## You are here

HomeGauss-Bonnet theorem

## Primary tabs

# 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.

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

53A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections