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Totally umbilical submanifolds in irreducible symmetric spaces

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Bang-Yen Chen  and
Paul Verheyen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
Paul Verheyen
Affiliation:
Department of Mathematics, Katholieke University Leuven, Celestijnenlaan 200B, B-3030 Leuven, Belgium
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Abstract

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A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.

MSC classification

Secondary: 53B20: Local Riemannian geometry 53C35: Symmetric spaces 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 3 , June 1982 , pp. 332 - 338
Copyright
Copyright © Australian Mathematical Society 1982

References

Chen, B. Y. (1979), ‘Extrinsic spheres in Riemannian manifolds’, Houston J. Math. 5, 319–324.Google Scholar
Chen, B. Y. (1980), ‘Classification of totally umbilical submanifolds in symmetric spaces’, J. Austral. Math. Soc. Ser. A 30, 129–136.CrossRefGoogle Scholar
Helgason, S. (1978), Differential geometry, Lie groups and symmetric spaces (Academic Press, New York).Google Scholar