By Vaisman

ISBN-10: 0824770633

ISBN-13: 9780824770631

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**Extra resources for A First Course in Differential Geometry**

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A) Show that ~ is an equivalence relation and E = Xj ~ equipped with the quotient topology has a natural structure of smooth manifold. (b) Show that the projection 7r : X —>• M, (x, u, a) i-> x descends to a submersion E^M. (c) Prove that (E, it, M, V) is naturally a smooth vector bundle. 12 (a) A section in a vector bundle E A- M defined over the open subset u e M is a smooth map s : U —> E such that s(p) e Ep = 7r -1 (p), Vp G U. Equivalently, this means that JTOS = lv. The space of smooth sections of E over U will be denoted by T(U,E) or C°°(U,E).

Again there is no difficulty to check the above definition is independent of the various choices. 8 Let E —> X be a rank k (complex) smooth vector bundle over the manifold X. e. , SN(X)} spans Ex. For each x € X set Sx = {v€CN; '£visi{x) = 0} i Note that dimS x = N - k. We have a map F : X -> GkiN(C) by x - • Sj-. (a) Prove that F is smooth. (b) Prove that E is isomorphic with the pullback F"Tkn. 9 Show that any vector bundle over a smooth compact manifold is ample. Thus any vector bundle over a compact manifold is a pullback of some tautological bundle!

3 Vector bundles The tangent bundle TM of a manifold M has some special features which makes it a very particular type of manifold. We list now the special ingredients which enter the special structure of TM since they will occur in many instances. Set for brevity E = TM and F = Km (m = dimM). (a) E is a smooth manifold and there exists a surjective submersion 7r : E —> M. For every U C M set E \v= n"\U). e. an open cover U of M and for every U €U a diffeomorphism 9u:E\u-*UxF, V^(P = TT{V),~~"(V)) such that (bl) <£p is a diffeomorphism Ep —• F for any p € U. ~~

### A First Course in Differential Geometry by Vaisman

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