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Gauss-Bonnet theorem

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The total curvature 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)


Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5
How to Cite This Entry:
Gauss-Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss-Bonnet_theorem&oldid=33081
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article