Gauss-Bonnet Theorem

Definition

Suppose $\dim (S)=2$. Let $K$ be a Gaussian Curvature of $S$. Then $\frac{1}{2\pi }{\int }_{S}KdA=2-2g=\chi (S)$ where $dA$ is the element of area of the surface, $g$ is the number of holes, and $\chi (S)$ is the Euler characteristic of $S$