Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-29T18:51:27.280Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of totally umbilical submanifolds in symmetric spaces

Part of: Global differential geometry Local 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Chen, B. Y. (1979), ‘Extrinsic spheres in Riemannian manifolds’, Houston J. Math. 5, 319–324.Google Scholar
Chen, B. Y. and Nagano, T. (1977), ‘Totally geodesic submanifolds of symmetric spaces. I’. Duke Jath. J. 44, 745–755.Google Scholar
Chen, B. Y. and Nagano, T. (1978), ‘Totally geodesic submanifolds of symmetric spaces. II’, Duke Math. J. 45, 405–425.CrossRefGoogle Scholar
Goldberg, S. I. (1969), ‘On conformally flat spaces with definite Ricci curvature’, Kodai Math. Sem. Rep. 21. 226–232.CrossRefGoogle Scholar
Helgason, S. (1968), Differential geometry, Lie groups and symmetric spaces (Academic Press, New York).Google Scholar
Miyazawa, T. and Chūman, G. (1972), ‘On certain subspaces of Riemannian recurrent spaces’, Tensor 23, 253–260.Google Scholar
Wolf, J. A. (1963), ‘Elliptic spaces in Grassmann manifolds’. Illinois J. Math. 7. 447–462.Google Scholar