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 $M$.