Gauss-Bonnet theorem for surfaces

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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 K denotes the Gaussian curvature at point p \in M,then:

\int_{p \in M} K dM = 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.