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 thecase that

$${\int }_{M}G𝑑A+{\int }_{\partial M}{g}_x𝑑s=2\pi \chi (M),$$

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 interms of geometrical information.