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On Fibre Bundles and Differential Geometry

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On Fibre Bundles and Differential Geometry 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.
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