Integral form of Euler characteristic

Question

There is a known formula for Euler characteristic in terms of Ricci scalar: \begin{equation} \chi(M)=\frac{1}{4\pi} \int_M \sqrt{g} \,R\,d^2x\,. \end{equation} I am sure that this formula holds for two dimensional manifolds, but what about higher dimensional ones? What integral can define Euler characteristic in higher dimensions?

Answer 1

You want the Chern-Gauss-Bonnet theorem. See also Chern-Weil theory.