Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces
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Abstract:
We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.References
- Shiing-shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747–752. MR 11027, DOI 10.2307/1969302
- Shiing-shen Chern, On the curvatura integra in a Riemannian manifold, Ann. of Math. (2) 46 (1945), 674–684. MR 14760, DOI 10.2307/1969203
- Daniel Henry Gottlieb, All the way with Gauss-Bonnet and the sociology of mathematics, Amer. Math. Monthly 103 (1996), no. 6, 457–469. MR 1390575, DOI 10.2307/2974712
- H. Hopf, Über die curvatura integra geschlossener hyperflächen, Math. Annalen 95 (1925), 340–376.
- —, Vektorfelder in $n$-dimensionalen manningfaltigkeiten, Math. Annalen 96 (1927), 225–250.
- John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
- M. Morse, Singular points of vector fields under general boundary conditions, Amer. J. Math. 51 (1929), 165–178.
- Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465–477. MR 341351
- Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
- Eberhard Teufel, Eine differentialtopologische Berechnung der totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie, Manuscripta Math. 31 (1980), no. 1-3, 119–147 (German, with English summary). MR 576494, DOI 10.1007/BF01303271
- E. Teufel, Integral geometry and projection formulas in spaces of constant curvature, Abh. Math. Sem. Univ. Hamburg 56 (1986), 221–232. MR 882416, DOI 10.1007/BF02941517
Additional Information
- Gil Solanes
- Affiliation: Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- Address at time of publication: Institut de Mathématiques de Bourgogne, 9 Avénue Alain Savary – BP 47870, 21078 Dijon Cedex, France
- Email: solanes@mathematik.uni-stuttgart.de, solanes@topolog.u-bourgogne.fr
- Received by editor(s): April 15, 2004
- Published electronically: April 22, 2005
- Additional Notes: This work was partially supported by MECD grant EX2003-0987 and MCYT grant BMF2003-03458
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 1105-1115
- MSC (2000): Primary 53C65
- DOI: https://doi.org/10.1090/S0002-9947-05-03828-6
- MathSciNet review: 2187647