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An integral formula for compact hypersurfaces in space forms and its applications

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Luis J. Alías
Luis J. Alías
Affiliation:
Departamento de Matemáticas Universidad de MurciaE-30100 Espinardo, MurciaSpain e-mail: [email protected]
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Abstract

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In this paper we establish an integral formula for compact hypersurfaces in non-flat space forms, and apply it to derive some interesting applications. In particular, we obtain a characterization of geodesic spheres in terms of a relationship between the scalar curvature of the hypersurface and the size of its Gauss map image. We also derive an inequality involving the average scalar curvature of the hypersurface and the radius of a geodesic ball in the ambient space containing the hypersurface, characterizing the geodesic spheres as those for which equality holds.

Keywords

Ricci curvaturescalar curvatureaverage scalar curvatureGauss map imagegeodesic sphere.

MSC classification

Secondary: 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 2 , April 2003 , pp. 239 - 248
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Deshmukh, S., ‘An integral formula for compact hypersurfaces in a Euclidean space and its applications’, Glasgow Math. J. 34 (1992), 309–311.CrossRefGoogle Scholar
[2]Deshmukh, S., ‘Hypersurfaces of non-negative Ricci curvature in a Euclidean space’, J. Geom. 45 (1992), 48–50.CrossRefGoogle Scholar
[3]Deshmukh, S., ‘Isometric immersion of a compact Riemannian manifold into a Euclidean space’, Bull. Austral. Math. Soc. 46 (1992), 177–178.CrossRefGoogle Scholar
[4]Deshmukh, S., ‘Compact hypersurfaces in a Euclidean space’, Quart. J. Math. Oxford Ser. (2) 49 (1998), 35–41.CrossRefGoogle Scholar
[5]Vlachos, T., ‘An integral formula for hypersurfaces in space forms’, Glasgow Math. J. 37 (1995), 337–341.CrossRefGoogle Scholar