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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$\begingroup$ It's in the last volume of Spivak's book, no? $\endgroup$– Mariano Suárez-ÁlvarezMar 20, 2012 at 23:25
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1$\begingroup$ What's wrong with Chern's paper? $\endgroup$– Igor RivinMar 20, 2012 at 23:26
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4$\begingroup$ Quote from Chern: "It helps to be vague with bundles." $\endgroup$– Will JagyMar 21, 2012 at 0:10
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3$\begingroup$ Check out Bryant's answer to this question: mathoverflow.net/questions/84521/… $\endgroup$– Ian AgolMar 21, 2012 at 13:03
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$\begingroup$ @Agol: yes, that's probably the best reference... $\endgroup$– Igor RivinMar 21, 2012 at 13:41
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3 Answers
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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.
answered Mar 21, 2012 at 1:41
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$\begingroup$ 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 2-plane bundles and using the splitting principle to finish. $\endgroup$– YouloushMar 22, 2012 at 1:19
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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 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)...
answered Mar 21, 2012 at 0:57
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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.
answered Mar 21, 2012 at 1:58