Gauss-Bonnet theorem

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

References

[1]	C.F. Gauss, , Werke , 8 , K. Gesellschaft Wissenschaft. Göttingen (1900)
[2]	O. Bonnet, J. École Polytechnique , 19 (1848) pp. 1–146
[3]	S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)
[4]	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
[5]	C.B. Allendörfer, A. Weil, "The Gauss–Bonnet theorem for Riemannian polyhedra" Trans. Amer. Math. Soc. , 53 (1943) pp. 101–129
[6]	J. Eells, "A generalization of the Gauss–Bonnet theorem" Trans. Amer. Math. Soc. , 92 (1959) pp. 142–153
[7]	L.S. Pontryagin, "On a connection between homologies and homotopies" Izv. Akad. Nauk SSSR Ser. Mat. , 13 (1949) pp. 193–200 (In Russian)

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5