ) U p q(C, M ) is defined. If the product is compatible 2 → 3 with the derivation laws, we have, on direct verification,

θ (α β) = θ α β + α θ β X ∧ X ∧ ∧ X for α U p(C, M ), β U q(C, M ) and X C. ∈ 1 ∈ 2 ∈

1.6 Exterior differentiation

We shall introduce an inner product in the A-module F p(C, M). For p every X C, the inner product is the homomorphism lX of F (C, M) p ∈1 into F − (C, M) defined by

(lXα)(Z, ... Zp 1) = α(X, Z1,..., Zp 1) − − p for every α F (C, M), Z1,... Zp 1 C. If α is alternate it is obvious ∈ − ∈ that lxα is also alternate. When α is of degree 0, lXα = 0. The inner product satisfies the following

1. laX = alX

l + = l + l , for a A, X, Y C. X Y X Y ∈ ∈

2. If θX is the Lie derivation,

11 θ l l θ = l , for X, Y, C. X Y − Y X [X Y] ∈

3. If α is alternate, lXlXα = 0.

4. Let M1, M2, M3 be three A-modules with a bilinear product com- patible with their laws D , D , D . Then we have, for α U p 1 2 3 ∈ (C, M ), β U q(C, M ) 1 ∈ 2 l (α β) = (l α) β + ( 1)pα l β. X ∧ X ∧ − ∧ X 1.6. Exterior differentiation 9

in fact, let Z1,..., Zp+q 1 C. Then − ∈

lX(α β)(Z1,..., Zp+q 1) = (α β)(X, Z1,..., Zp+q 1) ∧ − ∧ − = σ α(X, Zσ(1),..., zσ(p 1))β(Zσ(p),... Zσ(p+q 1)) − − Xσ ∈

+ τ α(Xτ(1),... Xτ(p))β(X, Zτ(p+1),... Zτ(p+q 1)) − Xτ ∈ where σ runs through all permutations of [1, p + q 1] such that − σ(1) < <σ(p 1) and σ(p) < <σ(p + q 1), while τ runs ··· − ··· − through those which satisfy τ(1) < < τ(p) and τ(p+1) < < ··· ··· τ(p + q 1). The first sum is equal to (l α) β and the second to − X ∧ ( 1)pα (l β). − ∧ X Theorem 2. Let D be a derivation law in an A-module M. Then there exists one and only one family of k-linear maps d : U p(C, M) + → U p 1(C, M)(p = 0, 1, 2,...) such that θ(X) = dl + l d for every X C. X X ∈ We call this map the exterior differentiation in the module of alter- nate forms on C. First of all, assuming that there exists such a map d, we shall prove 12 that it is unique. If d′ is another such map, we have

(d d′)l + l (d d′) = 0 − X X − Hence l (d d )α = (d d)l α for every α U p(C, M). We shall X − ′ ′ − X ∈ prove that dα = d α for every α U p(C, M) by induction on p. When ′ ∈ α is of degree 0, we have (d d)l α = 0 = l (d d )α. This being ′ − x X − ′ true for every X C, dα = d α. If the theorem were for p = q 1, then ∈ ′ − (d d)l α = 0 = l (d d )α. Again, since X is arbitrary, dα = d α ′ − X X − ′ ′ which proves the uniqueness of the exterior differentiation. The existence is also proved by induction. Let α U o(C, M) = M. ∈ Then we wish to define do such that lXdα = (dα)(X) = θXα (since l α = 0) = D α. Hence we can set (d(α)(X) = D α for every X C. X X X ∈ This is obviously A-linear since DX is A-linear in X. Let us suppose that d has been defined on U p(C, M) for p = 0, 1,... (q 1) such that the − formula is true. We shall define dα for α U p(C, M) by setting ∈ (dα(Z ,..., Z + ) = (θ α)(Z ,... Z + ) (dl α)(Z ,... Z + ) 1 q 1 Z1 2 q 1 − Z1 2 q 1 10 1. Differential Calculus

We have of course to show that the dα thus defined is an alternate form. That it is linear in Z2,... Zq+1 follows from the induction assump-

tion and the multilinearity of θZ1 α. We shall now prove that it is alter- nate. That means:

(dα)(Z1,..., Zq+1) = 0 whenever Zi = Z j with i , j.

ffi 13 Using the alternate nature of θZ1 α and dlZ1 α, we see that it su ces to prove that

= = (θZ1 α)(Z2,..., Zq+1) (dlZ1 α)(Z2,..., Zq+1) when Z1 Z2 = Now (θZ2 α)(Z2,..., Zq+1) (lZ2 θZ2 α)(Z3,..., Zq+1) = (θZ2 lZ2 α)(Z3,..., Zq+1) by (2) of Ch.1.6 = + (dlZ2 lZ2 α lZ2 dlZ2 α) (Z3,..., Zq+1) = (dlZ2 α)(Z2, Z3,..., Zq+1).

Since (dα) is alternate, linearity in Z1 follows from that in the other variables and additivity in Z1. This completes the proof of the theorem.

Remark. If we take A to be the algebra of differentiable functions on a manifold V, and M to be A itself then the exterior differentiation defined above coincides with the usual exterior differentiation.

1.7 Explicit formula for exterior differentiation

Lemma 2. The exterior differentiation defined above is given by

p+1 i+1 (dα)(Z ,..., Z + ) = ( 1) D α(Z ,... Z ,... Z + ) 1 p 1 − Zi 1 i p 1 Xi=1 i+ j + ( 1) α([Z , Z ], Z ,... Zˆ ,... Zˆ ,... Z + ) − i j 1 i j p 1 Xi< j

14 where the symbol over a letter indicates that the corresponding ele- ∧ ments is omitted. 1.8. Exterior differentiation and exterior product 11

If d is defined as above and α U p(C, M), it is easy to see that ∈ (l dα + dl α) = θ α for every Z C. Z1 Z1 Z1 1 ∈ By the uniqueness of exterior differentiation , we see that the above gives the formula for the exterior differentiation. We shall use this explicit formula only when the degree of α 2. ≤ Then we have the formula:

1. (dα)(X) = D α for α U 0(C, M), X C. X ∈ ∈ 2. (dα)(X, Y) = D α(Y) D α(X) α([X, Y]) for α U 1(C, M), X − Y − ∈ X, Y C. ∈ 3. (dα)(X, Y, Z) = D α(Y, Z) α([X, Y], Z) for α U 2(C, M), { X − } ∈ X, Y, Z C P ∈ where the summation extends over all cyclic permutations of (X, Y, Z).

1.8 Exterior differentiation and exterior product

We now investigate the behaviours of d with regard to the exterior prod- uct. Let M1, M2, M3 be three modules with derivation laws D1, D2, D3 and let M M M be a linear product compatible with the deriva- 1 × 2 → 3 tion laws. Then we have, for α U p(C, M ), β U q(C, M ) ∈ 1 ∈ 2 d(α β) = dα β + ( 1)pα dβ. ∧ ∧ − ∧ Again, we prove this by induction, but this on p+q. When p+q = 0, the above formula just expresses the compatibility of the product with 15 the derivation laws. Let us assume the theorem proved for p + q = r 1. − We have

d(α β)(Z ,..., Z + + ) = (l d(α β))(Z ,..., Z + + ) ∧ 1 p q 1 Z1 ∧ 2 p q 1

But, l d(α β) = θ (α β) d(l (α β)) Z1 ∧ Z1 ∧ − Z1 ∧ = θ α β + α θ β d(l α β + ( 1)pα l β) Z1 ∧ ∧ Z1 − Z1 ∧ − ∧ Z1 12 1. Differential Calculus

by(3) of Ch. 1.6 p 1 = θ α β + α θ β dl α β ( ) − l α Z1 ∧ ∧ Z1 − Z1 ∧ − − Z1 dβ ( 1)Pdα l β α dl β ∧ − − ∧ Z1 − ∧ Z1 by induction assumption = l dα β ( 1)pdα l β + α l dβ Z1 ∧ − − ∧ Z1 ∧ Z1 p 1 ( 1) − l α dβ − − Z1 ∧ = l (dα β) + ( 1)pl (α dβ) Z1 ∧ − Z1 ∧ = l ((dα β) + ( 1)p(α dβ)), Z1 ∧ − ∧ which proves our assertion.

Remark. If we take M2 = M3 = M and M1 = A with the product A M M defining the module structure, then, by the above formula, × → we obtain

d(aα) = (da) α + a dα for a U (C, A) = A ∧ ∧ ∈ ◦ = da α + a.dα  and α U p(C, M). ∧  ∈  16 Thus the exterior differentiation is not A-linear.

1.9 The curvature form

It is well-know that the exterior differentiation in the algebra of differen- tial forms on a manifold satisfies dd = 0. Let us compute in our general case value of ddα for α U 0(C, M) = M. Then one has ∈ ddα(X, Y) = D (dα)(Y) D (dα)(X) (dα)([X, Y]) X − Y − = D D α D D α D , α X Y − Y X − [X Y]

Let K(X, Y) = D D D D D , . Then it is obvious that X Y − Y X − [X Y] K(X, Y) is a k-endomorphism of M. But, it actually follows on trivial verification that it is an A- endomorphism. On the other hand, we have

i) K(X, Y) = K(Y, X) − 1.9. The curvature form 13

ii) K(X + X′, Y) = K(X, Y) + K(X′, Y), and iii) K(aX, Y)α = aK(X, Y)α for every α M. ∈ i) and ii) are trivial and we shall verify only iii).

K(aX, Y) = D D D D D , aX Y − Y aX − [aX Y] = aD D D (aD ) D , (Ya)X X Y − Y X − a[X Y] − = aD D (Ya)D aD D aD , + (Ya)D X Y − X − Y X − [X Y] X = aK(X, Y).

We have therefore proved that K is an alternate form of degree 2 17 over C with values in the module HomA(M, M).

2 Definition 4. The element K of U (C, HomA(M, M)) as defined above i.e., K(X, Y) = D D D D , X Y − Y X−[X Y] is called the curvature form of the derivation law D.

Examples. 1) Take the simplest case when M = A, with the canonical derivation law. Then

K(X, Y)u = XYu YXu [X, Y]u = 0 − − i.e., the curvature form is identically zero. 2) However, there are examples in which the curvature form is non- zero. Let A = k[x, y] with x, y transcendental over k. It is easy to see that ∂ ∂ C is the free module over A with P = , Q = as base. ∂x! ∂y! We take M to be A itself but with a derivation law different from the canonical one. By Th.1, Ch. 1.2, if we choose 1 M and take ω ∈ ∈ HomA(C, A), then there exists a derivation law D such that DX1 = ω(X). We shall define ω by requiring that ω(P) = y and ω(Q) = 1. Then it follows that DP(1) = y, DQ(1) = 1.

K(P, Q)(1) = D D (1) D D (1) D , (1) P Q − Q P − [P Q] 14 1. Differential Calculus

= y (Qy).1 y(D (1)) − − Q = 1. − 18 We now prove two lemmas which give the relation between Lie derivatives in two directions and the relation of Lie derivative to the exterior differentiation in terms of the curvature form.

p Lemma 3. θ θ α θ θ α = θ , α + K(X, Y)(α) for every α U X Y − Y X [X Y] ∈ (C, M).

In fact, when α is of degree 0, the formula is just the definition of the curvature form. In the general case, this follows on straight forward verification.

Lemma 4. θ dα dθ α = (l K) α for every α U p(C, M). X − X X ∧ ∈ It will be noted that K U 2(C, Hom (M, M)) and hence l K has ∈ A X values in HomA(M, M). Taking M1 = HomA(M, M), M2 = M, M3 = M in our standard notation, one has a bilinear product M M M 1 × 2 → 3 defined by (h, u) . The symbol used in the enunciation of the lemma → ∧ is with reference to this bilinear product.

Proof. As usual, we prove this by induction on p, the degree of α. When α is of degree 0, the formula reduces to

D (dα(u)) dα( X, u ) (dD α)(u) = (l K α)(u) for u C. X − − X X ∧ ∈ i.e., D D α  D  , α D D α = (l K (α)(u) X u − [X u] − u X X ∧ 19 which is but the definition of K. Assuming the truth of the lemma for forms of degree < p, we have

l θ dα l dθ α l ((l K) α) Y X − Y X − Y X ∧ = θ l dα l , dα + dl θ α θ θ α X Y − [X Y] Y X − Y X (l l K) α + (l K) (l α) − Y X ∧ X ∧ Y = θ dl α + θ θ α l , dα + dθ l α dl , α θ θ α − X Y X Y − [X Y] X Y − [X Y] − Y X (l l K) α + (l K) (l α) − Y X ∧ X ∧ Y = θ dl α + dθ l α + θ θ α θ θ α − X Y X Y X Y − Y X 1.9. The curvature form 15

θ , α (l l K) α + (l K) (l α) − [X Y] − Y X ∧ X ∧ Y = θ dl α + dθ l α + (l K) (l α) by lemma 3, Ch. 1.9 − X Y X Y X ∧ Y = 0 by induction assumption.



Remark. The maps θX, lX, d are of degrees 0, 1, 1 respectively con- − p sidered as homomorphisms of the graded module U ∗(C, M) = U p (C, M). If ϕ is a map U (C, M) U (C, M) of degree p, andPψ an- ∗ → ∗ other of degree q, we define the commutator [ϕ, ψ] = ϕψ ( 1)pqψϕ. − − We have, in this notation, the following formulae : i) [θX, lY ] = l[X,Y] by (2) of Ch.1.6

= ii) [d, lX] θX  by Th. 2, Ch. 1.6 iii) [θX,θY ] = θ[X,Y] for derivation laws of zero curvature iv) [θX, d] = 0 by lemma 3 and 4, Ch.1.9 v) [lX, lY ] = 0 by (3) of Ch.1.6

The operators θX and lX have been generalised ( see [ 15 ] to the 20 case M = A by replacing X by an alternate form γ on C with values in C. This generalisation has applications in the study of variations of complex structures on a manifold. A general formula for d2 is given by

Lemma 5. In our usual notation, d2α = K α. ∧ The meaning of has to be interpreted as in Lemma 4, Ch.1.9 . The ∧ proof is again by induction on the degree of α. If α is of degree 0,α M ∈ and we have to show that ddα(X, Y) = K(X, Y)α.Assuming the lemma verified for forms of degree < p, we get

l ddα = θ dα dl dα X X − X = dθ α + (l K) α dθ α + ddl α by Lemma 4, Ch. 1.9 X X ∧ − X X = (l K) α + K l α by induction assumption X ∧ ∧ X = l (K α). X ∧ Since X is arbitrary, ddα = K α and the lemma is proved. ∧ 16 1. Differential Calculus

Lemma 6.

dK = 0

dK(X, Y, Z) = (DX K)(Y, Z) K([X, Y], z) by (3) of Ch. 1.7 X { − } = DX K(Y, Z) K(Y, Z)DX K([X, Y], Z) X { − − } = D D D D D D D D , D D D + X Y Z − X Z Y − X [Y Z] − Y Z X X  D D D + D , D D , D + D D , + D , Z Y X [Y Z] X − [X Y] Z Z [X Y] [[X Y]Z] = 0

21 using Jacobi’s identity where the summations extend over cyclic permu- tations of X, Y, Z.

1.10 Relations between different derivation laws

We shall now investigate the relations between the exterior differenti- ations, curvature forms etc. corresponding to two derivation laws in the same module M. It has already been shown (Ch.1.2) that if D, D′ are two such derivation laws, then there exists h HomA(M, M)) such = + ff∈ that D′X DX hX. We denote the exterior di erential operator, cur- vature form etc. Corresponding to D′ by d′, K′ etc. Then d′ is given by d α = dα + h α. ( Here also, the Lambda-sign has to be inter- ′ ∧ preted as in lemma 4, Ch. 1.9 ). In fact, by definition, it follows that θ α = θ α + h α. Hence ′X X X ◦

dl α + h l α + l dα + l (h α) = θ′ α. X ∧ X X X ∧ X From Th. 2, Ch. 1.6 on the uniqueness of exterior differentiation, our 22 assertion follows. With regard to the curvature form, we have the following formula:

K′ = K + h h + dh. ∧ For

K′(X, Y) = (D + h )(D + h ) (D + h )(D + h ) D , h , X X Y Y − Y Y X Y − [X Y] − [X Y] 1.11. Derivation law in C 17

= K(X, Y) + h h h h + (D h ) (D h ) h , X Y − Y X X Y − Y Y − [X Y] = K(X, Y) + h h h h + dh(X, Y) X Y − Y X = K(X, Y) + (h h)(X, Y) + dh(X, Y). ∧ The classical notation for (h h) defined by (h h)(X, Y) = h(X)h(Y) ∧ ∧ − h(Y)h(X) is [ h, h ]. In that notation, we have

K′ = K + [h, h] + dh.

1.11 Derivation law in C

When A is the algebra of differentiable functions on a manifold V, any derivation law in the A-module C of differentiable vector fields on V is called a linear connection on the manifold. Let A be an algebra over k, D a derivation law in the A- module C of derivations of A. Let η : C C be the identity mapping. Then exterior → defferential of η is given by (dη)(X, Y) = D η(Y) D η(x) η([X, Y]) X − Y − = D Y D X [X, Y] for X, Y C. X − Y − ∈ The alternate linear form dη = T is called the torsion form of the 23 derivation law in C. Regarding the action of d on T, we have the Bianchi’s identity (dT)(X, Y, Z) = K(X, Y)Z, where the summation extends over all cyclic permutationsP of (X,.Y, Z). This is immediate from Lemma 5, Ch. 1.9 Let D be a derivation law in C. Then we can define a derivation law in the module of multilinear forms on C with values in an A- module M with derivation law D. For α F p(C, M), we define ∈ p (D α)(Z ,..., Z ) = D α(Z ,..., Z ) α(Z ,..., D Z ,..., Z ). X 1 p X 1 p − 1 X i p Xi=1 Moreover, we define for every α F p(C, M) ∈ p (D α)(Z ,..., Z + ) = D α(Z ,..., Z ) α(Z ,..., D Z ,..., Z + ) X 1 p 1 Z1 1 p − 2 Z1 i p 1 Xi=2 18 1. Differential Calculus

+ Obviously Dα F p 1(C, M). However, this operator D does not take ∈ alternate forms into alternate forms. We therefore set, for α U p(C, M) ∈ p+1 i+1 d′α = ( 1) (Dα)(Z , Z ,..., Z˜ ,..., Z + ). − i 1 i p 1 Xi=1

+ It is easy to see that d α U p 1(C, M). ′ ∈ Theorem 3. If the torsion form is zero, then d′ and the exterior deffer- 24 ential coincide.

In fact, it is easy to verify that lXd′ + d′lX = θX and the theorem then follows from Th.2, Ch.1.6 In particular, let K be the curvature form of the derivation law in C; since dK = 0, we have d′K = 0. It is easily seen that dK = (DXK)(Y, Z). where the summation extends over cyclic permutations Pof X, Y, Z. Hence we have the Second Bianchi Identity : If the torsion of the derivation law in C is 0, then (DX K)(Y, Z) = 0 where the summation is over all cyclic permutations ofP X, Y, Z.

1.12 Connections in pseudo-Riemannian manifolds

A differentiable manifold V together with a symmetric bilinear form is said to be Riemannian if the form is positive definite at all points. If the above form is only non-degenerate ( not necessarily positive definite ) at each point, the manifold is pseudo-Riemannian. Such a form defines a natural isomorphism of the module of vector fields on V onto its dual. Accordingly, in our algebraic set-up, we define a pseudo Rieman- nian form on C to be a symmetric bilinear form on C with values in A such that the induced map C C is bijective. → ∗ Theorem 4. If g is a pseudo-Riemannian form on C then there exists one and only one derivation law D on C such that

25 1) the torsion form of D is zero,

2) DXg = 0 for every X. 1.13. Formulae in local coordinates 19

Explicitly, 1) means that Xg(Y, Z) g(D Y, Z) g(Y, D Z) = 0 for − X − X every Y, Z C. ∈ In fact, by straight forward computation, it can be found that if there exists one such derivation law D, it must satisfy the equation

2g(D Y, Z) = Xg(Y, Z) Zg(Y, X) + Yg(Z, X) X − g(Y, [X, Z]) + g([Z, Y], X) g(Z, [Y, X]). − − Since the map C C induced from g is bijective, this equation deter- → ∗ mines D uniquely and conversely if we define D by this equation, it can be easily verified that D is a derivative law in C and that it satisfies the conditions of the theorem

1.13 Formulae in local coordinates

Finally we translate some of our formulae in the case of a differentiable manifold in terms of local coordinates. As far as local coordinates are concerned, we may restrict ourselves to an open subset V of Rn. Let U denote the algebra of differentiable functions on V and C the U -module of vector fields on V. Let (x1, x2,...n) be a system of coordinates. The ∂ partial derivatives P = have the following properties: i ∂xi

1) (Pi)i=1,2,...n is a base of C over U .

j 2) Pi x = δi j

3) [Pi, P j] = 0 for any i, j. 26

1 n i Also (dx ,... dx ) form a base for C ∗ over U dual to (Pi) i.e. (dx ) (P j) = δi j. Since the U -module C is free over U , all the associ- ated modules such as T p(C ), U p(C , U ) are all free over U . Thus U p(C , U ) has a basis consisting of elements dxλ(1) dxλ(p), λ S ∧···∧ ∈ where S where S is the set of all maps λ : [1, p] [1, n] such that → λ(1) < λ(2) < < λ(p). If M is an U -module, any alternate form p ··· λ U (C , M)) can be written in the form ω dx (1) dx(2) dxλ(p) ∈ λ S ∧ ∧ ∧···∧ P∈ 20 1. Differential Calculus

with ωλ M. ( The exterior product is with respect to the bilinear ∈ product M A M defining the structure of U -module ). × → Let M be a free U -module of finite rank with a derivation law D. Let eα(α = 1, 2,... m) be a base of M. We set = Γβ Γβ DPi eα iαeβ, iα U X ∈ Γβ The functions iα completely determine the derivation law. Con- β versely, given Γ U , we may define, for iα ∈ m α α u = ρ lα, ρ U . ∈ Xα=1 = α + αΓβ DPi u (piρ )eα ρ iαeβ Xα Xαβ and extend D to the whole of ξ by linearity. It is easy to see that is a 27 derivation law. The above becomes in the classical notation ∂ρα D u = e + ραΓβ e Pi ∂xi α i,α β Xα Xα,β From this we get if u M, then ∈ du = dx (D u) i ∧ Pi Xi Let ω U p(C, M). We have seen that ∈ ω = ωλ dx (1) dx (p) with ωλ M. ∧ ∧···∧ ∈ Xλ S ∈ = + Using the fact that θPi (α β) (θPi α) β α (θPi β), in order to ∧ ∧λ ∧ j λ = compute θXω, it is enough to compute θPi ω and θPi dx . But θPi ω

DPi ω∧ which we have fond out earlier. On the other hand,

(θ dx j(P ) = P ((dx j)(P )) dx j([P , P ]) Pi k i k − i k = 0. Chapter 2 Differentiable Bundles

2.1

We give in this chapter, mostly without proofs, certain definitions and 28 results on fibre bundles which we require in the sequel.

Definition 1. A differentiable principal fibre bundle is a manifold P on which a Lie group G acts differentiable to the right, together with a differentiable map p of P onto a differentiable manifold X such that P.B. for every x X, there exist an open neighbourhood U of x 0 ∈ 0 in X and a diffeomorphism γ of U G p 1(U) ( which is an open × → − submanifold of P) satisfying pγ(x, s) = x, γ(x, st) = γ(x, s)t for x U ∈ and s, t G. ∈ X shall be called the base , p the projection and P the bundle. For any x X, p 1(x) shall be called the fibre over x and for ξ P, the fibre ∈ − ∈ over pξ is the fibre through ξ. The following properties follow immediately from the definition: a) Each fibre is stable under the action of G, and G acts with out fixed points on P, i.e. if ξs = ξ for some ξ P and s G, then s = e; ∈ ∈ b) G acts transitively on each fibre, i.e. if ξ, η are such that pξ = pη, 29 then there exists s G such that ξ = η; ∈ c) For every x X, there exist an open neighbourhood V of x and 0 ∈ 0 21 22 2. Differentiable Bundles

a differentiable map σ : V P such that pσ(x) = x for every → x V. We have only to choose for V the neighbourhood U of con- ∈ dition (P.B), and define for x V,σ(x) = γ(x, e). A continuous ∈ (resp. differentiable) map σ : V P such that pσ(x) = x for every → x V is called a cross-section resp. differentiable cross-section ) ∈ over V.

d) For every x0 X, there exist an open neighbourhood V of x0 and ∈ 1 a differentiable map ρ of p− (V) into G such that ρ(ξs) = ρ(ξ)s for every ξ p 1(V) and s G. Choose for V as before the neighbour- ∈ − ∈ hood U of (P.B.). If π is the canonical projection U G G, the × → map ρ = π γ 1 of p 1(V) into G satisfies the required condition. It ◦ − − is also obvious that ρ is bijective when restricted to any fibre.

Conversely we have the following

Proposition 1. Let G be a Lie group acting differentiably to the right on a differentiable manifold P. Let X be another differentiable manifold and p a differentiable map P X. If conditions (b), (c), (d) are fulfilled, → then P with p : P X is a principal bundle over X. → For every x X, we can find an open neighbourhood V of x , a 0 ∈ 0 differential map σ : V P and a homomorphism ρ : p 1(V) G → − → 30 such that pσ(x) = x, ρ(ξs) = ρ(ξ)s and moreover ρσ(x) = e for every x V,ξ p 1(V). We define γ : U G P by setting γ(x, s) = ∈ ∈ − × → σ(x)s for x V, s G. If θ is the map p 1(V) V G defined by ∈ ∈ − → × θ : ξ (pξ,ρξ) it is easy to verify that θγ = γθ = Identity, using the → fact that ρσ(x) = e for every x U. Both θ and γ being differentiable, ∈ our assertion is proved.

2.2 Homomorphisms of bundles

Definition 2. A homomorphism h of a differentiable principal bundle P into another bundle P′ ( with the same group G) is a differentiable map h : P P such that h(ξs) = h(ξ)s for every ξ P, s G. → ′ ∈ ∈ 2.3. Trivial bundles 23

It is obvious that points on the same fibre are taken by h into points of P′ on the same fibre. Thus the homomorphism h induces a map h : X X such that the diagram → ′ h / P / P′

p p′    h /  X / X′ is commutative. The map h : X X is easily seen to be differentiable. → ′ This is called the projection of h.

Definition 3. A homomorphism h : P P is said to be an isomorphism → ′ if there exists a homomorphism h : P P such that hoh l h oh are ′ ′ → ′ ′ identities on P′, P respectively.

Proposition 2. If P and P′ are differentiable principal bundles with the 31 same base X and group G, then every homomorphism h : P P whose → ′ projection h¯ is a diffeomorphism of X onto X, is an isomorphism.

In the case when P and P′ have the same base X, an isomorphism h : P P for which h¯ is identity will be called an isomorphismoverX ¯ . → ′ 2.3 Trivial bundles

If G is a Lie group and X a differentiable manifold, G acts on the man- ifold X G by the rule (x, s)t = (x, st). X G together with the natural × × projection X G X is a principal bundle. Any bundle isomorphic to × → the above is called trivial principal bundle. Proposition 3. Let P be a principal bundle over X with group G. Then the following statements are equivalent: 1) P is a trivial bundle. 2) There exists a differentiable section of P over X. 3) There exists a differentiable map ρ : P G such that ρ(ξs) = ρ(ξ)s → for every ξ P, s G. ∈ ∈ 24 2. Differentiable Bundles

2.4 Induced bundles

Let P be a principal bundle over X with group G. Let q be a differen- tiable map of a differentiable manifold Y into X. The subset P of Y P q × consisting of points (y,ξ) such that q(y) = p(ξ) is a closed submanifold. There is also a canonical map p : P Y defined by P (y,ξ) = y. The q q → q 32 group G acts on Pq with the law (y,ξ)s = (y,ξs). It is easy to see that Pq together with pq is a principal bundle with base Y and group G. This is called the bundle induced from P by the map q. There then exists clearly a canonical homomorphism h : P P such that q → h Pq / P

pq p  h=q  Y / X is commutative. h is defined by h(y,ξ) = ξ. By proposition (3). Ch.2.3, the principal bundle Pq is trivial if and only if there exists a differentiable cross section for Pq over Y. This is equivalent to saying that there exists a differentiable map λ : Y P such that p = p λ, i.e., the diagram is → ◦ commutative.

P ? λ   p    /  Y q / X We now assume that q is surjective and everywhere of rank = dim X. If Pq is trivial we shall say that P is trivialised by the map q. Let q be a differentiable map Y X which trivialises P. Consider → the subset Y of Y Y consisting of points (y, y ) such that q(y) = q(y ). q × ′ ′ This is the graph of an equivalence relation in Y. Since q is of rank = dim X, Y is a closed submanifold of Y Y. q × 33 Let λ be a map Y P such that q = p λ. If (y, y ) Y , then → ◦ ′ ∈ q λ(y), λ(y ) are in the same fibre and hence there exists m(y, y ) G ′ ′ ∈ such that λ(y ) = λ(y)m(y, y ). Thus we have a map m : Y G ′ ′ q → such that for (y, y ) Y , we have λ(y ) = λ(y)m(y, y ). This map is ′ ∈ q ′ ′ 2.5. Examples 25 easily seen to be differentiable. Obviously we have, for (y, y ), (y , y ) ′ ′ ′′ ∈ Yq, m(y, y′)m(y′, y′′) = m(y, y′′).

Definition 4. Let Y be differentiable manifold and q a differentiable map of Y onto X everywhere of rank = dim X. The manifold Yq is defined as above. Any differentiable map m : Y G is said to be a multiplicator q → with value in G if it satisfies

m(y, y′) m(y′, y′′) = m(y, y′′) for (y, y′), (y′.y′′) Y . ∈ q We have seen that to every trivialisation of P by q corresponds a multiplicator with values in G. However, this depends upon the par- ticular lifting λ of q. If µ is another such lifting with multiplicator n, there exists a differentiable map ρ : Y G such that m(y , y)ρ(y) = → ′ ρ(y′)n(y′, y) for every (y, y′)eYq. Accordingly, we define an equivalence relation in the set of multiplicators in the following way: The multiplicators m, n are equivalent if there exists a differentiable map ρ : Y G such that m(y, y )ρ(y ) = ρ(y)n(y, y ) for every (y, y ) → ′ ′ ′ ′ ∈ Yq. Hence to every principal bundle P trivialised by q corresponds a class of multiplicators m(P). It can be proved that if m(P) = m(P′), then P and P′ are isomorphic over X. Finally, given a multiplicator m, there exists bundle a P trivialised by q for which m(P) = m. In fact, in the space Y G, introduce an equivalence relation R by defining 34 × (y, s) (y , s ) if (y, y ) Y and s = m(y , y)s. Then the quotient ∼ ′ ′ ′ ∈ q ′ ′ (Y G)/R can be provided with the structure of a differentiable principal × bundle over X trivialised by q. The multiplicator corresponding to the map λ : Y (Y G)/R defined by y (y, e) is obviously m. → × →

2.5 Examples

1) Given a principal bundle P over X, we may take Y = P and q = p. Then λ =Identity is a lifting of q to P. The corresponding multipli- cator m is such that y′ = ym(y, y′) where p(y) = p(y′)

2) Let (Ui)i I be an open cover of X such that there exists a cross section σ of P over∈ each U . Take for Y the open submanifold of X I (with i i × 26 2. Differentiable Bundles

I discrete) consisting of elements (x, i) such that x u . Define ∈ i q(x, i) = x. This is obviously surjective and everywhere of rank = dim X. Then P is trivialised by q, since the map λ(x, i) = σi(x) of Y P is a lifting of q. The manifold Y may be identified with → q the submanifold of X I I consisting of elements x, i, j such that × × x . If m is the corresponding multiplicator, we have λ(x, i) = ∈ ∪i ∩∪ j λ(x, j)m(x, j, i). This can be written as σi(x) = σ j(x)m ji(x) where the multiplicator m is looked upon as a family of maps m : U ji j ∩ U G such that m m (x) = m (x) for every x U U U . i → ji ik jk ∈ i ∩ j ∩ k Such a family of maps is called a set of transition functions. Two 35 sets of transition functions m , n are equivalent if and only if { i j} { i j} there exists a family of differentiable maps ρ : U G such that i i → m (x)ρ (x) = ρ (x)n (x) for every x U U . Conversely, given a ji i j ji ∈ i ∩ j set of transition functions m with respect to a covering U of V, we { i j} i can construct a bundle P over V such that P is trivial over each Ui and there exists cross - sections σi over Ui satisfying σi(x) = σ j(x)mi j(x) for every x U U ∈ i ∩ j Let G be the sheaf of germs of differentiable functions on X with values in G. The compatibility relations among transition functions

viz .m (x)m (x) = m (x) for every x U U U ji ik jk ∈ i ∩ j ∩ k only state that a set of transition functions is a 1 cocycle of the covering − (Ui)i I with values in the sheaf G. Two such cocycles are equivalent (in the sense∈ of multiplicators) if and only if they differ by a coboundary. In other words, the set of equivalent classes of transition functions for 1 the covering (Ui)i I is in one-one correspondence with H ((Ui)i I, G). It ∈ 1 ∈ will be noted that there is no group structure in H ((Ui)i I, G) in gen- eral. It can be proved by passing to the direct limit that ther∈ e is a one- on correspondence between classes of isomorphic bundles over X and elements of H1(X, G).

2.6 Associated bundles

36 Let P be a differentiable principal fibre bundle over X with group G. Let 2.6. Associated bundles 27

F be a differentiable manifold on which G acts differentiably to the right. Then G also acts on the manifold P F by the rule (ξ, u)s = (ξs, us) for × every s G. ∈ Definition 5. A differentiable bundle with fibre type F associated to P is a differentiable manifold E together with a differentiable map q : P F E such that (P F, q) is a principal fibre bundle over E with × → × group G.

Let G act on a differentiable manifold F to the right. Then we can construct a differentiable bundle associated to P with fibre F. We have (P F) only to take E = × under the action of G defined as above and q G to be the canonical projection P F E. The differentiable structure in × → E is determined by the condition: (P F, q) is a differentiable principal × bundle over E. Now let E be a differentiable bundle associated to P with fibre type F and group G. Then there exists a canonical map p : E X such E → that p q(ξ, u) = pξ for (ξ, u) P F where p, q are respectively the E ∈ × projections P X and P F E. X is therefore called the base → × → manifold of E and p the projection of E. For every x X, p 1(x) is E ∈ −E called the fibre over x. Let U be an open subset of X. A continuous ( resp. differentiable ) map σ : U E such that p σ(x) = x for every → E x U is called a section (resp differentiable section) of E over U. ∈ Let σ be a differentiable section of P over an open subset of U of X. This gives rise to a diffeomorphism γ of U F onto p 1(U) defined 37 × −E by γ(x, v) = q(σ(x), v) for x U, v F. On the other hand, we also ∈ ∈ have pEγ(x, v) = x. In particular , if P is trivial, there exists a global cross-section σ (Prop.3, Ch. 2.3) and hence γ is a diffeomorphism of U F onto E. × We finally prove that all fibres in E are diffeomorphic with F. In fact for every z P. the map F E (which again we denote by z) =∈ ff → 1 defined by z(v) q(z, v) is a di eomorphism of F Onto p−E (p(z)). Such a map z : F E is called a frame at the point x = p(z). Corresponding → to two different frames z, z′ at the same point x, we have two different diffeomorphisms z, z of F with p 1(x). If s G such that z s = z, then ′ −E ∈ ′ 28 2. Differentiable Bundles

1 the we have z(v) = z′(vs− ).

Examples. (1) Let V be a connected differentiable manifold and P the universal covering manifold of V. Let p : P V be a covering → map. Then the fundamental group π1 of V acts on P and makes of P a principal bundle over V with group π1. Moreover, any covering manifold is a bundle over V associated to the universal covering manifold of V with discrete fibre. On the other hand, any Galois covering of V may be regarded as a principal bundle over V with a quotient of π1 as group.

(2) Let B be a closed subgroup of a Lie group G. Then B is itself a Lie group and it acts to the right on G according to the following rule: G B G defined by (s, t) st. Consider the quotient space × → → V = G/B under the above action. There exists one and only one structure of a differentiable manifold on V such that G is a differen- 38 tiable bundle over V with group B ([29]). It is moreover easy to see that left translations of G by elements of G are bundle homomor- phisms of G into itself. The projections of these automorphisms to the base space are precisely the translations of the left coset space G/B by elements of G.

(3) Let G be a Lie group and B a closed subgroup. Let H be a closed subgroup of B. As in (2), B/H has the structure of a differentiable manifold and B acts on B/H to the right according to the rule:

1 1 q(b)b′ = b− q(b) = q(b′− b),

where b, b B and q is the canonical projection B B/H. We ′ ∈ → define a map r : G B/H G/H by setting r(s, bH) = sbH. It × → G is easy to see that this makes G B/H a principal bundle over . × H In other words, G/H is a bundle associated to G with base G/B and fibre B/H. 2.7. Vector fields on manifolds 29

2.7 Vector fields on manifolds

Let V be a differentiable manifold and U (V) the algebra of differen- tiable functions on V. At any point ξ V, a tangent vector U is a map ∈ U : U (V) R satisfying U( f + g) = U f + Ug; U f = 0 when f is → constant; and U( f g) = (U f )g(ξ) + f (ξ)(Ug) for every f, g U (V). The ∈ tangent vectors U at a point ξ form a vector space Tξ. A vector field X is an assignment to each ξ in V of a tangent vector Xξ at ξ. A vector field X may also be regarded as a map of U (V) into the algebra of real valued functions on V by setting (X f )(ξ) = Xξ f . A vector field X is a said to be differentiable if XU (V) U (V). Hence the set of differentiable vector 39 ⊂ fields on V is the module C (V) of derivations of U (V). If p is a differentiable map from a differentiable manifold V into another manifold V , we define a map p : U (V ) U (V) by setting ′ ∗ ′ → p f = f op. Furthermore, if ξ V, then a linear map of Tξ into T ξ ∗ ∈ p (which is again denoted by p) is defined by (pU)g = U(p∗g). Now let G be a Lie group acting differentiably to the right on a manifold V. As usual, the action is denoted (ξ, s) ξs. For every → (ξ, s) V G, there are two inclusion maps ∈ × V V G defined by η (η, s); and → × → G V G defined by t (ξ, t) → × →

These induce injective maps Tξ T ξ, , T T ξ, respectively. → ( s) s → ( s) The image of dξ Tξ in T ξ, is denoted by (dξ, s). The image of ∈ ( s) ds T in T ξ, is denoted by (ξ, ds). We set (dξ, ds) = (dξ, s) + (ξ, ds). ∈ s ( s) The image of dξ Tξ by the map η ηs of V into Vwill be denoted by ∈ → dξs. Similarly, the image of vector ds T by the map t ξt will be ∈ s → denoted ξds. Therefore the image of the vector (dξ, ds) T ξ, by the ∈ ( s) map (ξ, s) ξs is dξs + ξds. In particular, the group G acts on itself → and such expressions as dst and tds will be used in the above sense. The following formulae are easy to verify:

1) (dξs)t = (dξ)(st) 40

2) (ξds)t = ξ(dst) 30 2. Differentiable Bundles

3) (ξt)ds = ξ(tds) for ξ V and s, t G. ∈ ∈

Let G be a Lie group with unit element e. The space Te of vectors at e will be denoted by Y . A vector field X on G is said to be left invariant if sX = X for every s, t G. Every left invariant vector field t st ∈ is differentiable and is completely determined by its value at e. Given a Y , we define a left invariant vector field I by setting (I ) = sa. ∈ a a s This gives a natural isomorphism of Y onto the vector space of left invariant vector fields on V. There is also a similar isomorphisms of Y onto the space of right invariant vector fields. In the same way, when G acts on a manifold V to the right, for every a Y , we define a vector ∈ field Z on V, by setting (Z )ξ = ξ a for ξ V. Thus Y defines a vector a a ∈ space of vector fields on V.

2.8 Vector fields on differentiable principal bundles

Let P be a differentiable principal over a manifold V with group G. Then the projection p : P V gives rise to a homomorphism p : → ∗ U (V) U (P). Since p is onto , p is injective. This defines on → ∗ every U (P)- module a structure of an U (V)module. It is clear that any element h p U (V) is invariant with respect to the action of G on P. ∈ ∗ Conversely, if h U (P), whenever h is invariant with respect to G then ∈ h p∗U (V). In fact if f U (V) is defined by setting f (x) = h(z) where ∈ 1 ∈ 41 z is any element in p− (x), then f coincides locally with the composite of a differentiable cross- section and h, and is consequently differentiable. For every ξ P, there exists a natural linear map of Y into Tξ taking ∈ a Y onto ξa Tξ. It is easy to verify that the sequence of linear maps ∈ ∈

(0) Y Tξ T ξ (0) → → → p( ) →

is exact. The image of Y in Tξ, i.e. the space of vectors ξ a with a Y ∈ is the space of vectors tangential to the fibre ξG and will be denoted by Nξ.

Definition 6. A vector field X on P is said to be tangential to the fibres if for every x P, p(Xξ) = 0. ∈ 2.9. Projections vector fields 31

A vector field X is tangential to the fibres if and only if Xξ Nξ for ∈ every ξ P. It is immediate than an equivalent condition for a vector ∈ field X to be tangential to the fibres is that X(p∗U (V)) = (0). We denote the set of vector fields tangential to the fibres by N. Then N is an U (P)- submodule of C (P). We have the following

Proposition 4. If a1,..., ar is a base for Y , then Zal ,... Zar is a basis for N over U (P).

In fact, for every a Y , (Za)ξ = ξa Nξ, i.e., Za N. Moreover, ∈ r ∈ r∈ = = if f1,..., fr U (P) are such that fiZai 0, then fi(ξ)(ξai) 0 ∈ i=1 i=1 r P P for every ξ P. Since ξa form a bias for Nξ, f (ξ) = 0 for i = ∈ { i}i=1 i 1, 2,..., r and Zai ,... Zar are linearly independent. On the other hand, 42 r if X N, at each point ξ P, we can write Xξ = fi(ξ)(ξ)ξai, and ∈ ∈ i=1 r P = hence X fiZai where the fi are scalar functions on P. Let ξ P i=1 ∈ and g ,..., gP U (P) such that (Z )ξg = δ for every i. j; then the 1 r ∈ ai j i j functions fi are solutions of the system of linear equations:

r = Xg j f1(Zai g j). Xi=1

Since the coefficient are differentiable and the determinant Z g , | ai j| 0 in a neighbourhood of ξ, fi are differentiable at ξ.

2.9 Projections vector fields

Let P be a principal bundle over V and X a vector field on P. It is in general not possible to define the image vector field pX on V. This is however possible if we assume that for all ξ in the same fibre , the image pXξ in the same. This yields the following.

Definition 7. A vector field X on P is said to be projectable if p(Xξ) = p(Xξ ) for every ξ P, s G. 43 s ∈ ∈ 32 2. Differentiable Bundles

Proposition 5. A vector field X on P is projectable if and only if Xp∗U (V) p U (V). ⊂ ∗ This follows from the fact that

Xξ(p∗ f ) = p(Xξ) f = p(Xξs) f = Xξs(p∗ f )

for every f U (V). ∈ Proposition 6. A vector field X is projectable if and only if Xs X is − tangential to the fibre, ie. Xs X N for every s G. − ∈ ∈ In fact, if X is projectable , we have

p(Xs X)ξ = p(X 1 s Xξ) − ξs− − = p(X 1 ) p(Xξ) ξs− − = 0.

Hence (Xs X)ξ N(ξ) for every ξ P. − ∈ ∈ The converse is also obvious from the above.

Definition 8. The projection pX of a projectable vector field X is defined by (pX) ξ = pXξ for every ξ P. p ∈ Since p (pX) f = X(p f ) for every f U (V) we see that pX is a ∗{ } ∗ ∈ differentiable vector field on V. We shall denote the space of projectable vector field by ℘. It is easy to see that if X, Y ℘, then X + Y ℘ and ∈ ∈ p(X + Y) = pX + pY. Moreover ℘ is a submodule of C (P) regarded as an U (V)-module (but not an U (P)-submodule). For f U (V) and ∈ X ℘, we have p((p f )X) = f (pX). Thus p : ℘ C (V) is an U (V)- ∈ ∗ → homomorphism and the kernel is just the module N of vector fields on P tangential to the fibre. Furthermore, for every X, Y ℘, we have ∈ [X, Y] ℘ and p[X, Y] = [pX, pY]. ∈ Proposition 7. If V is paracompact, every vector field on V is the image 44 of a projectable vector field on P. i.e., ℘ C (V) is surjective. → Let x V and U a neighbourhood of x over which P is trivial. It is ∈ clear that any vector field X in U is the projection of a vector field X in 2.9. Projections vector fields 33

1 p− (U). Using the fact that V is paracompact we obtain that there exists a locally finite cover (Ui)i I of V and a family of projectable vector fields X ℘ such that pX = X∈ on U where X coincides on U with a given i ∈ i i i i i vector field X on V. Let (ϕi)i I be a differentiable partition of unity for ∈ V with respect to the above cover. Then X = (p∗ϕi)Xi is well-defined i I ∈ and is in ℘ with projection pX = ϕi pXi = XP. i I P∈ Example. Let G be a Lie group acting differentiably to the right on a differentiable manifold V. The action of G on V is given by the map p : V G V. Consider the manifold V G with the above projection × → × onto V. G acts to the right on V G by the rule × 1 (x, s)t = (xt, t− s) for every x V, s, t G. ∈ ∈ The map γ : V G V G defined by γ(x, s) = (xs, s 1) for × → × − x V, s G is a diffeormorphism. We also have ∈ ∈ γ(x, st) = γ(x, s)t

This show that V G is a trivial principal bundle over V with group × G and projection p. A global cross -section is given by σ : x (x, e). → Let Y = Te be the space of vectors at e and Ia the left invariant 45 vector field on G whose value at e is a. Let (0, Ia) be the vector field on V G whose value at (x, s) is (x, sa). Then for every a Y , (0, I ) is × ∈ a projectable; as a matter of fact , it is even right invariant. For,

(O, Ia)(x,s)t = (x, sa)t 1 = (x, t− sa)

= (0, I ) 1 a (x,t)t− s = (0, Ia)(x,s)t

We moreover see that p(0, Ia) = Za since we have p(x, sa) = xsa = (Za)xs. We define a bracket operation [a, b] in Y by setting [Ia, Ib] = I[a,b]. Then we have in the above situation

[(0, Ia), (0, Ib)] = (0, I[a,b]). 34 2. Differentiable Bundles

Then it follows that

[Za, Zb] = [p(0, Ia), p(0, Ib)]

= p[(0, Ia), (o, Ib)]

= Z[a,b]. Chapter 3 Connections on Principal Bundles

3.1

A connection in a principal bundle P is, geometrically speaking, an as- 46 signment to each point ξ of P of a tangent subspace at ξ which is sup- plementary to the space Nξ. This distribution should be differentiable and invariant under the action of G. More precisely,

Definition 1. A connection Γ on the principal bundle P is a differentiable tensor field of type (1, 1) such that

1) Γ(X) N for every X C (P) ⊂ ∈ 2) Γ(X) = X for every X N ∈ 3) Γ(X) =Γ(X)s for every s G. ∈

Thus, at each point ξ, Γ is a projection of Tξ onto Nξ. Condition 3) is equivalent to Γ(dξ)s =Γ(dξs) for every ξ P, dξ Tξ and s G. ∈ ∈ ∈ Examples. 1) If G is a discrete group, the submodule N of U (P) is (0). Thus any projection of C (P) onto N has to be 0. Hence the only connection on a Galois covering manifold is the (0) tensor.

35 36 3. Connections on Principal Bundles

2) Let V be a differentiable manifold and G a Lie group. Consider the trivial principal bundle V G over V. The tensor Γ defined by × Γ(dx, ds) = (x, ds)

is easily seen to be a connection.

47 3) Let P be a Lie group and G a closed subgroup. Consider the principal bundle P over P/G. Let ℘ be the tangent space at e to P, and Y that at e to G. Then Y can be identified with a subspace of ℘. For a Y , ∈ the vector field Za is the left invariant vector field Ia on P. We shall now assume that in ℘, there exists a subspace M such that

1) ℘ = Y M ⊕ 2) for every s G, s 1Ms M. ∈ − ⊂ (M is only a subspace supplementary to Y , which is invariant under the adjoint representation of G in ℘. Such a space always exists if we assume that G is compact or semisimple).

Under the above conditions, there exists a connection Γ and only one on P such that Γ(M) = (0) and Γ(tb) = t(Γb) for every b ℘ and t G. ∈ ∈ The kernel of Γ is the submodule of C (P) generated by the left invariant vector field I on P with a M. Moreover, Γ is left invariant under the a ∈ action of G. Conversely, every left invariant connection corresponds to such an invariant subspace M supplementary to Y in ℘. Theorem 1. If V is paracompact, for every differentiable principal bun- dle P over V, there exists a connection on P.

We have seen that there exists a connection on a trivial bundle. Us- ing the paracompactness of V, we can find a locally finite open cover (Ui)i I of V such that P is trivial over each Ui and such that there ex- ∈ 1 ist tensor fields Γi of type (1, 1) on P whose restrictions to P− (Ui) are 48 connections. Set θ =Γ Γ for every i, j I. Then the θ satisfy the i j i − j ∈ i j following equations:

θ d ξ Nξ,θ (ξ a) = 0, and θ (dξs) = (θ dξ)s i j ∈ i j i j i j 3.2. Horizontal vector fields 37

1 for every ξ p (U U ), dξ Tξ, a Y , s G. ∈ − i ∩ j ∈ ∈ ∈ Now, let (ϕi)i I be a partition of unity with respect to the above cov- ∈ ering, i.e., support of ϕi Ui and ϕi = 1. Denote p∗ϕi byϕ ˜i. Then ⊂ 1 ϕ˜ i is a partition of unity with respectP to the covering p− (Ui).ϕ˜ kθik is a tensor on P for every k and hence

ζi = ϕ˜kθik Xk is a tensor having the following properties:

ζ dξ Nξ, ζ ξ a = 0 and ζ (dξs) = (ζ dξ)s. i ∈ i i i The tensor Γ ζ is easily seen to be a connection on p 1(U ). Since i − i − i ϕ˜ is a partition of unity, it follows that Γ ζ , Γ ζ coincide on p 1(U k i− i j− j − i∩ U ) for every i, j I. Therefore the tensor field Γ on P defined by j ∈ Γ=Γ ζ on p 1(U ) is a connection on P. i − i − i 3.2 Horizontal vector fields

Let Γ be a connection on P. By definition, Γ is a map C (P) N. The → kernel J of this map is called the module of horizontal vector fields. It is clear that Γ maps ℘ into itself. Hence we have ℘ = N (℘ J). It ⊕ ∩ is easily seen that the vector fields belonging to the U (V)-module ℘ ζ ∩ are invariant under the action of G. We have defined the projection p of 49 ℘ onto C (V). The kernel of this projection is N since the restriction of p to ℘ J is bijective. ∩ We shall now see that the module C (F) of all vector fields on P is generated by projectable vector fields.

Theorem 2. Let V be a paracompact manifold and P a principal bundle over V. Then the map η : U (P) ℘ J J defined by η( fs Xs) = U⊗(V) ∩ → ⊗ fsXs is bijective. P P The proof rests on the following

Lemma 1. C (V) is a module of finite type over U (V). 38 3. Connections on Principal Bundles

In fact, by Whitney’s imbedding theorem ([28]) there exists a reg- ular, proper imbedding of each connected component of V in R2n+1 where n = dim V. This gives us a map f : V R2n+1 defined by → y ( f, (y),..., f2n+1(y)) which is of maximal rank. Let (U j) j J be → ∈ an open covering of V such that on every U j, (d fi)i=1,2,...2n+1 contains a base for the module of differential forms of degree 1 on U j. Let S be the set of maps α : [1, n] [1, 2n + 1] such that α(i + 1) < α(i) for → i = 1, 2,... (n 1). We shall denote by Uα the union of the open sets − of the above covering in which d fα1 ... d fαn are linearly independent. Thus we arrive at a finite cover (Uα)α C having the above property. Us- ing a partition of unity for this cover,∈ any form ω on V can be written as a linear combination of the d fi. Using the map f , we can introduce a 50 Riemannian metric on V which gives an isomorphism of C (V) onto the module of differential forms of degree 1. This completes the proof of the lemma.

Proof of the theorem. Let X1,... X2n+1 be a set of generators for the module C (V) and (Uα)α S a finite covering such that for every ∈ α, Xα(1),... Xα(n) form a base for the module of vector fields on Uα. We shall now prove that the map η is injective.

Let (ϕα)α S be a partition of unity for this covering and Xs be vector ∈ n = = i fields ℘ ζ such that pXs Xs. Then ϕαXs gα,sXα(i), with ∈ ∩ i=1 n P i = i gα,s U (V) and hence ϕαXs gα,sXα(i) using the structure of ∈ i=1 2n+P1 U (V)-module on C (P). If u = fs Xs with fs U (P) is such s=1 ⊗ ∈ = = P i = that η(u) 0, then η(ϕα u) 0 for every α. Therefore fα gα,s 0 on s = i P Uα and consequently on V. But ϕα u fs gα,β Xα(i); it follows that s,i ⊗ P u = ϕα u = 0. The proof that η is surjective is similar. P Corollary. If V is paracompact, the module of horizontal vector fields on P is generated over U (P) by the projectable and horizontal vector fields. 3.3. Connection form 39

3.3 Connection form

Le P be a principal bundle over V and Γ a connection on P. If dξ Tξ, ∈ then Γ dξ Nξ and a ξa is an isomorphism of Y onto Nξ. We define ∈ → a differential form γ on P with values in Y by setting γ(dξ) = a where 51 Γ(dξ) = ξ a. In order to prove that γ is differentiable, it is enough to prove that γ takes differentiable vector fields into differentiable func- tions with values in Y . We have γ(Z ) = a for every a Y and a ∈ γ(X) = 0 for every X J. Since the module C (P) is generated by J and ∈ the vector fields Za, γ(X) is differentiable for every differentiable vector field X on P. Thus corresponding to every connection Γ on P, there ex- ists one and only one form with values in Y such that Γ(dξ) = ξγ(dξ) for every d ξ Tξ. It is easily seen that γ satisfies ∈ 1) γ(ξa) = a for every ξ P and a Y ∈ ∈ 1 2) γ(dξs) = s γ(dξ)s for dξ Tξ and s G. − ∈ ∈ A Y - valued form on P satisfying 1) and 2) is called a connection form. Given a connection form γ on P, it is easy to see that there exists one and only one connection Γ for which γ is the associated form, i.e., Γ(dξ) = ξγ(dξ) for every dξ Tξ. ∈

3.4 Connection on Induced bundles

Let P, P′ be two principal bundles over V, V′ respectively. Let h be a homomorphism of P′ into P. If γ is a connection form on P, the form h∗γ on P′ obviously satisfies conditions (1) and (2) and is therefore a connection form on P′. In particular, if P is the bundle induced by a map q : V V, then ′ ′ → γ induces a connection on P′. Let P be a differentiable principal bundle over V, and q a map 52 Y V which trivialises P, ρ being a lifting of q to P with m as the → multiplicator. As in Chapter 2.4, we denote by Y the subset of Y Y q × consisting of points (y, y′) such that q(y) = q(y′). Now ω = ρ∗γ is a dif- ferential form on Y. We have ρ(y ) = ρ(y) m(y, y ) for every (y, y ) Y . ′ ′ ′ ∈ q 40 3. Connections on Principal Bundles

Differentiating we obtain for every vector (dy, dy ) at (y, y ) Y ′ ′ ∈ q

ρ(dy′) = ρ(dy)m(y, y′) + ρ(y)m(dy, dy′)

Since ω(dy′) = γ(ρdy′) we have

1 1 γ(ρdy′) = m(y, y′)− ω(dy)m(y, y′) + γ(ρ(y)m(y, y′)m(y, y′)− m(dy, dy′)) 1 1 = m(y, y′)− ω(dy)m(y, y′) + m(y, y′)− m(dy, dy′)

Hence m(dy, dy ) = m(y, y )ω(dy ) ω(dy)m(y, y ). ′ ′ ′ − ′ Conversely if ω is a differential form on Y satisfying

m(dy, dy′) = m(y, y′)ω(dy′) ω(dy)m(y, y′) − for every (y, y ) Y , then there exists one and only one connection form ′ ∈ q γ on P such that ω = ρ∗γ. In particular, when the trivialisation of P is in terms of a covering (Ui)i I of V with differentiable sections σi(Ch.2.5), the connection form ∈ ff = γ on P gives rise to a family of di erential forms ωi σi∗γ on Ui. From 53 the equations σi(x) = σ j(x)m ji(x) defining the transition functions m ji, we obtain on differentiation,

1 1 ω (dx) = m (x)− ω (dx)m (x) + m (x)− m (dx) for x U , i ji j ji ji ji ∈ i i.e., m (dx) = m (x)ω (dx) ω (dx)m (x). ji ji i − j ji

Conversely given a family of differentiable forms ωi on the open sets of a covering (Ui)i I of V satisfying the above, there exists one and ∈ = only one connection form γ on P such that ωi σi∗γ for every i.

3.5 Maurer-Cartan equations

To every differentiable map f of a differentiable manifold V into a Lie group G, we can make correspond a differential form of degree 1 on V 1 with values in the Lie algebra Y of G defined by α(ξ)(dξ) = f (ξ)− f (dξ) 1 for ξ V and dξ Tξ. We shall denote this form by f d f . This is ∈ ∈ − easily seen to be differentiable. 3.5. Maurer-Cartan equations 41

If we take f : G G to be the identity map, then we obtain a → canonical differential form ω on G with values in Y . Thus we have ω(ds) = s 1ds Y . Also ω(tds) = ω(ds) for every vector ds of − ∈ G. That is, ω is a left invariant differential form. Moreover, it is easy to see that any scalar left invariant differential form on G is obtained by composing ω with elements of the algebraic dual of Y . In what follows, we shall provide A Y with the canonical derivation law (Ch.1.2). ⊗ Proposition 1 (Maurer-Cartan). The canonical form ω on G satisfies dω + [ω,ω] = 0. 54

We recall that the form [ω,ω] has been defined by [ω,ω](X, Y) = [ω(X),ω(Y)] for every vector fields X, Y C (G). Since the module of ∈ vector fields over G is generated by left invariant vector fields, it suffices to prove the formula for left invariant vector fields X = I , Y = I a, b a b ∈ Y . We have

d(ω)(I , I ) = I ω(I ) I ω(I ) ω([I , I ]) a b a b − b a − a b

But [Ia, Ib] = I[a,b] and ω[Ia, Ib] = ω(I[a,b]) = [a, b] = [ω(Ia),ω(Ib)] since ω(Ia) = a,ω(Ib) = b. Corollary. If f is a differentiable map V G, then the form α = f 1d f → − satisfies dα + [α,α] = 0.

1 In fact, α(dξ) = f − (ξ) f (dξ) = ω( f (dξ)) = ( f ∗ω)(dξ) and hence α = f ∗ω. The above property of α is then an immediate consequence of that of ω. Conversely, we have the following

Theorem 3. If α is a differential form of degree 1 on a manifold V with values in the Lie algebra Y of a Lie group G satisfying dα + [α,α] = 0, then for every ξo V and a differentiable map f : U G such that 1 ∈ → f − (ξ) f (dξ) = α(dξ) for every vector dξ of U.

Consider the form β = p α p ω on V G where p : V G V 55 1∗ − 2∗ × 1 × → and p : V G G arc the two projections and ω the canonical left 2 × → invariant form on G. If β is expressed in terms of a basis a ,... a of { 1 r} 42 3. Connections on Principal Bundles

Y , the component scalar differential forms βi are everywhere linearly 1 independent. This follows from the fact that on each p1− (ξ), β reduces ff to p2∗ω. We shall now define a di erentiable and involutive distribution on the manifold V G. Consider the module M of vector fields X on × V G such that β (X) = 0 for every β . We have now to show that × i i X, Y M [X, Y] M. But this is an immediate consequence of the ∈ ⇒ ∈ relations d α = [α,α] and dω = [ω,ω]. − By Frobenius’ theorem (see [11]) there exists an integral submani- fold W (of dim = dim V) of V G in a neighbourhood of (ξo, e). Since , × 1 = , for every vector (ξo, a) 0 tangent to p1− ξo, β(ξo, a) a 0 there exist a neighbourhood U of ξ and a differentiable section σ into V G over o × U such that σ(U) W. Define f (ξ) = p σ(ξ). By definition of W, it ⊂ 2 follows that βσ(dξ) = 0. This means that

αp1σ(dξ) ωp2σ(dξ) = 0 − 1 i.e., α(dξ) = ω(p2σ(dξ)) = ( f ∗ω)dξ = f − (ξ) f (dξ).

Remark. When we take G = additive group of real numbers, the above theorem reduces to the Poincare’s theorem for 1-forms. Concerning the uniqueness of such maps f , we have the

Proposition 2. If f1, f2 are two differentiable maps of a connected man- 1 = 1 56 ifold V into a Lie group G such that f1− d f1 f2− d f2, then there exists an element s G such that f = f s. ∈ 1 2 Define a differentiable function s : V G by setting → 1 s(ξ) = f (ξ) f − (ξ) for every ξ V. 1 2 ∈ Differentiating this, we get

s(dξ) f2(ξ) + s(ξ) f2(dξ) = f1(dξ).

Hence s(dξ) = 0. Therefore s is locally a constant and since V is connected s is everywhere a constant. Regarding the existence of a map f in the large, we have the follow- ing 3.6. Curvature forms 43

Theorem 4. If α is a differential form of degree 1 on a simply connected manifold V with values in the Lie algebra Y of a Lie group G, such that dα + [α,α] = 0, then there exists a differentiable map f of V into G such 1 that f − d f = α. The proof rests on the following

Lemma 2. Let mi j be a set of transition functions with group G of a simply connected manifold V with respect to a covering (Ui). If each mi j is locally a constant, then there exists locally constant maps λ : U G i i → such that m (x) = λ (x) 1λ (x) for every x U U . i j i − j ∈ i ∩ j In fact, there exists a principal bundle P over V with group G con- sidered as a discrete group and with transition functions mi j (Ch.2.5). P is then a covering manifold over V and hence trivial. Therefore the 57 maps λ : U G exist satisfying the conditions of the lemma (Ch.2.5). i i → Proof of the theorem. Let (Ui)i I be a covering of V such that on each ∈ 1 Ui, there exists a differentiable function fi satisfying fi(x)− fi(dx) = α(dx) for every x U . We set m (x) = f (x) f (x) 1. It is obvious that ∈ i i j i j − the mi j form a set of locally constant transition functions. Let λi be the maps U G of the lemma. Then λ (x) f (x) = λ (x) f (x) for every i → i i j j x Ui U j. The map f : V G which coincides with λi fi on each Ui ∈ ∩ 1 → is such that f − d f = α.

3.6 Curvature forms

Definition 2. Let γ be a connection form on a principal bundle P over V. Then the alternate form of degree 2 with values in the Lie algebra Y of the structure group G defined by K = dγ + [γ, γ] is said to be the curvature form of γ. A connection on P is said to be flat if the form K 0. ≡ Theorem 5. The following statements are equivalent: a) The connection γ is flat i.e., dγ + [γ, γ] = 0. b) If X, Y are two horizontal vector fields on p, so is [X, Y]. 44 3. Connections on Principal Bundles

c) For every x V, there exists an open neighbourhood U of x and a o ∈ o differentiable section σ on U such that σ∗γ = 0. Proof. a) b) is obvious from the definition. ⇒ b) c) ⇒ 

58 Let a1, a2,... an be a basis of Y and let γ = γiai where γi are ff = scalar di erential forms. Since γi(Za j) δi j, it followsP that γ1,...γr { } form a set of differential forms of maximal rank r. From b), it follows that the module J of horizontal vector fields is stable under the bracket operation. Hence J forms a distribution on P which is differentiable and involutive. Let ξ P such that p(ξ ) = X and let W be an integral o ∈ o o manifold W of dim = dim V in a neighbourhood of ξo. As in Th.3, Ch.3.5, W is locally the image of a differentiable section σ over an open neighbourhood of xo. Since all the tangent vectors of W are horizontal, we have γσ(dx) = 0 for every vector dx on U. c) a) ⇒ Let σ be a differentiable section over an open subset U of V such that σ γ = 0. For every ξ p 1(U), define a differentiable function ∗ ∈ − ρ : p 1(U) G by the condition ξ = σ(p(ξ))ρ(ξ). Differentiating this, − → we obtain dξ = σp(dξ)ρ(ξ) + σp(ξ)ρ(dξ) 1 1 Hence γ(dξ) = ρ− (ξ)γ(σp(dξ))ρ(ξ) + ρ(ξ)− ρ(dξ) by conditions (1) 1 and (2) for connection forms. i.e. γ(dξ) = ρ(ξ)− ρ(dξ) and a) follows from cor. to prop.1, Ch.3.5.

59 Theorem 6. If there exists a flat connection on a principal bundle P over a simply connected manifold V, then P is a trivial bundle. Moreover, a differentiable cross-section σ can be found over V such that σ∗γ = 0.

By Th.4, Ch.3.5, there exists an open covering (Ui)i I and cross sec- tions σ : U P such that σ γ = 0. Let σ (x)m (x) =∈σ (x) where the i i → i∗ i i j j corresponding transition functions are mi j. Then we have

γ(σi(dx)mi j(x) + σi(x)mi j(dx)) = γσ j(dx). 3.6. Curvature forms 45

Therefore, mi j(dx) = 0, i.e. the mi j are locally constant. Lemma 2, Ch.3.5 then gives a family λ of locally constant maps U G such that i i → = 1 mi j λi− λ j

It is easily seen that σ λ 1 = σ λ 1 on U U . Define a cross i i− j −j i ∩ j = 1 section σ on V by setting σ σiλi− on every Ui. Then we have = 1 γ(σ(dx)) γ(σi(dx)λi− (x)) = 1 λi(x)(γσi(dx))λi− (x) = 0 for x U . ∈ i

Hence the bundle is trivial and σ∗γ = 0.

Proposition 3. If X is a vector field on P tangential to the fibre, then 60 K(X, Y) = 0 for every vector field Y on P.

In fact, since ℘ = ℘ J N and ℘ generates C (P), it is enough to ∩ ⊕ prove the assertion for Y N and Y ℘ ζ. In the first case X = Z and ∈ ∈ ∩ a Y = Zb, we have

K(Za, Zb) = (dγ + [γ, γ])(Za, Zb) = Z γ(Z ) Z γ(Z ) γ[Z , Z ] + [γ(Z ), γ(Z )] a b − b a − a b a b = γ(Z , ) + [a, b] − [a b] = 0

In the second case, Y is invariant under G and it is easy to see that [Za, Y] = 0. Since γ(Y) = 0, we have

K(Za, Y) = (dγ + [γ, γ])(Za, Y) = Z γ(Y) Yγ(Z ) γ[Z , Y] + [γ(Z ), γ(Y)] a − a − a a = 0.

Proposition 4.

1 K(d ξs, d ξs) = s− K(d ξ, d ξ)s for ξ P, d ξ, dαξ Tξ and s G. 1 2 1 2 ∈ 1 ∈ ∈ 46 3. Connections on Principal Bundles

By Prop.3, Ch.3.6, it is enough to consider the case when d ξ, d ξ 1 2 ∈ Jξ. Extend d1ξ, d2ξ to horizontal vector fields X1, X2 respectively. Then we have

K(d1ξs, d2ξs) = K(X1 s, X2 s)(ξ) = γ[X s, X s](ξ) − 1 2 = γ([X , X ]s)(ξ) − 1 2 = γ([X , X ]ξ s) − 1 2 1 = s− γ([X , X ]ξ)s − 1 2 1 = s− K(X1, X2)(ξ)s 1 = s− (d ξ, d ξ)s for every s G. 1 2 ∈ 61

3.7 Examples

1. Let dim V = 1. Then C (V) is a face module generated by a single vector field. But C (V) is U (V)-isomorphic to J ℘ and hence ∩ J = U (P) J ℘ is also U (P) - free, and of rank 1. Since K ∩ UN(V) is alternate, K 0. In other words, if the base manifold of P is a ≡ curve, any connection is flat.

2. Let G be a closed subgroup of a Lie group H and let Y , F be their respective LIe algebras. We have seen that (Example 3, Ch.3.1) if M is a subspace of F such that F = M Y and s 1Ms M ⊕ − ⊂ for every s G, then the projection Γ : F M gives rise to a ∈ → connection which is left invariant by element of G. Denoting by I the left invariant vector fields on H whose values at e is a F , a ∈ we obtain

K(I , I ) = γ([I , I ]) + [γ(I ), γ(I )] a b − a b a b = γI , + [Γ a, Γ b] − [a b] e e = Γ ([a, b]) + [Γ (a), Γ (b)]. − e e e 3.7. Examples 47

62 Thus K is also left invariant for elements of G. Moreover Γ is flat if and only if Γe is a homomorphism of J onto U . This again is true if and only if M is an ideal in J .

Chapter 4 Holonomy Groups

4.1 Integral paths

We shall consider in this chapter only connected base manifolds. 63

Definition 1. A path ψ in a manifold W is a continuous map ψ of the unit interval I = [0, 1] into W. A path ψ is said to be differentiable if ψ can be extended to a differentiable map of an open neighbourhood of I into W. ψ(0) is called the origin and ψ(1) the extremity, of the path. The path t ψ(1 t) is denoted by ψ 1. If γ is a connection on a principal → − − bundle P over X, then a differentiable path ψ in P is said to be integral 1 if ψ∗γ = 0. If ψ is integral, so is ψ− .

Let ψ be a path in X. A path ψˆ in P such that po ψˆ = ψ is called a lift of ψ. It is easy to see that if ψˆ is an integral lift of ψ, so is ψˆ s, where 1 ψˆ s is defined by ψˆ s(t) = ψˆ(t)s. If ψˆ is an integral lift of ψ, then ψˆ − is an 1 integral lift of ψ− .

Theorem 1. If ψ is a differentiable path in V with origin at x V, then ∈ for every ξ P such that p(ξ) = x, there exists one and only one integral ∈ lift of ψ with origin ξ.

Let I′ be an open interval containing I to which ψ can be extended. 64 The map ψ : I V defines an induced bundle Pψ with base I and a ′ → ′ 49 50 4. Holonomy Groups

canonical homomorphism h of Pψ into P such that the diagram

h Pψ / P

q p

   ψ /  I′ / V

is commutative. The connection γ on P also induces a connection h∗γ on Pψ. Since I′ is a curve, by Ex.1 of Ch.3.7 and Th.5 of Ch.3.6, there exists a section σ such that the inverse image by σ of h∗γ is zero. i.e. σ h γ = 0. Define ψˆ : I P by setting ψˆ = hσ on I . Then it is ∗ ∗ ′ → ′ obviously an integral lift of ψ. If s G is such that ξ = ψˆ(0)s, then ψˆ s ∈ is an integral lift of ψ with origin ξ. Ifϕ, ˆ ψˆ are two such lifts, then we may define a map s : I G such → that ψˆ(t) = ϕˆ(t)s(t) for every t I. ∈ Sinceϕ, ˆ ψˆ are differentiable, it can be proved that s is also differ- entiable using the local triviality of the bundle. Differentiation of the above yields ψˆ(dt) = ϕˆ(dt)s(t) + ϕˆ(t)s(dt). Hence

γ(ϕ ˆ(dt)s(t)) + γ(ϕ ˆ(t)s(dt)) = 0.

Using conditions (1) and (2) of connection form, we get s(dt) = 0. 65 Hence s is a constant. Sinceϕ ˆ(0) = ψˆ(0), the theorem is completely proved.

4.2 Displacement along paths

Let Px be the fibre at x and ψ a path with origin x and extremity y. For every ξ P , there exists one and only one integral lift ψˆ of ψ whose ∈ x origin is ξ. The extremity of ψˆ is an element of the fibre at Py. We shall denote this by τψξ. Then τψ is said to be a displacement along ψ. Any displacement along the path ψ commutes with the operations of G in the sense that τψ(ξs) = (τψξ)s for every ξ P ; therefore τψ is ∈ x 4.3. Holonomy group 51

differentiable and is easily seen to be a bijective map Px Py. Trivially, 1 → (τ ) = τ 1 . It is also obvious that the displacement is independent of ψ − ψ− the parameter, i.e., if θ is a differentiable map of I into I such that

1) θ(0) = 0,θ(1) = 1;

2) ψ′θ = ψ

τ = τ then ψ′ ψ Given a differentiable path ψ, we may define a new path ψ′ by suit- ably changing the parameter so that ψ′ has all derivatives zero at origin and extremity. By the above remark, we have τ 1 = τ . ψ− ψ

4.3 Holonomy group

Definition 2. A chain of paths in V is a finite sequence of paths ψ , { p ...,ψ where ψ is a path such that ψ + (0) = ψ(1) for i p 1. 1} i i 1 ≤ − We define the origin and extremity of the chain ψ to be respectively 66 ψ1(0) and ψp(1). Given two chains ψ = ψp, ψp 1,...ψ1 ,ϕ = ϕq,ϕq 1,...ϕ1 such that origin of ψ = { − } { − } extremity of ϕ, we define ψϕ = ψ ,...ψ ,ϕ ,...ϕ . ϕ 1 is defined to { p 1 q 1} − be ϕ 1,...ϕ 1 and it is easily seen that (ψϕ) 1 = ϕ 1ψ 1. { 1− q− } − − 1− A displacement τψ along a chain ψ = ψ ,...ψ in the base mani- { p 1} fold V of a principal bundle P is defined by τψ = τψ ,... τψ . p◦ ◦ 1 Let x, y V. Define Φ(x, y) to be the displacements τψ where ψ is ∈ a chain with ψ(0) = x, ψ(1) = y. It can be proved, by a suitable change of parameters of the paths ψi (ch. 4.2), that there exists a differentiable path ϕ such that τϕ = τψ. The following properties are immediate con- sequences of the definition:

1) α Φ(x, y) = α 1 Φ(y, x). ∈ ⇒ − ∈ 2) α Φ(x, y), β Φ(y, z) = βα Φ(x, y). ∈ ∈ ⇒ ∈ 3) Φ(x, y) is non empty since V is connected. 52 4. Holonomy Groups

We shall denote Φ(x, x) by Φ(x). Φ(x) is then a subgroup of the group of automorphisms of Px commuting with operations of G. This is called the holonomy group at x with respect to the given connection on the principal bundle P. If we restrict ourselves to all chains homotopic to zero at x, then we get a subgroup Φr(x) of Φ(x), viz., the subgroup of displacements τψ where ψ is a closed chain at x homotopic to zero. This is called the restricted holonomy group at X. This is actually a normal 67 subgroup of Φ(x), for if ψ is homotopic to 0 and ϕ any continuous path 1 ϕψϕ− is again homotopic to zero. Proposition 1. For every x V, there exists a natural homomorphism ∈ Φ Φ of the fundamental group 1(V, x) onto (x)/ r(x). Q Let ψ be a closed path at x. Then there exists a differentiable path ψ′ homotopic to ψ. We map ψ on the coset of Φr(x) containing τψ1. If 1 ϕ and ψ are homotopic, so are ϕ′ and ψ′ i.e. ψ′ϕ′− is homotopic to = 1 Φ zero. Then τψ ϕ 1 τψ, τϕ− r. Thus we obtain a canonical map ′ ′− ∈ θ : (V, x) Φ(x)/Φ (x) which is easily seen to be a homomorphism 1 → r of groups.Q That this is surjective is immediate. In general, θ will not be injective. As the fundamental group of manifolds which are countable at is itself countable, so isΦ(x)/Φ (x). ∞ r Hence from the point of view of structure theory, a study of Φr(x) is in most cases sufficient.

4.4 Holonomy groups at points of the bundle

If x, y are two points of the connected manifold V and ϕ a path join- ing x and y, then there exists an isomorphism Φ(y) φ(x) defined by → τψ τϕ 1τψτϕ for every path closed at y. The image of Φr(y) under → − this isomorphism is contained in Φr(x). However, this isomorphism is not canonical depending as it does on the path ϕ. The situation can be improved by association to each point of the bundle P a holonomy group which is a subgroup of G. Let Ax be the group of automorphisms of Px commuting with oper- 68 ations of G. Let ξ P , Then we define an isomorphism λξ : A G ∈ x x → by requiring αξ = ξλξ(α) for every α A . That λξ is a homomorphism ∈ x 4.5. Holonomy groups for induced connections 53 and is bijective is trivial. At every ξ P, we define the holonomy group ∈ ϕ¯(ξ) to be λξΦ(pξ). Similarly the restricted holonomy group Φr(ξ) at ξ is λξΦr(pξ).

Theorem 2.

For any two points ξ, η P, Φ(ξ), Φ(η) are conjugate subgroups of ∈ G. Moreover, if ξ, η lie on an integral path, then Φ(ξ) = Φ(η). If ξ, η are on the same integral path, then η = τϕξ for some path ϕ with origin pξ = x and extremity pη = y. It is sufficient to prove that Φ(ξ) Φ(η). Let ψ be a closed chain at x. Then ⊂

ξ(λξτψ) = τψξ = 1 τψτϕ− η = 1 = 1 τϕ− τψ′ η where ψ′ ϕψϕ− = 1 τϕ− (ηλη(τψ′ )) = ξλη(τψ′ )

Hence λη(τψ ) = λξ(τψ) Φ(η) and our assertion is proved. ′ ∈ Finally, if ξ, η are arbitrary, we may join pξ, pη by a path ϕ. Letϕ ˆ be an integral lift of ϕ through ξ and η′ its extremity. Then there exists s G such that η = ηs. We have already proved that Φ(ξ) = Φ(ηs). 69 ∈ ′ Now

(ηs)(ληs(α)) = α(ηs)

= (ηλη(α))s

1 Hence Φ(ξ) = s− Φ(η)s, which completes the proof of theorem 2.

4.5 Holonomy groups for induced connections

Let h be a homomorphism of a principal bundle P′ over V′ into a princi- pal bundle P over V. Let h be the projection of h. Let γ be a connection form on P and h γ the induced connection on P . If ξ P and ∗ ′ ′ ∈ ′ h(ξ′) = ξ, then Φ(ξ′) Φ(ξ). Moreover, Φ(ξ′) is the set of λξ(τh ) ⊂ ψ′ 54 4. Holonomy Groups

where ψ′ is a closed chain at pξ′ = x′. For, let s Φ(ξ′) corresponding = ∈ to the closed chain ψ′ at x′. Then ξ′ s τψ′ ξ′. = ξs h(τψ′ ξ′) = τψξ where ψ = hψ′

= ξλξ(τψ).

Therefore s = λξ(τψ) and it is obvious that any such λξ(τψ) belongs to Φ(ξ′). In particular, let V′ be the universal covering manifold of V, h the covering map, P′ a principal bundle over V′ and h a homomorphism 70 P P whose projection is h. Then for ξ P , Φ(ξ ) is the set of ′ → ∈ ′ ′ λξτψ where ψ is the image under h of a closed chain at pξ′. But V′ being simply connected, ψ is any closed chain at ξ homotopic to zero and hence Φ(ξ′) = Φr(ξ).

4.6 Structure of holonomy groups

Theorem 3.

For every ξ P, Φ (ξ) is an arcwise connected subgroup of G. ∈ r If ψ is a closed path at x V which is homotopic to zero, then it can ∈ be shown that there exists a differentiable map ϕ : I I V(I being ′ × ′ → ′ a neighbourhood of I) such that ϕ(t, 0) = ψ(t) and ϕ(t, 1) = x for every t I. This can be lifted into a mapϕ ˆ : I I P such that ∈ ′ × ′ → 1) pϕˆ(t,θ) = ϕ(t,θ) for every t,θ I; ∈ 2) For every θ I, t ϕˆ(t,θ) is an integral path; ∈ → 3)ϕ ˆ(0,θ) = ξ with p(ξ) = x.

In fact, ϕ : I I V induces a bundle Pϕ over I I . Let γϕ be the ′ × ′ → ′ × ′ induced connection on Pϕ. For every θ I , the path t (t,θ) in I I ∈ ′ → ′ × ′ can be lifted to an integral pathσ ˆ θ in Pφ with origin at ((0,θ),ξ) Pϕ. ∈ The origin depends differentiably on θ and therefore the map σ of I I ′ × ′ 4.7. Reduction of the structure group 55 defined by σ(t,θ) = σθ(t) is differentiable. Hence ρ σ = ϕˆ satisfies the ◦ conditions 1), 2) and 3). Now we define a function s : I G by requiring that ξs(θ) = → ϕˆ(1,θ). Obviously s(1) = e and s(0) is the element of the restricted 71 holonomy group at ξ corresponding to the path ψ. It is also easy to verify that s(θ) is a differentiable function of θ using the local triviality of P. Therefore any point of Φr(ξ) can be connected to e by a differentiable arc.

Corollary. Φr(ξ) is a Lie subgroup of G. This follows from the fact that any arcwise connected subgroup of a Lie group is itself a Lie group [31].

4.7 Reduction of the structure group

Theorem 4.

The structure group of P can be reduced to Φ(ξ) for any ξ P. ∈ For this we need the following

Lemma 1. Let M(ξ) be the set of points ξ′ of P which are extremities of integral paths emanating from ξ. There exists an open covering (Ui)i I of V and cross-sections σ over U such that σ (x) M(ξ) for every∈ i i i ∈ x U . ∈ i Let x = pξ and x V. Consider the path ψ connecting x and x . 0 ∈ 0 Then the integral lift ψˆ of ψ with origin ξ has extremity ξ0 with pξ0 = x0. Let U be a neighbourhood of x0 in which are defined n linearly = independent vector fields X1,... Xn where n dim V. Let X1, X2,... Xn 1 = be horizontal vector fields on p− (U) such that pXi Xi for every i. By the theorem on the existence of solution for differential equations, there exists a neighbourhood of (0, 0,...) in R Rn in which is defined × a differentiable map ξ having the properties

1) ξ(0, a) = ξ0 72

2) ξ′(t, a) = tai(Xi)ξ(t,a). P 56 4. Holonomy Groups

From the uniqueness of such solutions, we get ξ(λt, a) = ξ(t, λa) for sufficiently small values of λ, t and a. Thus there exists a differen- tiable map ξ having the properties 1) and 2) above on [0, 2] W where n × W is a neighbourhood of (0,...) in R . Since the Xi are horizontal, ξ(t, a) M(ξ). Now consider the differentiable map g : W P defined ∈ → by g(a) = ξ(1, a). The map p.g is of maximal rank. Moreover since dim V = n, this is locally invertible. In other words, in a neighbourhood U of X we have a differentiable map f : U W such that p g f = ′ 0 ′ → ◦ ◦ Identity, i.e., gof is a cross section over U, and

( gof ) (y) = ξ(1, f (y)) M(ξ) for every y U′. ∈ ∈ Proof of theorem 4. We have only to take for transition functions the functions mi j such that σ (x) = σ (x)m (x) for every x U U i j ji ∈ i ∩ j where the σi have the properties mentioned in the lemma. Then it is obvious that m (x) Φ(σ (x )) = Φ(ξ) by theorem 2, Ch. 4.3. i j ∈ i 0

Now let V be simply connected. Then Φ(ξ) = Φr(ξ). Since Φ(ξ) 73 is a Lie subgroup of G, it follows [11] that the m ji are differentiable functions as maps: U U Φ(ξ) also. If ρ are the diffeomorphisms i ∩ j → i U G p 1(U ) defined by ρ (ξ, s) = σ (ξ)s for ξ U , s G. i × → − i i i ∈ i ∈ Consider the set W consisting of ρ (ξ, s) with ξ U , s Φ(ξ). It is i i ∈ i ∈ clear that W p 1(U ) = W p 1(U ). Moreover if we provide each i ∩ − j j ∩ − i Wi with the structure of a differentiable manifold by requiring that the ρ be a diffeomorphism, then the differentiable structures on W W i i ∩ j agree. In other words, M(ξ) = Wi has a differentiable structure (with i I which it is a submanifold of P).S∈ Φ(ξ) acts on M(ξ) differentiably to the right and makes of it a principal bundle over V with projection p. We denote the inclusion map M(ξ) P by f . → Proposition 2. Let dη be a vector at the point η of the manifold M(ξ) and Y (ξ) the Lie subalgebra of Y corresponding to Φ(ξ), then γ( f dη) ∈ Y (ξ). 4.8. Curvature and the holonomy group 57

Since the connection form is 0 on horizontal vectors, it is enough to consider the case when f dη is tangential to the fibre, i.e. is of the form sa with a Y (ξ), s Φ(ξ). In this case γ(sa) = a Y (ξ) which proves ∈ ∈ ∈ the assertion. If we define γ∗(dη) = γ( f dη) for every vector dη of M(ξ) then γ∗ is again a connection form on the principal bundle M(ξ). For every point η M(ξ), the holonomy group corresponding to γ is Φ(ξ). ∈ ∗ 4.8 Curvature and the holonomy group

Theorem 5.

For ξ P, Φ (ξ) = (e) if and only if the curvature form is identically 74 ∈ r zero. Consider the universal covering manifold V˜ of V with covering map ˜ = h. Let P Ph be the induced bundle with h as the homomorphism P˜ P. Let ξ˜ P˜ such that hξ˜ = ξ. Then by Ch.4.5., the holonomy → ∈ group Φ(ξ˜) with respect to the induced connectionγ ˜ = h∗γ on P˜ is Φr(ξ). Therefore, if Φr(ξ) = (e), M(ξ˜) is the image of a cross-sectionσ ˜ of P˜ over V. By prop. 4, Ch.3.6,γ ˜σ˜ (d x˜) = 0 for everyx ˜ V˜ . Hence ∈ by th.5, Ch.3.6 the curvature is 0. Since V˜ is locally diffeomorphic with V the curvature form of γ is 0. Conversely, if K = 0, K˜ = 0 and by th.6, ch.3.6, there exists a sectionσ ˜ of P˜ over V˜ such thatγ ˜σ˜ = 0 and such thatσ ˜ (x ˜) = ξ˜. Hence the holonomy group at ξ˜ reduces to e , i.e., { } Φ (ξ) = e . r { } Theorem 6 (Ambrose-Singer; [1]).

The Lie algebra of the restricted holonomy group Φ (ξ) at ξ P is r ∈ the subspace of the Lie algebra G of G generated by the values K(d1η, d η) of the curvature form with d η, d η Tη, η fM(ξ). We may 2 1 2 ∈ ∈ assume that V is simply connected in which case Φ(ξ) = Φr(ξ), the general case being an easy consequence. Moreover, since it is enough to take d1η, d2η to be horizontal, we may consider the values of K on the manifold M(ξ). We may therefore restrict ourselves to the case when Φr(ξ) = G, and M(ξ) = P. Let I be the subalgebra of G generated by 58 4. Holonomy Groups

the values of K. Let L be the set of vector fields X on P such that γ(X) is a function with values in I . This is an U (P)- submodule. It is easily 75 seen that L has everywhere rank = dim V + dim I , using the fact that γ(Z ) = a I for a I . Moreover, a ∈ ∈ γ([X, Y])ξ = K(X, Y)ξ Xγ(Y)ξ + Yγ(X)ξ + [γ(X), γ(Y)]ξ − − I for every X, Y L and ξ P. ∈ ∈ ∈

By Frobenius’ theorem, there exists an integral manifold for the dis- tribution given by L . Let L be the maximal integral manifold for L containing ξ. Since the integral paths for γ with origin at ξ are integral with respect to L we must have L = P. Hence dim I = dim G , or I = G . Finally it remains to prove that the subspace M of U generated by the values of K is itself a Lie subalgebra. Since K(d ηs, d ηs) = s 1K(d η, d η)s I , it follows that s 1Ns 1 2 − 1 2 ∈ − ⊂ N for every s G. Therefore for every X I , [X, N] N. In particular, ∈ ∈ ⊂ [M , M ] M . ⊂ We now give a geometric interpretation of K(d ξ, d ξ) for ξ P. 1 2 ∈ Firstly, we may assume d1ξ, d2ξ to be both horizontal vectors (Prop. 3, Ch.3.6). Let x = pξ, d1 x = pd1ξ and d2 x = pd2ξ. Extend d1 x, d2 x = to vector fields X1 and X2 on V such that [X1, X2] 0. Let X1, X2 be the corresponding projectable, horizontal vector fields on P. Let η → Fi(η, t) be the automorphism of parameter t defined on a neighbourhood of ξ by the vector field Xi(i = 1, 2). We set ξ1(θ) = F1(θ,ξ),ξ2(θ) = F (θ,ξ (θ)),ξ (θ) = F ( θ,ξ (θ)) and ξ (θ) = F ( θ,ξ (θ)). The chain 2 1 3 2 − 2 4 1 − 3 76 of paths F (t/θ,ξ), F (t/θ,ξ (θ)), F ( t/θ,ξ (θ)), F ( t/θ,ξ (θ)) is not 1 2 1 1 − 2 2 − 3 closed in general though its projection on V is closed (since [X1, X2] = 2 2 0). Let s(θ ) be the element of G. such that ξ4(θ) = ξs(θ ). Then [X1, X2]ξ = ξs′(0) and we have

[X1, X2]ξ = γ([X1, X2]ξ)

= K(X ξ, X ξ) − 1 2 = K(d ξ, d ξ). − 1 2 4.8. Curvature and the holonomy group 59

Chapter 5 Vector Bundles and Derivation Laws

5.1

Let P be a differentiable principal bundle over a manifold V with group 77 G. Consider a vector space L of finite dimension over the field R of real numbers. Let s s be a linear representation of G in L.G acts → L differentiably on L (regarded as a manifold ) to the right by the rule = 1 vs s−L v. Let E be a fibre bundle associated to P with fibre L and q the map P L E. For every ξ P, the map v q(ξ, v) is a bijection of L onto × → ∈ → the fibre of E at pξ, or a frame of L. On each fibre E at x V of the x ∈ associated bundle E, we may introduce the structure of a vector space by requiring that the frame defined by any point ξ p 1(x) be linear. It ∈ − is clear that such a structure does exist and is unique. Let U be an open subset of V over which P is trivial and ρ a diffeomorphism U L 1 × → p− (U) defined by ρ(x, v) = q(σ(x), v) where σ is a differentiable cross section of P over U. Then we have

1) p ρ(x, y) = x

2) ρ(x, v + v′) = ρ(x, v) + ρ(x, v′)

3) ρ(x, λv) = λρ(x, v)

61 62 5. Vector Bundles and Derivation Laws

for every x U, v, v L and λ R. ∈ ′ ∈ ∈ Conversely, let E be a differentiable manifold and p : E Va 1 → 78 differentiable map. Assume each p− (x) = Ex to be a vector space over R. The manifold E (together with p) is called a differentiable vector bundle (or simply a vector bundle ) over V if the following condition is satisfied. For every x V, there exist an open neighbourhood V of x 0 ∈ 0 and a differentiable isomorphism ρ of U Rn onto p 1(U) such that × − 1) pρ(x, v) = x

2) ρ(x, v + v′) = ρ(x, v) + ρ(x, v′) 3) ρ(x, λv) = λρ(x, v) for every x V, v, v L and λ R. ∈ ′ ∈ ∈ Proposition 1. Every vector bundle of dimension n over V is associated to a principal bundle over V with group GL(n, R). In fact, let E be a vector bundle of dimension n over V and q : E V the projection of E. For every x V, define Px to be the set → n ∈ of all linear isomorphisms of R onto Ex and P = Px. The group x V G = GL(n, R) acts on P by the rule (ξs)v = ξ(sv)S for∈ every v Rn. ∈ Define p : P V by p(P ) = x. By definition of vector bundles, → x there exist an open covering Uα V and a family of diffeomorphisms n 1 ⊂ ρα : Uα R q− (Uα) satisfying conditions 1), 2) and 3). For every × → 1 α, the map γα : Uα G p (Uα) defined by γα(x, s)v = ρα(x, sv) × → − for x Uα, s G is bijective. We put on P a differentiable structure ∈ ∈ by requiring that all γα be diffeomorphisms. It is easy to see that on the overlaps the differentiable structures agree. Moreover,

γα(x, st)v = ρα(x, (st)v)

= ρα(x, s(tv))

= γα(x, s)(tv) n 79 for x Uα, s, t G and v R . Thus P is a principal bundle over V with ∈ ∈ ∈ group G. Let q be the map of P Rn onto E defined by q (ξ, v) = ξv for ′ × ′ ξ P, v Rn. It is easy to see that P Rn is a principal bundle over E ∈ ∈ × with projection q′. Hence E is an associated bundle of P. 5.2. Homomorphisms of vector bundles 63

5.2 Homomorphisms of vector bundles

Definition 1. Let V and V′ be two differentiable manifolds, E a vector bundle over V and E′ a vector bundle over V′.A homomorphism h of E into E is a differentiable map h : E E such that such that, for ′ → ′ every x V, p h(E ) reduces to a point x V and the restriction h of ∈ ′ x ′ ∈ ′ x h to Ex is a linear map of Ex into E′x.

The map h : V V defined by the condition p h = hp is differen- → ′ ′ tiable and is called the projection of h. If V = V′, by a homomorphism of E into E′, we shall mean hereafter a homomorphism having the iden- tity map V V as projection. In that case, h : E E is injective → → ′ or surjective according as all the maps h : E E are injective or x x → ′x surjective. A vector bundle of dimension n over V is said to be trivial if it is isomorphic to the bundle V Rn (with the natural projection V Rn V). × × → Let P be a principal bundle over V to which the vector bundle E is 80 associated. If P is trivial, so is E and every section of P defines an isomorphism of V L onto E. × Let E, E′, E′′ be vector bundles over V. Then a sequence of homo- morphisms h k E′ E E′′ −→ −→ is said to be exact if, for every x V, the sequence ∈

hx kx E′ E E′′ x −−→ x −→ x is exact

5.3 Induced vector bundles

Let E be a vector bundle over a manifold V and q a differentiable map of a manifold Y into V. Let E be the subset of the product Y E consisting q × of elements (y, η) such that q(y) = p(η). Define p : E Y by setting q → p (y, η) = y. If p (y, η) = p (y , η ), then y = y and p 1(y)(y Y) by ′ ′ ′ ′ ′ ′ ′− ∈ 64 5. Vector Bundles and Derivation Laws

setting

(y, η) + (y, η′) = (y, η + η′) λ(y, η) = (y, λη) for η, η E and λ R. It is clear that this makes of E a differentiable ′ ∈ ∈ q vector bundle over Y. This is called the bundle over Y induced by q. Let P be a principal bundle over X and E a vector bundle associated to 81 P. If Pq is the principal bundle over Y induced by q (Ch. 2.4) then Eq is associated to Pq.

5.4 Locally free sheaves and vector bundles

Let M be a differentiable manifold and ε a sheaf over M. If V U are ⊂ two open sets in M,ϕ denotes the restriction map ε(U) ε(V) and VU → for every x U,ϕ denotes the canonical map of ε(U) into the stalk ε ∈ xU x at x. Let U be the sheaf of differentiable real valued functions on M. For every open set U M,ε(U) is the algebra of real valued differentiable ⊂ functions on U. A sheaf ε over M is called a sheaf of U - modules if, for every open set U M, U (U) is an U (U)-module and if the ⊂ restriction maps satisfy the condition:

ϕVu( f σ) = (ϕVU f )(ϕVU σ) whenever V U are open sets in M, f U (U) and σ ε(U)., ⊂ ∈ ∈ Let ε and ε′ be two sheaves of U -modules over M.A homomor- phism h of ε into ε′ is a family of maps hU : HomU (U)(ε(U),ε′(U))(U open subset of M) such that if V U are two open sets in M,ϕ h = ⊂ VU U hV ϕVU. Let E be a differentiable vector bundle over M. The sheaf of dif- ferentiable sections of E is a sheaf ε of U -modules over M; for every open set U M,ε(U) is the U (U)- module of differentiable sections of ⊂ E over U.

Definition 2. A sheaf ε of U -modules over M is said to be free of rank 82 n if ε is isomorphic to the sheaf U n of differentiable maps M Rn. → 5.4. Locally free sheaves and vector bundles 65

A sheaf ε of U -modules over M is said to be locally free of rank n if every point x M has a neighbourhood U such that ε restricted to U is ∈ a free sheaf of rank n.

For every vector bundle E over M, the sheaf ε of differentiable sec- tions of E is locally free of rank= dimension of E. Proposition 2. A sheaf ε of U -modules over M is locally free of rank n if and only if for every x M, there exists a neighbourhood U of x ∈ and elements σ ,σ ,...σ ε(U) such that for every open set V U, 1 2 n ∈ ⊂ (ϕVUσ1,ϕVUσ2,...ϕVUσn) is a base of ε(V) over U (V). That a locally free sheaf has such a property is an immediate con- sequence of the definition. Conversely, if such elements σ1,σ2,...σn over U exist, then, for every open set V U,the map h of U (V)n into ⊂ V ε(V) defined by hV ( f1, f2,... fn) = f1σ1 + f2σ2 + ... fnσn is bijective n and the maps hV define an isomorphism of the sheaf U of differentiable maps U Rn onto the sheaf ε restricted to U. → Let E and E′ be two vector bundles over M. To every homomor- phism h : E E there corresponds in an obvious way a homomor- → ′ phism τh : ε ε and the assignment E ε of locally free sheaves to → ′ → vector bundles is a functor T from the category of vector bundles over M into the category of locally free sheaves over M. Moreover, if h k 0 E′ E E′′ 0 → −→ −→ → is an exact sequence, then the sequence 83

τh τk 0 TE′ Te TE′′ 0 → −−→ −−→ → is also exact (in the first sequence, 0 denotes the vector bundle of dimen- sion 0 over M). We shall now define a functor from the category of locally free sheaves into the category of vector bundles over M. Let ε be a locally free sheaf over M. Let U be the stalk at x M for the sheaf U and m x ∈ x the ideal of germs of f U such that f (x) = 0. Then we have the exact ∈ x sequence 0 m ε R 0. → x → x → → 66 5. Vector Bundles and Derivation Laws

If εx is the stalk at x for the sheaf ε, then we have correspondingly the exact sequence

0 m ε ε ε /m ε 0. → x x → x → x x x →

Let Ex = εx/mxεx, E = Ex and define p : E M by the con- x M → dition p(E ) = (x). Let U beS∈ an open subset of M and σ ,σ ,...σ x 1 2 n ∈ ε(U) satisfying the condition of prop.2. Then, for every x U,ϕ σ , ∈ xU 1 ϕxU σ2, ...ϕxU σn is a basis of εx over Ux. Let σi∗(x) be the image of ϕxU σi in the quotient space Ex. Then σ1∗(x),σ2∗(x),...σn∗(x) also is a n 1 base of EX over R. The map ρU : U R p− (U) E defined by = × → ⊂ ρ(x, a1, a2,... an) aiσi∗(x) is obviously a bijection. There exists on ff 84 E one and only one diP erentiable structure such that every map ρU is a diffeomorphism. With this structure, E is a differentiable vector bundle T εover M. Let ε,ε be two locally free sheaves of U modules over ∗ ′ − M. To every homomorphism h : ε ε corresponds in an obvious way → ′ a homomorphism T h : T ε T ε and the assignment ε T ε is a ∗ ∗ → ∗ ′ → ∗ functor T ∗ from the category of locally free sheaves to the category of vector bundles over M. Moreover, if

0 ε′ ε ε′′ 0 → → → → is an exact sequence of locally free sheaves, then

0 T ∗ε′ T ∗ε T ∗ε′′ 0 → → → → is an exact sequence of vector bundles.

Proposition 3. For every vector bundle E over M, there exists a canon- ical isomorphism of E onto T T E. For every locally free sheaf of U ∗ − modules ε over M, there exists a canonical isomorphism of ε onto TT ∗ε. (We shall later identify E with T ∗T E and E with TT ∗E means of these isomorphisms.)

For instance, if ε = TE and E = T ε, the isomorphism E E ′ ∗ → ′ maps u E into σ (x) where σ is a differentiable section over an ∈ x ∗ open neighbourhood of x such that σ(x) = u. In particular T defines a 5.5. Sheaf of invariant vector fields 67 bijection of the set of classes of isomorphic vector bundles over M onto the set of classes of isomorphic locally free sheaves of U modules over − M. Many vector bundles are defined in differential geometry from a lo- 85 cally free sheaf, using the functor T ∗. For instance the tangent bundle for a manifold M corresponds the sheaf C over M such that C (U) is the module of differentiable vector fields over the open set U M, ⊂ 5.5 Sheaf of invariant vector fields

Let P be a differentiable principal bundle over a manifold V with group G. Let be the sheaf over V such that for every open set U V, (U) is J ⊂ J the spec of invariant differentiable vector fields on p 1(U) P. Clearly − ⊂ each (U) is a module over the algebra U (U) of differentiable func- J tions on U and is sheaf of U -modules. J Let U be an open subset of V such that P is trivial over U and such that the module C (U) of vector fields on U is a free module over U (U). Let X1.X2,... Xm be a base of C (U) over U (U). Let I1, I2,... In be a base of right invariant vector fields on G. Then ((Xi, 0), (0, I j)) for i = 1, 2,... m, j = 1, 2,... n is a base of the U (U)-module of vector fields on U G invariant under G acting to the right. This base satisfies × the condition of Prop.2. Therefore the sheaf of invariant vector fields on U G is a free sheaf of rank = m + n over the sheaf U , restricted × to U. Hence, is a locally free sheaf of U modules of rank equal to J dim V + dim G. By Chap.5.4, there exists a vector bundle j = T of dimension = ∗J dim V + dim G canonically associated to . Any point in the fibre J of 86 J x J at x V may be interpreted as a family of vectors L along the fibre P ∈ x satisfying the condition

Lξ = L(ξ)s for ξ P , s G. s ∈ x ∈ Let K be the sheaf of U -modulo over V such that, for every open subset U V, K (U) is the sub-module of (U) consisting of invariant ⊂ J vector fields tangential to the fibres. Then K is locally free of rank = dim G and there exists a canonical injection K . The vector → J 68 5. Vector Bundles and Derivation Laws

bundle K = T ∗K associated to K is a vector bundle of dimension = dim G over V. Any point in the fibre K of K at x V may be x ∈ regarded as a vector field of the manifold Px invariant under the action of G. For every open set U V, any X (U) is projectable and the ⊂ ∈ J projection defines a homomorphism P Hom ( (U), C (U)). The U ∈ U (U) J family of homomorphisms p gives rise to a homomorphism p : U J → C . Since p is surjective when U is paracompact, p : C is U J → surjective. On the other hand, since the kernel of pU is K (U), the kernel of p is the image of K . Therefore we have the exact sequence of → J locally free sheaves over V:

0 K C 0. → →J→ → This gives rise to an exact sequence of vector bundles over V:

0 K J C 0 → → → → 87 where C denotes the tangent bundle of V.

Theorem 1. The vector bundle K is a vector bundle associated to P with typical fibre Y , the action of G on Y being given by the adjoint representation.

Let E = (P Y )/G the vector bundle over V canonically associated × to P and q the map P Y E. We shall define a canonical isomorphism × → of E onto K. Let U be an open subset of V and σ ε(U) a differentiable ∈ 1 section of E on U. We define a differentiable mapσ ˜ of p− (U) into Y 1 by the condition q(ξ, σξ˜ ) = σ(pξ) for every ξ p− (U). Thenσ ˜ (ξs) = 1 1 ∈ s− σ˜ (ξ)s for every s G and ξ p− (U). Therefore the vector field σ 1 ∈ σ = ∈ ˜ X on p− (U) defined by Xξ ξ(σξ) belongs to K (U). It is easy to verify that the map σ Xσ is an isomorphism λ of the U (U)-module → U ε(U) onto the U (U)-module K (U). The family λU (U open in V) is an isomorphism of the sheaf ε of differentiable sections of E onto the sheaf K . Hence (λ ) defines an isomorphism λ of E onto K. Let (ξ, a) U ∈ P Y . Then λq(ξ, a) = ϕ X where U is an open neighbourhood of x × xU and X K (U) a vector field satisfying the condition Xξ = ξa. ∈ 5.5. Sheaf of invariant vector fields 69

The isomorphism λ will be used later to identify the bundle E = (P Y )/G with K. The vector bundle K is called the adjoint bundle of × P.

Definition 3. Let 88 h k 0 E′ E E′′ 0 → −→ −→ → be an exact sequence of vector bundles over the manifold V.A splitting µ λ of this exact sequence is an exact sequence 0 E E E → ′′ −→ −→ ′ → 0, such that λh is the identity on E′ and kµ the identity on E′′. Any homomorphism λ (resp µ) of E into E′ (resp of E′′ into E) such that λh (resp kµ) is the identity determines a splitting. We shall now interpret the splitting of the exact sequence

(S) 0 K J C 0. → → → → For every open set U V, the module (U) (resp. K (U)) will be ⊂ J identified with the module of differentiable sections of J (resp. . K) over U.

Theorem 2. There exists one and only one bijection ρ of the set of con- nections on P onto the set of splittings J K of (S ) such that, if Γ is a → connection on P, Γ(X) = (ρΓ) X ◦ If Γ is a connection on P, then, for every open set U V, the restric- 1 ⊂ tion of the tensor Γ to U′ = p− (U) is a projection of the module C (U′) of vector fields on U′ onto the module N(U′) of vector fields on U′ tan- 89 gential to the fibres which is invariant under the action of G and induces a projection λ : (U) K (U). The family (λ ) is a homomor- U J → U phism of the sheaf onto the sheaf K and defines a homomorphism J ρΓ : J K which is a splitting of (S ). For every invariant vector field → X on P, we have (ρΓ) X = λ (X) =Γ(X). Since two homomorphisms ◦ V λ, λ : J K such that λ X = λ X for every differentiable section ′ → ◦ ′ ◦ X of J coincide, the map ρ is completely determined by the condition Γ(X) = (ρΓ) X. Moreover, since two connections Γ and Γ such that ◦ ′ Γ(X) =Γ′(X) for every invariant vector field X coincide, ρ is injective. It 70 5. Vector Bundles and Derivation Laws

remains to prove that ρ is surjective. Let λ be a splitting of (S ). For every open set U V, Let λ be the projection of (U) onto K (U) defined ⊂ U J by setting λ X = λ X for every X (U). Assume P to be trivial U ◦ ∈ J over U and let U = p 1(U). The injection )(U) C (U ) defines ′ − J → ′ an isomorphism of the U (U )-module U (U ) (U) onto C (U ). ′ ′ J ′ UN(U) The injection K (U) N(U ) defines an isomorphism of the U (U )- → ′ ′ module U (U′) K (U) onto N(U′). Therefore λU is the restriction UN(U) of a projection λ of C (U′) onto N(U′) which, regarded as tensor over U′ is invariant under the action of G. Hence λ is the restriction to (V) V J of a projection Γ : C (P) N(P) which is a connection on P. For every → invariant vector field X on P, Γ(X) = λ x = λ X. Therefore λ = ρΓ. V ◦ λ µ Remark. Let 0 K J C 0 be the splitting of (S ) corre- ← ←− ←− ← 90 sponding to a connection Γ on P. For every vector field X on the base manifold, regarded as a section of C, the invariant vector field µ X ◦ is the horizontal vector field on P (for the connection Γ), having X as projection (cf. Atiyah [2]).

5.6 Connections and derivation laws

Let V and V′ be two differentiable manifolds, and E, E′ two differen- tiable vector bundles of dimension n over V and V′ respectively. Let h be a homomorphism of E into E with projection h : V V. Assume ′ ′ → the map h : E′ E to be bijective for every x′ V′. If σ is a x′ → h(x′) ∈ section of E over V, then exists one and only one section σ′ of E′ over V′ such that hσ′ = σh. If σ is differentiable, so is σ′. Since h is a differentiable map of V′ into V, any module over the algebra U (V′) of differentiable functions over V′ can be regarded as a module over the algebra U (V) of differentiable functions over V. Then the map σ σ → ′ is a homomorphism of ε(V) into ε(V′) regarded as U (V)-modules. In particular, let P be a principal differentiable bundle with group G over V and let p : P V be the projection. Let s s be a linear → → L representation of G in a vector space L. Assume E to be a vector bundle associated to P with typical fibre L and q to be the map P L E. × → 5.6. Connections and derivation laws 71

Taking V = P, E = P L, h = q, h = p, for every section σ of E ′ ′ × the section σ is defined by qσ (ξ) = σp(ξ) for ξ P. Let L (P) be 91 ′ ′ ∈ the space of differentiable maps of P into L. Since to any section σ′ of E = P L corresponds a mapσ ˜ L (P) such that σ (ξ) = (ξ, σ˜ (ξ)), ′ × ∈ ′ we obtain a homomorphism λ : σ σ˜ of the U (V)-module ε(V) into → L (P) such that q(ξ, σξ˜ ) = σp(ξ) for every ξ P. ∈ Definition 4. A differentiable function on P with values in the vector space L is said to be a G-function if f (ξs) = s 1 f (ξ) for every ξ P, s −L ∈ ∈ G.

We shall denote the space of G-functions P L by L (P). → G Proposition 4. The homomorphism λ : ε(V) L (P) is injective and → λε(V) = LG(P).

If σ ε(V) and λσ = 0, then σ(ξ) = 0 for every ξ P. Therefore ∈ ∈ σ = 0 since p is surjective. On other hand, if σ ε(V) andσ ˜ = λσ, = = = ∈= 1 we have q(ξs, σ˜ (ξs)) σp(ξs) σp(ξ) q(ξ, σξ˜ ) q(ξs, s−L σ˜ (ξ)) for every ξ P and s G. Henceσ ˜ L (P). Conversely, let f L (P). ∈ ∈ ∈ G ∈ G The map ξ q(ξ, f ξ) maps each fibre into a point of E and therefore → X can be written σ , with σ ε(V). We have f = λσ. p ∈ We shall now show how a connection on P gives rise to a derivation law in the module of sections of the vector bundle E over V. By means of the map ε(V) L (P), the derivation law in the U (V)-module ε(V) → will be deduced from a derivation law in the U (P)-module L (P). 92 To the representation s s of G in L corresponds a representation → L in L of the Lie algebra Y of left invariant vector fields on G. For every a Y , define a by setting a v = av v for every v L, the vectors ∈ L L − ∈ at 0 of L being identified with elements of L. Then it is easy to see that [a, b] = a b b a . Hence a a is a linear representation of Y in L L L − L L → L L. Let Y (P) be the space of differentiable functions on P with val- ues in Y . The linear map a a of Y into hom (L, L) defines an → L R 72 5. Vector Bundles and Derivation Laws

U (P)-linear map α α of Y (P) into hom (L (P), L (P)) where → L U (P) (α f )(ξ) = α(ξ) f (ξ) for ξ P,α Y (P) and f L (P). L L ∈ ∈ ∈ We have already seen (Ch. 1) that there are canonical derivation laws in the modules Y (P) and L (P) and hence in homU (P)(L (P), L (P)) also. It is easy then to see that

(Xα) f = X(α f ) α (X f ) L L − L for every α Y (P) and f L (P). We now define in L (P) a new ∈ ∈ derivation law in terms of a given connection form γ on the bundle by setting DX f = X f + γ(X)L f for every X C (P) and f L (P). This differs from the canoni- ∈ ∈ cal derivation law in L (P) by the map γ : X γ(X) of C (P) into → L 93 homU (P)(L (P), L (P)). The curvatures form the canonical derivation law has been shown to be zero in Ch.1.9. Hence if K is the curvature form of the derivation law and the curvature form of the connection K from γ, we have

K(X, Y) = D D D D D . X Y − Y X − [X Y] = Xγ(Y) Yγ(X) γ([X, Y]) + [γ(X) , γ(Y) ] L − L − L L L = Xγ(Y) Yγ(X) γ([X, Y]) + [γ(X), γ(Y)] L − L − L L = (dγ(X, Y))L + [γ(X), γ(Y)]L

= K(X, Y)L

Theorem 3. For a given connection form γ on P, there exists one and only one derivation law D in the module of sections on the vector bundle E on V such that for every section σ and every projectable vector field X on P, we have = + D∼pXσ Xσ˜ γ(X)Lσ˜ The proof is an immediate consequence of the two succeeding lem- mas

Lemma 1. If X is a vector field on P tangential to the fibres and f ∈ L (P), then DX f = 0. 5.6. Connections and derivation laws 73

In fact, if X = Z where a Y , we have a ∈ = + = + DZa f Za f (γ(Za))L f Za f aL f

Hence 94 = + DZa f (ξ) (ξa) f aL f (ξ). 1 If g(s) = f (ξs) = s− f (ξ), then (ξa) f = ag = aL f (ξ). Therefore = − DZa f 0 and the lemma is proved since the module of vector fields tangential to the fibres is generated by the vector fields Z (a Y ). a ∈ Lemma 2. If X is projectable and f L (P), then D f L (P). ∈ G X ∈ G By lemma 1, it is enough to consider the case when X is a horizontal projectable vector field. Then γ(X) = 0 and hence we have

(DX f )(ξs) = (X f )(ξs) = Xξs f = (Xξ s) f = Xξh,

1 where h(ξ) = f (ξs) = s− f . Therefore

1 1 (DX f )(ξs) = s− (Xξ f ) = s− (DX f )(ξ) for s G, i.e., D f L (P). ∈ X ∈ G The converse cannot be expected to be true in general. For, if E is a trivial vector bundle over V with group reduced to e , it is clear that { } we can have only the trivial connections in P, whereas there are nonzero derivation laws in V. However, we have the

Theorem 4. Let G be the group of automorphisms of a vector space L, P a principal differentiable bundle with group G over a manifold V 95 and E a differentiable vector bundle over V associated to P with typical fibre L. Let p be the projection of P onto V. For every derivation law D in the U (V)-module ε(V) of differentiable sections of E, there exists one and one connections form γ on P such that

= + (C) D∼pXσ Xσ˜ γ(X)Lσ˜ for every σ ε(V) and every projectable vector field X on P. ∈ 74 5. Vector Bundles and Derivation Laws

We denote as before by q the map of P L onto E. For every σ × ∈ ε(V) and every vector dx at a point x V, let Ddx be the value (DXσ)(x), ∈ = where X is any vector field on V such that Xx dx. Let dξ be a vector with origin at ξ P . We define a map r(dξ) of ε(V) into the fibre E ∈ x x by setting

r(dξ)σ = D ξσ q(ξ, dξσ˜ ) pd − for every σ ε(V). Since r(dξ)( f σ) = f (x)(r(dξ)σ) for every f ∈ ∈ U (V) and every σ ε(V), r(dξ)σ depends only upon σ(x) or uponσ ˜ (ξ). ∈ Therefore, there exists an endomorphism α of L such that r(dξ)σ = q(ξ,ασ˜ (ξ)) for every σ ε(V). Since G is the group of automorphisms ∈ of L, there exists an element γ(dξ) in the Lie algebra Y of G such that r(dξ)σ = q(ξ, γ(dξ) σ˜ (ξ)) for every σ ε(V), and γ is clearly a form L ∈ 96 degree 1 on P with values in y satisfying the condition (C). From (C) we deduce that γ(X) is differentiable whenever X is a differentiable pro- jectable vector field on P. Therefore γ is differentiable. It is the easy to verify that γ is a connection form on P. Any connection form on P satisfying the condition (C) coincides with γ since the representation of Y in L is faithful and since a form on P is determined by its values on projectable vector fields.

5.7 Parallelism in vector bundles

Let E be a vector bundle over V associated to the principal bundle P with typical fibre L, γ a connection form on P, D the corresponding derivation law in the module of sections ε(V) of E, q the usual map P L E,ϕ × → a differentiable path in V with origin at x V, andϕ ˆ an integral lift of ∈ ϕ in P with respect to γ. For every y E , there exists v L such that ∈ x ∈ q(ϕ ˆ(0), v) = y. Then it is easy to see that the path q(ϕ ˆ(t), v) in E depends only on y and ϕ. The path q(ϕ ˆ(t), v) is called the integral lift of ϕ with origin y. The vectors q(ξ(t), v) are sometimes said to be parallel vectors along ϕ. Let σ be a section of the bundle E and dx a vector at x V. We ∈ would like to define the derivation of σ with respect to dx in terms of parallelism. Let ϕ be a differentiable path with origin x and ϕ′(0) = dx 5.7. Parallelism in vector bundles 75 andϕ ˆ an integral lift of ϕ in P. Let v(t) = σ˜ (ϕ ˆ(t)) so that q(ϕ ˆ(t), v(t)) = σ(ϕ(t)). Define y(t) = q(ϕ ˆ(0), v(t)). Then we have 1 Ddxσ = lim y(t) y(0) . t 0 t { − } → In fact, if 97

ξ = ϕˆ(0) and dξ = ϕˆ ′(0), we have 1 1 lim (y(t) y(0)) = q(ϕ ˆ(0), lim v(t) v(0) ) t 0 ( t − ) t 0 t { − } → → 1 = q(ϕ ˆ(0), lim σ˜ (ϕ ˆ(t) σ˜ (ϕ ˆ(0)) ) t 0 t { − } → = q(ϕ ˆ(0), dξσ˜ )

= q(ϕ ˆ(0), dξσ˜ + γ(dξ)Lσ˜ )

= Ddxσ.

For every Y E , let (ξ, v) P L such that q(ξ, v) = Y. Then ∈ x ∈ × q(dξ, v) T . This gives in particular, a map of the space ζξ of hori- ∈ Y zontal vectors with origin at ξ into TY , which is clearly injective. If we compose this with the projection, we obtain the bijection p : ζξ T . → x We define a vector at Y in E to be horizontal if it is of the form q(dξ, v) with q(ξ, v) = Y and dξ S ξ. It is easy to see that this definition does ∈ not depend on the particular pair (ξ, v). With this definition, for every integralϕ ˆ in P and every u L, the path Y(t) = q(ϕ ˆ(t), v) has horizontal ∈ agents at all points. The space of horizontal vectors at Y is a subspace of T supplementary to the subspace of vectors T tangents to the fibres Y ∈ Y through Y.

Theorem 5. Let σ be a section of E over V. Then σ(dx) is horizontal if 98 and only if Ddxσ = 0. Let dξ be a horizontal tangent vector at ξ p 1(x) whose projection ∈ − is dx. Then we have

Ddxσ = q(ξ, dξσ˜ + γ(dξ)σ ˜ ) = q(ξ, dξσ˜ ). 76 5. Vector Bundles and Derivation Laws

Butσ ˜ has been defined by

q(ξ, σ˜ (ξ)) = σ(pξ).

Hence q(ξ, σ˜ (dξ)) + q(dξ, σ˜ (ξ)) = σ(pdξ) = σdx. If σ(dx) is horizontal, so is q(ξ, σ˜ (dξ)). But pE.q(ξ, σ˜ (dξ)) = x, i.e.,σ ˜ (dξ) = σ˜ (ξ). Since dξσ˜ = σ˜ (dξ) σ˜ (ξ), it follows that dξσ˜ = 0. Therefore − q(ξ, dξσ˜ ) = 0. The converse is also immediate. This theorem enables us to define integral paths in E with respect to a given derivation law in ε(V).

5.8 Differential forms with values in vector bundles

Definition 5. Let E be a differentiable vector-bundle over the manifold V. A differential form α of degree n on V with values in the vector bundle E is an n-form on the module C (V) of differentiable vector fields on V 99 with values in the module ε(V) of differentiable sections of the bundle E.

For every n-tuple d x, d x,... d x of vectors at x V, the value 1 2 n ∈ α(d1 x, d2 x,... dn x) of α belongs to Ex. The U (V)-module of forms of degree n on V with values in E will be denoted by εn(V). Let P be a differentiable principal bundle over V, with group G and projection p. Assume E to be associated to P, with typical fibre L and let q be the map P L E. For every integer n 0, we shall define × → ≥ an isomorphism λ of εn(V) into the space L n(P) of differential forms of degree n on P with values in the vector space L. For n = 0, λ has been defined in 5.6. Assume n > 0. For every α εn(V), we define a form ∈ α˜ = λα on P with values in L by the condition

q(ξ, α˜(d1ξ, d2ξ,... dnξ)) = α(pd1ξ, pd2ξ,... pdnξ)

100 for every sequence d ξ of n vectors with origin at ξ P. If X , X ,... X i ∈ 1 2 n are n projectable vector fields on P, then

(λα)(X1, X2,... Xn) = λ(α(pX1, pX2,... pXn)) 5.9. Examples 77

Therefore, λα is differentiable and belongs to L n(P). It is imme- diately seen that λ is an injective homomorphism of εn(V) into L n(P) regarded as U (V)-modules. Moreover, ifα ˜ belongs to the image of λ in L n(P), then :

1 1)α ˜(d1ξs, d2ξs,... dnξs) = s− α˜ (d1ξ, d2ξ,... dnξ)s for every sequence d ξ of n vectors with origin at ξ P and every i ∈ s G, ∈

2)α ˜ (d1ξ, d2ξ,... dnξ) = 0 for every sequence d ξ of n vectors with origin at ξ P such that one i ∈ of the diξ has projection 0.

Definition 6. A form α L n(P) satisfying the conditions 1) and 2) is ∈ said to be a G- form of degree n on P with values in L.

n Let LG (P) be the set of G-forms of degree n. It is easy to see that n n LG (P) is a submodule of L (P) over U (V). Moreover LG (P) is the image of the homomorphism λ : εn(V) L n(P). → 5.9 Examples

1. Let γ, γ′ be two connection forms on the principal bundle P over V. Let β = γ γ . Then we have β(dξ s) = s 1β(dξ)s; and − ′ − β(dξ) = γ(dξ) γ (dξ) = 0 if pdξ = 0. In other words, β L 1(P) − ′ ∈ G with respect to the adjoint representation of G in Y . Hence β = α˜ where α is a differential form of degree 1 on V with values in the adjoint bundle of P which is a vector bundle associated to P with typical fibre Y . This gives a method of finding all connec- tion forms from a given one. This is particularly useful when G is abelian, in which case the adjoint representation of G in Y is triv- ial and consequently α may be considered as a differential form on V with values in Y

2. Let K be the curvature form of the connection form γ on a princi- 101 pal bundle P. Then 78 5. Vector Bundles and Derivation Laws

1 K(d1ξs, d2ξs) = s− K(d1ξ, d2ξ)s

and K(d ξ, d ξ) = 0 if either d ξ or d ξ Nξ 1 2 1 2 ∈ Therefore K is an alternate G-form of degree 2 on P with values in Y and corresponds to a form of degree 2 on V with values in the associated adjoint bundle.

5.10 Linear connections and geodesics

Let C be the vector bundle of tangent vectors on V. We consider C as a vector bundle associated to the principal bundle P of tangent frames on V with typical fibre Rn. There exists an one-one correspondence between connections on P (which are called linear connections on V) and derivations laws in the module C (V) of vector fields on V. The torsion form is an alternate form of degree 2 with values in the tangent bundle C. Given any linear connection on V, we have the notion of geodesics on V. (In Ch.5.10 to Ch.5.12, we consider paths with arbitrary intervals of definition). In fact, if ϕ is a differentiable path in V, then there exists a canonical lift of ϕ in C defined by t ϕ (t). We shall denote this lift → ′ by ϕ′.

Definition 7. A path ϕ in V is said to be a geodesic if the canonical lift 102 ϕ is integral. (In other words, the vector ϕ (t) at ϕ (t) C should be ′ ′′ ′ ∈ horizontal for the given linear connection).

Lemma 3. Let ϕ be a path in V and ϕˆ an integral lift of ϕ in P. Then ϕ is a geodesic if and only if there exists v Rn such that q(ϕ ˆ(t), v) = ϕ (t) ∈ ′ for every t I. ∈ The proof is trivial. It will be noted that the notion of a geodesic depends essentially on the parametrisation of the path. However, a geodesic remains a geodesic for linear change of parameters. Let Y C be a vector at a point x V and θ the horizontal vector at ∈ ∈ Y Y whose projection is Y. Then θ is a vector field on C called the geodesic 5.11. Geodesic vector field in local coordinates 79 vector field. The geodesic vector field is differentiable according to the following computation of θ in local coordinates.

5.11 Geodesic vector field in local coordinates

Let C be the tangent bundle of a manifold V with a derivation law D. Let p be the projection C V. If Y is a vector at x = pY V and → ∈ U a neighbourhood of pY wherein a coordinate system (x1,... xn) is ∂ defined, we denote the vector fields on U by P . Let yi = dxi be ∂xi i ff j = j a family of di erential forms on U dual to Pi, i.e., y (Pi) δi . The j 1 y may also be regarded as scalar functions on p− (U) C. Thus for 1 2 n 1 2 n ⊂ 1 Y P, (y , y ,... y , x , x ,... x ) form a coordinate system in p− (U). ∈ i Set Qi = ∂/∂y . Then we assert that the geodesic vector field θ is given 1 in p− (U) by θ = yiP Γk yiy jQ i − i, j K Xi Xi, j,k Γk where the i, j are defined by 103 = Γk DPi P j i, jPk. Xk In fact, using Th.5, Ch.5.7, we see that the value of the differential j + kΓ j i forms dy y i,k(x)dx on the vector is zero. It is easy to observe i,k = Pi Γk i j = that θY y (Pi)Y i, jy y Qk is horizontal and that pθY Y. This i − i, j,k shows thatPθ as definedP above is the geodesic vector field.

5.12 Geodesic paths and geodesic vector field

Proposition 5. ϕ is geodesic path in V if and only if the canonical lift = ϕ′ of ϕ is integral for θ, i.e., ϕ′′(t) θϕ′(t) for every t.

In fact, if ϕ is a geodesic, then we have pϕ′(t) = ϕ(t) and pϕ′′(t) = ϕ(t) for every t I. Hence ϕ (t) = θϕ . The converse is trivial in as ∈ ′′ ′′(t) much every path integral for θ is also integral for the connection. 80 5. Vector Bundles and Derivation Laws

Proposition 6. If ψ is an integral path for θ then ϕ = pψ is a geodesic ϕ′(t) = ψ(t)

By assumption ψ′(t) = θψ(t) and if ϕ(t) = pψ(t), we have ϕ′(t) = pψ′(t) = pθψ(t), i.e., ψ is the canonical lift of ϕ and is integral. 104 The geodesic vector field θ on C generates a local one-parameter group of local automorphisms of C. We say that the linear connection on V is complete if θ generates a one-parameter group of global auto- morphisms of V. It is known that in a compact manifold, any vector field generates such a one-parameter group of automorphisms. On other hand, if the local one-parameter group generated by θ in C is represented locally by 2n functions ϕi(y1,... yn, x1,... xn, t) of (2n + 1) variables, then it is easy to observe that the first n of these functions are linear in y1,..., yn. From this it is immediate that any linear connection on a compact manifolds is complete. The nomenclature ‘complete’ is due to the fact that a Riemannian connection is complete if and only if the Riemannian metric is complete ([27]). Let Γ be a complete linear connection and θ the geodesic vector field. Let tθ be the automorphism of C corresponding to the parameter t in the one-parameter group generated by θ. Paths which are integral for θ are of the form t tθY (i.e., the orbit of Y under tθ). Given any → vector Y at a point x on the manifold V,ϕ(t) = p(tθY) is the geodesic curve defined by Y. For every x V, the map ρ : C V defined by ∈ x → setting ρ(Y) = p(1θY) is a differentiable map of maximal rank at 0x and therefore defines a diffeomorphism of an open neighbourhood of 0 in the vector space Cx onto an open neighbourhood of x in V([26]). Chapter 6 Holomorphic Connections

6.1 Complex vector bundles

Definition 1. A complex vector bundle is a differentiable vector bundle 105 E over a manifold V with a differentiable automorphism J : E E → such that

1) JE E for every x V; x ⊂ x ∈ 2) J2Y = Y for every Y E. − ∈ Let P be a differentiable principal bundle over V with group G. Let us assume given a left representation s s of G in a complex vector → L space L, such that each sL is a complex automorphism of L. Then a vector bundle E associated to P with typical fibre L can be made into a complex vector bundle by setting J (ξ, v) = q(ξ, √ 1v) for every ξ P q − ∈ and v L, q being the usual projection P L E. Conversely any ∈ × → complex vector bundle E can be obtained in the above way. In fact, for x V we define P to be the vector space of all linear isomorphisms ∈ x α : Cn E such that α( √ 1v) = Jα(v) for v Cn. Then as in Ch. 4, → x − ∈ one can provide P = P with the structure true of a principal bundle ∪ x over V with group GL(n, C).E is easily seen to be a complex bundle associated to P with fibre Cn with respect to the obvious representation of GL(n, C) in Cn. If E is a complex vector bundle, the U (V) module ε(V) of differ- 106 − 81 82 6. Holomorphic Connections

entiable sections of E can be provided with the structure of an UC(V) − module UC(V) being the algebra of complex-valued differentiable func- tions on V) by setting ( f +ig)σ = f σ+Jgσ for f, g U (V). Conversely ∈ if I is an endomorphism of the U (V) module of sections of a differ- − entiable vector bundle E over V such that I2σ = σ for every section − σ, then there exists one and only one automorphism J of E such that J(σx) = (Iσ)x (for every section σ and x V) and J makes of E a ∈ complex vector bundle.

6.2 Almost complex manifolds

Definition 2. An almost complex manifold is a differentiable manifold V for which the tangent bundle has the structure of a complex vector bundle, i.e., there exists an U (V) endomorphism I of C (V) such that − I2 = (Identity). − The UC(V)- module CC(V) of complex vector fields (defined as for- mal sums X + iX , X , X C (V)) can be identified with the module 1 2 1 2 ∈ of derivations of UC(V) over C. The U (V) endomorphism I of C (V) − can then be extended to an UC(V)- endo-morphism of CC(V) by setting I(X + iX ) = IX + iIX for X , X C (V). 1 2 1 2 1 2 ∈ Definition 3. A vector field X CC(V) is said to be of type (1, 0) (resp. . ∈ of type (0, 1)) if IX iX (resp. . IX = iX). − −

We shall denote by C(1,0)(V), C(0,1)(V) the UC(V)- modules of vector fields of type (1, 0) and type (0, 1) respectively. 107 Clearly one has CC(V) = C , (V) C , (V). Moreover, any vector (1 0) ⊕ (0 1) field X of type (1, 0) can be expressed in the form X iIX with X 1 − 1 1 ∈ C (V). On the other hand, X iIx is of type (1, 0) for every X CC(V). − ∈ Similarly every vector field of type (0, 1) can be expressed in the form X + iIX with X C (V) and X + iIX is of type (0, 1) for every X 1 1 1 ∈ ∈ UC(V). Let V be an almost complex manifold and E a complex vector bun- dle over V.A complex form of degree p on V with values in E is a multilinear form of degree p on the UC(V) - module CC(V) with values 6.3. Derivation law in the complex case 83 in ε(V). Any real form can be extended in one and only one way to a complex form. On the other hand, every complex form is an extension of a real form. Every holomorphic manifold V carries with it a canonical almost complex structure. With reference to this a vector field on V is of type (1, 0) if and only if its expression in terms of any system (Z1,..., Zn) of ∂ ∂ local coordinates does not involve ,..., . Hence it follows that ∂Z¯1 ∂Z¯n [C , (V), C , (V)] C , (V). Conversely an almost complex struc- (1 0) (1 0) ⊂ (1 0) ture on V comes from a holomorphic structure on V if [C(1,0)(V), C(1,0) (V)] C , (V) cf. Newlander and Nirendberg, Annals of Math. 1957. ⊂ (1 0) Let X, Y CC(V). Then X iIX, Y iIY C , (V) and the above ∈ − − ∈ (1 0) condition requires that [X iIX, Y iIY] C , (V). In other words, − − ∈ (1 0) [X, Y] [IX, IY] = I[IX, Y] I[X, IY]. − − − F(X, Y) = [X, Y] + I[IX, Y] + I[X, IY] [IX, IY] − is easily seen to be CC(V) bilinear. Hence we have: 108 − An almost complex structure on V is induced by a holomorphic structure if and only if F(X, Y) = 0 for every X, Y CC(V). ∈ 6.3 Derivation law in the complex case

We wish to study derivation laws in the module of sections of a complex vector bundle over an almost complex manifold (mostly holomorphic manifold). This can be done in the algebraic set-up of Ch.1. Let A be a commutative algebra over the field of real numbers and AC its complexification. Let C be the AC module of derivations of AC. 2 We assume given on C and AC - endomorphism I such that I = − (Identity). If we define X¯ C by Xa¯ = Xa¯ for a A, then we also ∈ ∈ assume that I satisfies IX¯ = IX¯ . X C is said to be of type (1, 0) (resp. . ∈ type (0, 1)) if IX = iX (resp. . IX = iX). Let M be an A- module. A − multilinear form α of degree p on C with values in M is said to be of type (r, s) if r + s = p and α(X1,... Xp) = 0 whenever either more than ′ rXi′ s are of type (1, 0) or more than sXi s are of type (0, 1). We shall denote the submodule of F p(C, M) consisting of all forms of type (r, s) 84 6. Holomorphic Connections

by F r,s(C, M). Then it is easy to see that F p(C, M) = F r,s(C, M), r+s=p the sum being direct. Let us denote the module of alternateP forms of r,s type (r, s) on C with values in M by U (C, M). Let M1, M2, M3 be three AC- modules with a bilinear product M1 M2 M3. It is then r,s r ,s× → 109 easy to see that if α U (C, M1), β U ′ ′ (C, M2), then α β r+r ,s+s ∈ ∈ ∧ ∈ U ′ ′ (C, M3). We shall now make the additional assumption that F[X, Y] = [X, Y]+ I[IX, Y] + I[X, IY] [IX, IY] = 0 for every X, Y C. (This corresponds − ∈ to the case when the base manifold V is holomorphic). A derivation law p D in an AC module M gives rise to an exterior derivation d in U (C, M). Using the explicit formula for d (Ch.1.7) and the fact that F 0, it is ≡ easily proved that for α U r,s(C, M), dα is the sum of a form d α of ∈ ′ type (r + 1, s) and a form d′′α of type (r, s + 1). The curvature form 2 2,0 1,1 K U (C, Hom C (M, M) is a sum of three components K , K and ∈ A K0,1. For α U p,q(C.M) we have d2α = K α (Lemma 5, Ch. 1.9) and ∈ ∧ calculating the components of type (p, q + 2) we obtain

2 0,2 d′′ α = K α. ∧ Now we make the further assumption that K0,2 = 0. It follows that 2 p,q d′′ = 0. For every integer p, form the sum U (C, M) = Tp. Then q we have a complex P

d′′ d′′ d′′ U p,0(C, M) U p,1(C, M) U p,q(C, M) −−→ −−→· · · −−→· · ·

The complex Tp of course depends on the derivation law D. Let Dˆ be another derivation law in M. Then Dˆ = D + ω where ω ∈ 110 HomC(C, HomAC(M, M)) (see Ch. 1.10). We shall say that two deriva- tion laws D, Dˆ are equivalent if ω is of type (1, 0). The boundary op- erator d′′ in the complex Sp remains the same when D is replaced by an equivalent derivation law Dˆ . For, if D = Dˆ + ω, we have (Ch.1.10) dˆα = dα + ω α for α U p,q(C, M). Hence on computing the compo- ∧ ∈ nents of type (p, q+1), one obtains dˆ′′α = d′′α. Thus we have associated to every equivalence class of derivation laws, a complex Tp (for every integer p). 6.4. Connections and almost complex structures 85

A derivation law D in M induces a derivation law D in HomAC (M, M) = End M(Ch.1.3). Then it is easy to see that K(X, Y)ρ = K(X, Y) ρ ρ K(X, Y) ◦ − ◦ for X, Y C, ρ end M. Hence if K0,2 = 0, K0,2 is also = 0. Moreover, ∈ ∈ if D, Dˆ are equivalent, so are D, Dˆ . Hence to an equivalence class of derivation laws in M corresponds a complex T p (for every integer p):

d′′ d′′ d′′ U p,o(C, End M) U p,1(C, End M) U p,q(C, End M) −−→ −−→· · · −−→· · · Now K1,1 is an element of U 1,1(C, EndM). Since we have dK = 0 0,2 1,1 1,1 (Lemma 6, Ch.1.9) and K = 0, we get d′′K = 0, i.e., K is a cocycle of the complex T . If D is replaced by an equivalent derivation law Dˆ , we have (Ch.1.10) Kˆ = K + dω + ω ω where Dˆ = D + ω. 111 1,0 1,1 1,1 ∧ But ω U (C, EndM). Hence Kˆ = K + d′′ω. In other word, the ∈ 1,1 cohomology class of K in the above complex T 1 is uniquely fixed.

6.4 Connections and almost complex structures

Let P be a differentiable principal bundle over an almost complex man- ifold V with complex Lie group G. Let Y be the Lie algebra of G 112 identified with the space of real tangent vectors at e. We denote by IG the endomorphism Y Y given by the complex structure of G and by → I the almost complex structure of V. Obviously we have

1 1 s(I a)− s = I (sas− ) for s G, a Y . G G ∈ ∈ Let ξ P and pξ = x V. Then we have the exact sequence ∈ ∈ 0 Nξ Tξ T 0 → → → x → where Nξ is the space of vectors at ξ tangential to the fibre. If a connec- tion is given on P, then we may define as almost complex structure on P by carrying over the complex structure on Tx to S ξ (the space of hor- izontal vectors at ξ) and that of Y to Nξ (by virtue of the isomorphism a ξa). This is easily seen to define an almost complex structure on P → so that we have the 86 6. Holomorphic Connections

Proposition 1. Let γ be a connection form on a differentiable principal bundle P over an almost complex manifold V with complex Lie group G. Then there exists one and only one almost complex structure on the manifold P such that

1) γ(Ipdξ) = IGγ(dξ)

2) pI dξ = Ipdξ for ξ P. p ∈ In virtue of the general procedure given in Ch.6.2, γ is defined on complex vector fields by setting γ(iX) = I γ(X) for X C (P). The G ∈ connection form is obviously of type (1, 0) with respect to the induced almost complex structure on P. Moreover, the P G P given by the × → action of G on P is almost complex. We shall call a principal bundle P over an almost complex manifold V with complex Lie group G an almost complex principal bundle if P has an almost complex structure such that the projection p and the map (ξ, s) ξsof P G P are both almost → × → complex (i.e., compatible with the almost complex structure). We have seen above that P with the almost complex structure induced by a con- nection form γ on P is an almost complex principal bundle. Conversely, let P be an almost complex principal bundle. If γ = γ1,0 + γ0,1 is a connection form on P then γ1,0 is again a connection form. In fact,

1 γ1,0(ξa) = γ1,0((ξa + iI ξa) + (ξa iI ξa)) 2 p − p 1 = γ(ξa iI ξa) 2 − p 1 1 = (a I γ(I ξa)) = (a I γ(ξI a)) 2 − G P 2 − G G = a

1,0 1 1,0 113 Similarly it may be shown that γ (dξs) = s− γ (dξ)s. Moreover we have 1 γ1,0(I dξ) = γ1,0(I dξ + idξ + I dξ idξ) P 2 P P − 1 = γ(I dξ + idξ) 2 P 6.4. Connections and almost complex structures 87

1,0 = IGγ (dξ).

On the other hand, pIPdξ = Ipdξ by assumption. Hence by prop.1, Ch.6.4, γ1,0 induces the given almost complex structure on P. Let γ, γˆ be two connection forms on an almost complex principal bundle P. Then γ γˆ is a G-form (Ch.5) of degree 1 on P with values − in Y . It may be identified with a differential form of degree 1 on V with values in the adjoint bundle. Using the fact that every vector field on V of type (1,0) is the projection of a vector field on P of type (1,0) it is easy to see that this identification is type-preserving. In other words, the type of G-forms on P with values in Y depends only on the almost complex structure on V. We say that γ, γˆ are equivalent if γ γˆ is of − type (1,0). The following proposition in then immediate.

Proposition 2. Two connections γ, γˆ on P induce the same almost com- plex structure on P if and only if γ and γˆ are equivalent.

On the other hand we have seen that every almost complex structure on P which makes of it an almost complex principal bundle is induced 114 by a connection of type (1, 0). We have thus set up one-one correspon- dence between almost complex bundle structure on P and equivalence classes of connections. We will hereafter assume that the base manifold V is holomorphic and investigate when a connection form γ induces a holomorphic struc- ture on P. Then curvature form K of γ is a G-form of degree 2 with values in Y . Let K = K2,0 + K1,1 + K0,2 be its decomposition. Proposition 3. A connection form γ on P induces a holomorphic struc- ture on P if and only if K0,2 = 0.

In fact, if γ induces a holomorphic structure on P, then d′′γ = 0. Since K = dγ + [γ, γ] and γ is of type (1, 0) for the induced complex structure, K0,2 = 0. Conversely, let K0,2 = 0. We have then to show that

F (X, Y) = [X, Y] + I [ X, Y] + I [X, I Y] [I X, I Y] = 0 P P P P P − P P for any two vector fields X, Y on P. Since FP is a tensor, it is sufficient to prove that FP(X, Y) = 0 for projectable vector fields X, Y. Then IPX, IPY 88 6. Holomorphic Connections

F X Y are also projectable, and we have P P(X, Y) = FV (P , P ) = 0 since V is holomorphic. On the other hand,

γ(F(X, Y)) = γ([X, Y] [I X, I Y]) + I γ([I X, Y] + [X, I Y]) − P P G P P = γ([X, Y] [I X, I Y] + [iI X, Y] + [X, iI Y]) − P P P P = γ([X + iIPX, Y + iIPY]).

0,2 115 But since K = 0, K(X + iIPX, Y + iIPY) = 0. i.e., (X + iI X)γ(Y + iI Y) (Y + iI Y)γ(X + iI X) γ([X + iI X, Y + P P − P P − P iIPY]) = 0. Since γ is of type (1, 0), we obtain γ(F(X, Y)) = 0 which gives F(X, Y) = 0.

Definition 4. A differentiable principal bundle P over a holomorphic manifold V with a complex Lie group G is said to be a holomorphic principal bundle if P is a holomorphic manifold such that the projection p : P V and the map (ξ, s) ξs of PχG P are holomorphic. → → → If a connection form γ exists on P with K0,2 = 0, then the induced almost complex structure on P is holomorphic and its is obvious that this makes of P a holomorphic principal bundles.

6.5 Connections in holomorphic bundles

Let P be a holomorphic principal bundle over V with group G. The definitions of holomorphic vector bundles with respect to a holomorphic representation of G in a complex vector space L are given in analogy with the differentiable case. All the results of Chapter 2 can be carried over to holomorphic bun- dles with obvious modifications. Let 0(U) be the algebra of holomor- phic functions on an open subset U of V and 0 the sheaf of holomorphic functions on V. The notion of a sheaf of 0- modules is defined as in Chapter 4. Moreover, there exists a functor from the category of holo- 116 morphic vector bundles over V to the category of locally free sheaves of 0 modules which takes exact sequences to exact sequences, and vice versa. 6.5. Connections in holomorphic bundles 89

Let G be a complex Lie group, P a holomorphic principal bundle with group G over a holomorphic manifold V, and E a holomorphic vector bundle over V associated to P with typical fibre L. Any connec- tion form γ on P defines a derivation law in the module of sections of E (Ch.5).

Proposition 4. If the connection form γ is of type (1, 0), then a differen- tiable section σ of E over an open subset U of V is holomorphic if and only if d′′σ = 0.

Ifσ ˜ is the G-function on P with values in L corresponding to σ under our usual isomorphism, it is easy to prove (as in the differentiable case) that σ is holomorphic if and only ifσ ˜ is holomorphic. But this is equivalent to saying that d′′σ˜ = 0 for the canonical derivation law in the UC(P) module of L-valued functions on P. Since the derivation law D induced by γ in L (P) was defined by DX f = X f + γ(X)L f (in the usual notation) and γ is of type (1, 0), the d′′ for D and the canonical derivation law are the same. Since the isomorphism α α˜ is type-preserving, we → have (d′′α˜ ) = d˜′′α˜ where d˜′′ corresponds to the derivation law D in LG(P). Hence d′′σ˜ = 0. Let U be an open subset of V over which E is trivial and let σ1,..., σ be holomorphic sections on U which form a basis for the U (U) n − module ε(U) of sections of E over U. Then for any section σ = fiσi 117 over U we have P

d′′σ = (d′′ fi) σi + fi d′′σi (Ch.1.8) X ∧ X ∧ = d′′ fi σi Prop. 4; Ch. 6.5 X ∧ = (d′′ fi)σi X In other words, d′′ depends only on the manifold V and not on the 2 principal bundle P. Moreover it is obvious that d′′ σ = 0. However, this can be proved algebraically. In fact, if γ, γˆ are of type (1, 0), so is γ γˆ = ω. From this it follows that that induced − derivation laws are equivalent and the corresponding d′′ are the same (Ch. 6.3). Furthermore, since K0,2 = 0 (which is a consequence of prop. 90 6. Holomorphic Connections

2 3. Ch.6.4 γ being of type (1, 0)) we have seen d′′ = 0 in Ch.6.3. We have therefore a complex

d d d εp,0(U) ′′ εp,1(U) ′′ εp,q(U) ′′ −−→ −−→· · · −−→· · · for every open subset U of V, where εp,q(U) is the module of differential forms of type (p, q) on U with values in E. This defines a complex S P(E) of sheaves over V:

d d d εp,0 ′′ εp,1 ′′ εp,q ′′ −−→ −−→· · · · · · −−→· · · · · · where εp,q denotes the sheaf of differential forms of type (p, q) on V 118 with values in the vector bundle E. Let U p,q be the sheaf of complex valued differentiable functions of type (p, q) on V, 0 the sheaf of holo- morphic functions on V,εh the sheaf of holomorphic sections of E. Then p,q p,q ε is isomorphic to the sheaf U εh and therefore is a fine sheaf. ⊗0 Moreover if 0p is the sheaf of holomorphic forms of degree p on V then the sequence

p p,0 d′′ p,1 0 0 εh ε ε → ⊗0 → −−→ →···

is exact Dolbeault theorem [12]). Therefore the complex T p(ε) is a fine p p resolution of the sheaf 0 εh and if S (ε) is the complex of sections ⊗0

d d d εp,0(V) ′′ εp,1(V) ′′ εp,q(V) ′′ −−→ −−→· · · −−→· · · Then Hq(S p(ε)) Hq(C, 0p ε ) ≃ ⊗ h for every pair of integers p, q > 0. Now assume E to be the adjoint bundle ad (P) to P. We shall con- struct a canonical cohomology class in H1(S 1(ad(P))). Let γ be a con- nection form of type (1, 0) on P and K = K2,0 + K1,1 its curvature form. As a G-form on P with values in the Lie algebra Y , K corresponds to an 1,1 alternate form χ of degree 2 on V with values in ad(P). Since d′′K = 0 6.7. Atiyah obstruction 91

1,1 1,1 (Ch. 6.3), we have d′′χ = 0. In other words, χ as a 1-cocycle of the complex S 1(ad(P)). 119 Moreover ifγ ˆ is another connection of type (1, 0), χˆ1,1 differs from χ1,1 by a coboundary of the complex S 1(ad(P)). We have therefore as- sociated a unique 1-cohomology class in S 1(ad(P)) to the holomorphic bundle P. To the cohomology class of χ1,1 corresponds an element of 1 1 H (V, 0 (ad(P))h). This will be referred to as the Atiyah class of the ⊗0 principal bundle P. Regarding the existence of a holomorphic connection on the bundle P (i.e., a connection such that γ is a holomorphic form) , we have the

Theorem 1. There exists a holomorphic connection in a holomorphic bundle P if and only if the Atiyah class a(P) of the bundle is zero.

In fact, if γ is a connection form on P of type (1, 0), γ is holomorphic 0,2 1,1 if and only if d′′γ = 0. Since K = 0, we see that d′′γ = K = 0 and hence χ1,1 = 0. Conversely, if a(P) = 0 and γ a connection form of type (1, 0) on P, we have χ1,1 0. Hence there exists a form α (adP)1,0(V) such ∼ ∈ that d α = χ1,1. Thenγ ˆ = γ α˜ is a connection form on P such that ′′ − d′′γˆ = 0.

Corollary. There always exists a holomorphic connection on a holo- morphic bundle P over a Stein manifold V.

1 In fact, the sheaf 0 (adG )n is a coherent sheaf and therefore ⊗0 a(P) = 0, since V is a Stein manifold. By Theorem 1, there exists a holomorphic connection on P.

6.7 Atiyah obstruction

Let 0 F 1 F F 0 be an exact sequence of locally free 120 → → → ′′ → sheaves of 0 modules over a holomorphic manifold V. Since F ′ is locally free, the corresponding sequence

0 Hom0(F ′′, F ′) Hom0(F , F ′) Hom0(F ′, F ′) 0 → → → → 92 6. Holomorphic Connections

is exact. This gives rise to the exact sequence

o o 0 H (V, Hom0(F ′′, F ′)) H (V, Hom0 F ′, F ′)) → →···→ 1 H (V, Hom0(F ′′, F ′) → →··· o H (V, Hom0(F ′, F ′)) is then the module of sections of the sheaf Hom0 (F ′, F ′). The image of the identity section of Hom0(F ′F ′) is called the obstruction to the splitting of the given sequence. Let , T , C be the sheaves of holomorphic vector fields on P Kh h h tangential to the fibres, of all holomorphic invariant vector fields on P and of all holomorphic vector fields on V respectively. Then we have the exact sequence

0 T C 0 → Kh → h → n → Let b(P) be the obstruction to the splitting of this exact sequence. 1 b(P) is then a class in H (V, Hom0(C , )). We shall call this the h Kh Atiyah obstruction class.

121 Theorem 2. The necessary and sufficient condition for a holomorphic connection to exist on P is that b(P) = 0.

In fact, b(P) = 0 if and only if the above sequence splits and the rest of the proof is exactly as for the differentiable case.

Theorem 3. There exists a canonical isomorphism ρ of the sheaf = 1 R 0 (adP)h onto Hom0(Ch, h) such that ρa(P) = b(P). ⊗0 K −

We shall define ρ : (U) Hom0 (C (U), (U)) for every U R → (U) h Kh open subset U of V. Every ω (U) is a differential form on U with ∈ R 1 values i the adjoint bundle andω ˜ is a G-form on p− (U) with values in 1 Y . For every X Ch(U) and ξ p− (U), we define (ρU (ω)(X))ξ = 1 ∈ ∈ ξ(ωXξ iI ωXξ). It is easily seen that ρ is an isomorphism and that 2 − G U it defines an isomorphism

ρ : Hom0(C , ) R→ h Kh 6.8. Line bundles over compact Kahler manifolds 93

Let U be a covering of V by means of open sets U , over each { i} i of which P is holomorphically trivial. We shall compute a(P), b(P) as cocycles of the above covering with values in the sheaves , Hom0(C , R h h) with respect to the above covering. Let γi be holomorphic con- K 1 1,1 nections on p− (Ui) and γ a differentiable connection on P. K is the element d γ in (adP)1,1(V). If γ = γ + α over p 1(U ) with α ′′ i i − i i ∈ (adP)1,0(V), then K1,1 is the d image of α on U . For, d γ = d α˜ = ′′− i i ′′ ′′ i d α˜ (in our usual notation). Hence a(P) is represented in H1(X, ) 122 ′′ i R with respect to the above covering by the cocycle α α = (γ γ ) i − j − i − j on U U . i ∩ j On the other hand,in the exact sequence

0 Hom0(C , ) Hom0(I , ) Hom0( , ) 0 → n Kh → h Kh → Kh Kh → the identity section of Hom0( h, h) can be lifted on Ui into the map K K 1 Γ Hom0(T , ) where Γ (X)ξ = ξ(γ (Xξ) iI γ(Xξ)) for X i ∈ h Kh i 2 i − G ∈ T ,ξ P. Hence the obstruction class is represented by λ on U U h ∈ i j i ∩ j defined by

1 λ , (pdξ) = ξ((γ γ ) (dξ) iI (γ γ )(dξ)) i j 2 i − j − G i − j for every dξ Tξ. This represents the cocycle of the class b(P) for the ∈ covering U . Obviously ρ(a(P)) = b(P). { i} − 6.8 Line bundles over compact Kahler manifolds

Let P be a holomorphic principal bundle over a manifold V with group G = C∗(C∗ = GL(1, C)). We shall compute the Atiyah class a(P) for such a bundle. Let U be a covering of V over each of which P is { i} trivial with holomorphic sections σi on Ui and a set of holomorphic transition functions m . Since the adjoint representation is trivial, the { i j} adjoint bundle is also trivial and the G-functions and G-forms on P are respectively functions and forms on P which are constant on each fibre. Let γ be a connections form of type (1, 0) on P. It is easy to see that 123 d (σ ˜ γ) = d γ. But σ γ σ γ = m 1dm = d(log m ). Thus the ′′ i∗ ′′ ∗j − i∗ i− j i j i j 94 6. Holomorphic Connections

forms d(log mi j) form a cocycle representing the Atiyah class for the covering U . { i} Finally, when the base manifold is compact Kahler H1(V, = 01) K may be identified ([30]) with a subspace of H2(V, C) by Dolbeault’s the- orem. The sequences

e2πi (1) 0 Z 0 0∗ 0 → → −−−→ → and

d (2) 0 C 0 01 0 → → −→ →

where Z, C are constant sheaves and 0∗ is the sheaf of nonzero holomor- phic functions, can be imbedded in the commutative diagram

/ / / / 0 / Z / 0 / 0∗ / 0 1 (3) j i 2πi d log    /  /  /  / 0 / C / 0 / 0′ / 0 where j, i are respectively the inclusion and identity maps. We get con- sequently the commutative diagram

1 α / 2 H (V, 0∗) / H (V, Z)

δ β  γ  1 / 2 H (V, 0′) / H (V, C)

124 where α is the connecting homomorphism of the exact sequence (1), β, δ maps induced by diagram (3), and γ the injection given by Dolbeault’s theorem ([29]). If mi j are the multiplicators for a covering (Ui) of { 1} V, δ(m ) is given by d log(m ). The bundle P may be regarded as i j 2πi i j 1 an element of H (V, 0∗) and its image by α is the first integral Chern class of P and its image by β is the Chern class C(P) with complex coefficients. By the commutativity of the diagram, we have a(P) = 2πiC(P). 6.8. Line bundles over compact Kahler manifolds 95

In particular, we obtain the result that if there exists a holomorphic connection in a line bundle over a compact Kahler manifold, then its Chern class with complex coefficients = 0. It has been proved under more general assumptions on the group of the bundle that the exis- tence of holomorphic connections implies the vanishing of all its Chern classes [2].

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