# Gauss-Bonnet theorem for surfaces without boundary

If $S$ is a compact, orientable surface without boundary, then

 $\int_{S}K=2\pi\,\chi(S),$

where $K$ is the Gaussian curvature of $S$ and $\chi(S)$ its Euler-Poincaré characteristic. (http://planetmath.org/EulerrCharacteristic)

Title Gauss-Bonnet theorem for surfaces without boundary GaussBonnetTheoremForSurfacesWithoutBoundary 2013-03-22 16:37:30 2013-03-22 16:37:30 Simone (5904) Simone (5904) 8 Simone (5904) Theorem msc 53A05