# Gauss-Bonnet theorem

In mathematics, 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 ∂*M*. Denote by *K* the Gaussian curvature at points of *M*, and by *k*_{g} the geodesic curvature at points of ∂*M*. Then

- ∫
_{M'\'}*d*A*+ ∫*M_{∂}s_{g}d*= 2π χ(*M'')

*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 ∫_{∂M} *k*_{g} d*s* 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 generalisation to *n* dimensions was found in the 1940s, by Allendoerfer, Weil, and Chern.