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Hello, I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler characteristic when coupled to the fundamental class. Thanks in advance.

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It's in the last volume of Spivak's book, no? – Mariano Suárez-Alvarez Mar 20 '12 at 23:25
What's wrong with Chern's paper? – Igor Rivin Mar 20 '12 at 23:26
Quote from Chern: "It helps to be vague with bundles." – Will Jagy Mar 21 '12 at 0:10
Check out Bryant's answer to this question:… – Ian Agol Mar 21 '12 at 13:03
@Agol: yes, that's probably the best reference... – Igor Rivin Mar 21 '12 at 13:41
up vote 6 down vote accepted

For a complete proof of the Gauss-Bonnet-Chern for arbitrary vector bundles (not just tangent bundles) see Section 8.3.2 of these notes. The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern.

For a purely topological proof, see Section 5.3 of these notes.

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I fixed the broken links above. – Liviu Nicolaescu Mar 21 '12 at 14:23
Thanks! Looks perfect, although the proof in the first ref is basically the same as what I had previously seen, though my reference had a simpler approach, examining the case of 2-plane bundles and using the splitting principle to finish. – Youloush Mar 22 '12 at 1:19

One reference that seems fairly good and that I just found by googling those key words is

The first time I learnt this, however, was with these lecture notes: F. Mercuri, P. Piccione, D. V. Tausk, Notes on Morse theory, Publicações Matemáticas do IMPA, Rio de Janeiro, 2001, ISBN 85-244-0178-8; which maybe a little hard to find, but are very nicely written and I like them very much. Though, the proof you are looking for should be widely available elsewhere (google gives thousands of results, and I only looked at the first ones)...

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Chern's original paper may be found at: Chern, Shiing-Shen (1945), "On the curvatura integra in Riemannian manifold", Annals of Mathematics 46 (4): 674–684; this citing quoted from the Wikipedia entry, though I have a copy of the original paper somewhere in the piles in my office.

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