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(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a Riemannian manifold $M$ with Gaussian curvature of points $G$ and geodesic curvature of points $g_x$ on the boundary $\partial M$, it is the case that $$\int_M GdA + \int_{\partial M}g_xds = 2\pi\chi(M),$$ where $\chi(M)$ is the Euler characteristic of the manifold. The Gauss-Bonnet theorem can provide metrics for topological invariants such as the Pfaffian.
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(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a Riemannian manifold $M$ embedded in $\mathbb{R}^3$ with Gaussian curvature of points $G$ and geodesic curvature of points $g_x$ on the boundary $\partial M$, it is the case that $$\int_M GdA + \int_{\partial M}g_xds = 2\pi\chi(M),$$ where $\chi(M)$ is the Euler characteristic of the manifold. The Gauss-Bonnet theorem can provide metrics for topological invariants such as the Pfaffian.
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