# Gauss-Bonnet Theorem

## Theorem

Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\partial M$.

Let $\Kappa$ be the Gaussian curvature of $M$.

Let $k_g$ be the geodesic curvature of $\partial M$.

Then :

- $\displaystyle \int_M \kappa \, \mathrm d A + \int_{\partial M} k_g \, \mathrm d s = 2 \pi \chi\left({M}\right)$

where:

- $\mathrm d A$ is the element of area of the surface
- $\mathrm d s$ is the line element along $\partial M$
- $\chi\left({M}\right)$ is the Euler characteristic of $M$.

## Proof

## Source of Name

This entry was named for Carl Friedrich Gauss and Pierre Ossian Bonnet.

## Historical Note

Carl Friedrich Gauss was aware of the Gauss-Bonnet Theorem but never published it.

It was Pierre Ossian Bonnet who first published, in $1848$, a special case.