# Gauss-Bonnet theorem

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The total curvature $\omega$ of a two-dimensional compact Riemannian manifold $V^2$, closed or with boundary, and the rotation $\tau$ of its smooth boundary $\partial V^2$ are connected with the Euler characteristic $\chi$ of $V^2$ by the relation

$$\omega+\tau=2\pi\chi.$$

Here

$$\omega=\int\limits_{V^2}KdS,$$

where $K$ is the Gaussian curvature and $dS$ is the area element;

$$\tau=\int\limits_{\partial V^2}k_gdl,$$

where $k_g$ is the geodesic curvature and $dl$ is the line element of the boundary. The Gauss–Bonnet theorem is also valid for a manifold with a piecewise-smooth boundary; in that case

$$\tau=\int k_gdl+\sum_i(\pi-\alpha_i),$$

where $\pi-\alpha_i$ is the rotation of the boundary at a vertex. The theorem is valid, in particular, on regular surfaces in $E^3$. The Gauss–Bonnet theorem was known to C.F. Gauss ; it was published by O. Bonnet  in a special form (for surfaces homeomorphic to a disc).

For a non-compact complete manifold $V^2$ without boundary, an analogue of the Gauss–Bonnet theorem is the Cohn-Vossen inequality :

$$\int\limits_{V^2}KdS\leq2\pi\chi.$$

The Gauss–Bonnet theorem and the given inequality are also valid for convex surfaces and two-dimensional manifolds of bounded curvature.

The Gauss–Bonnet theorem can be generalized to even-dimensional compact Riemannian manifolds $V^{2p}$, closed or with boundary:

$$\int\limits_{V^{2p}}\Omega dS+\int\limits_{\partial V^{2p}}\phi dl=\frac{(2\pi)^p}{(2p-1)!}\chi,$$

where $dS$, $dl$ denote the volume elements in $V^{2p}$ and $\partial V^{2p}$, while $\Omega$ is some polynomial in the components of the curvature tensor of $V^{2p}$, and $\phi$ is some polynomial in the components of the curvature tensor and the coefficients of the second fundamental form of $\partial V^{2p}$ . The Gauss–Bonnet theorem has also been generalized to Riemannian polyhedra . Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric , , .