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Classification of totally umbilical submanifolds in symmetric spaces

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
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Abstract

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A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

MSC classification

Secondary: 53B20: Local Riemannian geometry 53C35: Symmetric spaces 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 129 - 136
Copyright
Copyright © Australian Mathematical Society 1980

References

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