# Gauss-Bonnet theorem for surfaces

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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.