Hello, I am looking for a proof for the ChernGaussBonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via ChernWeil 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.

$\begingroup$ It's in the last volume of Spivak's book, no? $\endgroup$ – Mariano SuárezÁlvarez Mar 20 '12 at 23:25

1$\begingroup$ What's wrong with Chern's paper? $\endgroup$ – Igor Rivin Mar 20 '12 at 23:26

3$\begingroup$ Quote from Chern: "It helps to be vague with bundles." $\endgroup$ – Will Jagy Mar 21 '12 at 0:10

3$\begingroup$ Check out Bryant's answer to this question: mathoverflow.net/questions/84521/… $\endgroup$ – Ian Agol Mar 21 '12 at 13:03

$\begingroup$ @Agol: yes, that's probably the best reference... $\endgroup$ – Igor Rivin Mar 21 '12 at 13:41
For a complete proof of the GaussBonnetChern 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 ChernWeil theory, but the language is more modern.
For a purely topological proof, see Section 5.3 of these notes.


$\begingroup$ 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 2plane bundles and using the splitting principle to finish. $\endgroup$ – Youloush Mar 22 '12 at 1:19
One reference that seems fairly good and that I just found by googling those key words is https://web.archive.org/web/20100524152105/http://www.math.upenn.edu/~alina/GaussBonnetFormula.pdf.
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 8524401788; 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)...
Chern's original paper may be found at: Chern, ShiingShen (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.