(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 the case that $$ \int_M G \, dA + \int_{\partial M}g_x ds = 2\pi\chi(M), $$ 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 in terms of geometrical information.