TITLE: On the generalized Gauss-Bonnet theorem
QUESTION [20 upvotes]: 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 $E$.
$$∫_{M}\mathrm{Pf}\Big(\frac{−F^{∇}}{2\pi}\Big)=χ(M^{2n}).$$

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?

REPLY [5 votes]: Mathai and Quillen have a theorem that computes the Euler characteristic as an integral of a form defined using a section of and a connection on the tangent bundle. If one scales the section by a factor $t$, then at $t=0$ one gets the Gauss-Bonnet theorem, and as $t\to\infty$ the integral localizes near the zero set of the section and becomes the Poincar\'e-Hopf theorem.