# Gauss-Bonnet theorem

(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional compact Riemannian manifold $M$ with boundary, Gaussian curvature of points $G$ and geodesic curvature of points $g_{x}$ on the boundary $\partial M$, it is the case that

 $\int_{M}G\,dA+\int_{\partial M}g_{x}ds=2\pi\chi(M),$

where $\chi(M)$ is the Euler characteristic of the manifold, $dA$ denotes the measure with respect to area, and $ds$ denotes the measure with respect to arclength on the boundary. This theorem expresses a topological invariant in terms of geometrical information.

Title Gauss-Bonnet theorem GaussBonnetTheorem 2013-03-22 16:36:37 2013-03-22 16:36:37 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 53A05