# Gauss-Bonnet theorem for surfaces

From Diffgeom

Template:Curvature result 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 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.