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.

Importance

As a tool for area computation

The Gauss-Bonnet theorem can be used to compute areas of regions in surfaces having constant curvature. For instance, it can be used to obtain a formula for the area of a geodesic triangle on the two-dimensional sphere, embedded in three-space.

As a topological control on the Gaussian curvature

The Gauss-Bonnet theorem tells us that given a manifold, we cannot give it both a metric with everywhere positive Gaussian curvature, and a metric with everywhere negative curvature.

It also tells us that if we fix the total volume of the manifold, then there is only one possible value for the curvature, if it is constant.