Gauss–Bonnet theorem

The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).

Suppose $M$ is a compact two-dimensional orientable Riemannian manifold with boundary $\partial M$. Denote by $K$ the Gaussian curvature at points of $M$, and by $k_{g}$ the geodesic curvature at points of $\partial M$. Then

$\int _{M}K\;dA+\int _{\partial M}k_{g}\;ds=2\pi \chi (M)$

where $\chi (M)$ is the Euler characteristic of $M$.

The theorem applies in particular if the manifold does not have a boundary, in which case the integral $\int _{\partial M}k_{g}\;ds$ can be omitted.

If one bends and deforms the manifold $M$, its Euler characteristic will not change, while the curvatures at given points will. The theorem requires, somewhat surprisingly, that the total integral of all curvatures will remain the same.

A generalization to $n$ dimensions was found in the 1940s, by Allendoerfer, Weil, and Chern. See generalized Gauss-Bonnet theorem and Chern-Weil homomorphism.