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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.

Srivatsan
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Qiao
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  • For other references, see my question here: http://math.stackexchange.com/questions/64305/texts-on-principal-bundles-characteristic-classes-intro-to-4-manifolds-gauge – Jesse Madnick Dec 29 '11 at 17:58
  • Here is a paper on the relationship between Poincaré-Hopf and Gauss-Bonnet-Chern: http://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 07:56

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