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:
 * $\ds \int_M \kappa \rd A + \int_{\partial M} k_g \rd s = 2 \pi \map \chi M$

where:
 * $\d A$ is the element of area of the surface
 * $\d s$ is the line element along $\partial M$
 * $\map \chi M$ is the Euler characteristic of $M$.

Also see

 * Chern-Gauss-Bonnet Theorem
 * Riemann-Roch Theorem
 * Atiyah-Singer Index Theorem