The Poincaré–Hopf Theorem and the Chern–Gauß–Bonnet Theorem

The topological observations about de Rham cohomology are continued in this chapter. They lead to an elegant and far-reaching formula about zeros of sections in vector bundles. Their number (weighted by a sign) is identified with the integral of a polynomial function of the curvature of the Levi–Civita connection. This combination of the classical Theorems of Poincaré–Hopf and Chern–Gauß–Bonnet is deduced using a method by Mathai and Quillen.