Gauss-Bonnet theorem for surfaces

Statement

The Gauss-Bonnet theorem states that the average value of Gaussian curvature over a volume-normalized compact orientable two-dimensional Riemannian manifold is proportional to the Euler characteristic of the manifold. Specifically, if $K$ denotes the Gaussian curvature at point $p\in M$,then:

$\int _{p\in M}KdM=2\pi \chi (M)$

Here $\chi (M)$ denotes the Euler characteristic, which is a purely topological notion.

Note that the Gauss-Bonnet theorem works only for orientable manifolds since it crucially depends on an embedding in 3-space.