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I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From Calculus to Cohomology by Madsen & Tornehave. I know the statement of the theorem is as follows:

Let $M$ be an even-dimensional compact, oriented smooth manifold, $F^{∇}$ be the curvature of the connection $∇$ on a smooth vector bundle $E$.

$$\int_M Pf \left( \frac{−F^\nabla}{2 \pi} \right) = \chi(M^{2n}) .$$

My questions are: how does this relate to counting (with multiplicities) the number of zeros of generic sections of the vector bundle? Also, are there other good references for learning this topic? Thanks.

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For other references, see my question here: math.stackexchange.com/questions/64305/… –  Jesse Madnick Dec 29 '11 at 17:58
    
Here is a paper on the relationship between Poincaré-Hopf and Gauss-Bonnet-Chern: arxiv.org/abs/1302.6895. Also see Berline-Getzler-Vergne (Heat Kernels and Dirac operators), the chapter on characteristic classes. –  Kofi May 12 '13 at 7:56

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