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Generalized Minkowski formulae for compact submanifolds of a Riemannian manifold

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Geoffrey Howard Smith
Geoffrey Howard Smith
Affiliation:
The School of Mathematical Sciences, The N.S.W. Institute of TechnologyP.O. Box 123 Broadway, N.S.W. 2007Australia
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Abstract

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In 1903 H. Minkowski obtained two integral formulae for closed convex surfaces in three dimensional Euclidean space. In this paper we obtain generalised Minkowski formulae on compact orientable immersed submanifolds of an arbitrary Riemannian manifold. By successive specialisation we indicate how known integral theorems can be obtained as particular cases of our result.

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 3 , June 1984 , pp. 378 - 388
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Chen, B. Y., Geometry of submanifolds (Marcel Dekker Inc., 1973).Google Scholar
[2]Chen, B. Y. and Yano, K., ‘Integral formulas for submanifolds and their applications,’ J. Differential Geometry 5 (1971), 467–477.CrossRefGoogle Scholar
[3]Hsiung, C. C., ‘Some integral formulas for closed hypersurfaces,’ Math. Scand. 2 (1954), 286–294.CrossRefGoogle Scholar
[4]Hsiung, C. C., ‘Some integral formulas for closed hypersurfaces in Riemann space,’ Pacific J. Math. 6 (1956), 291–299.CrossRefGoogle Scholar
[5]Hsiung, C. C., Liu, J. and Mittra, S., ‘Integral formulas for closed submanifolds of a Riemannian manifold,’ J. Differential Geometry 12 (1977), 133–151.CrossRefGoogle Scholar
[6]Katsurada, Y., ‘Generalised Minkowski formula for closed hypersurfaces in Riemann space,’ Ann. Mat. Pura Appl. (4) 57 (1962), 283–294.CrossRefGoogle Scholar
[7]Kobayashi, S. and Nomizu, K., Foundations of differential geometry (Interscience Publishers 1963, 1969 (2 volumes)).Google Scholar
[8]MacMahon, P. A., Combinatory analysis (Chelsea Publishing Company, 1960).Google Scholar
[9]Minkowski, H., ‘Volumen und Oberfläche,’ Math. Ann. 57 (1903), 447–495.CrossRefGoogle Scholar
[10]Rund, H., ‘Integral formulae on hypersurfaces in Riemannian manifolds,’ Ann. Mat. Pura. Appl. (4) 88 (1971), 92–122.CrossRefGoogle Scholar
[11]Smith, G. H., ‘Conformal variation of a hypersurface of constant mean curvature in an Einstein space,’ J. Austral. Math. Soc. Ser. A 33 (1982), 229–234.CrossRefGoogle Scholar