# Online Encyclopedia

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

where χ(*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 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.