Gauss-Bonnet theorem for surfaces
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 denotes the Gaussian curvature at point ,then:
Here 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.