Gauss-Bonnet theorem
The total curvature of a two-dimensional compact Riemannian manifold
, closed or with boundary, and the rotation
of its smooth boundary
are connected with the Euler characteristic
of
by the relation
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Here
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where is the Gaussian curvature and
is the area element;
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where is the geodesic curvature and
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
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where is the rotation of the boundary at a vertex. The theorem is valid, in particular, on regular surfaces in
. 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 without boundary, an analogue of the Gauss–Bonnet theorem is the Cohn-Vossen inequality [3]:
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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 , closed or with boundary:
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where ,
denote the volume elements in
and
, while
is some polynomial in the components of the curvature tensor of
, and
is some polynomial in the components of the curvature tensor and the coefficients of the second fundamental form of
[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 |
Gauss-Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss-Bonnet_theorem&oldid=13846