View source for 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 [[#References|[1]]]; it was published by O. Bonnet [[#References|[2]]] 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 [[#References|[3]]]:

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

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 $\partial V^{2p}$ [[#References|[4]]]. The Gauss–Bonnet theorem has also been generalized to Riemannian polyhedra [[#References|[5]]]. Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric [[#References|[4]]], [[#References|[6]]], [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  , ''Werke'' , '''8''' , K. Gesellschaft Wissenschaft. Göttingen  (1900)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Bonnet,   ''J. École Polytechnique'' , '''19'''  (1848)  pp. 1–146</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.E. Cohn-Vossen,   "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Sharafutdinov,   "Relative Euler class and the Gauss–Bonnet theorem"  ''Siberian Math. J.'' , '''14''' :  6  (1973)  pp. 930–940  ''Sibirsk Mat. Zh.'' , '''14''' :  6  pp. 1321–1635</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.B. Allendörfer,   A. Weil,   "The Gauss–Bonnet theorem for Riemannian polyhedra"  ''Trans. Amer. Math. Soc.'' , '''53'''  (1943)  pp. 101–129</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Eells,   "A generalization of the Gauss–Bonnet theorem"  ''Trans. Amer. Math. Soc.'' , '''92'''  (1959)  pp. 142–153</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.S. Pontryagin,   "On a connection between homologies and homotopies"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''13'''  (1949)  pp. 193–200  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''5''' , Publish or Perish  (1975)  pp. 1–5</TD></TR></table>