Hostname: page-component-6bf8c574d5-8gtf8 Total loading time: 0 Render date: 2025-02-23T02:10:17.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Submanifolds and the length of the second fundamental tensor

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Thomas Hasanis
Thomas Hasanis
Affiliation:
Department of Mathematics University of IoanninaIoannina, Greece
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 sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

Keywords

second fundamental tensortotally umbilical submanifolds.

MSC classification

Secondary: 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Chen, B.-Y., Geometry of submanifolds (Marcel Dekker, New York, 1973).Google Scholar
[2]Chen, B.-Y. and Okumura, M., ‘Scalar curvature, inequality and submanifold’, Proc. Amer. Math. Soc. 38 (1973), 605–608.CrossRefGoogle Scholar
[3]Erbacher, J., Isometric immersions of Riemannian manifolds into space forms (Thesis, Brown University, 1970).Google Scholar
[4]Hasanis, Th., ‘Isometric immersions into spheres’, J. Math. Soc. Japan 33 (1981), 551–555.CrossRefGoogle Scholar
[5]Hasanis, Th., ‘Submanifolds and a pinching problem on the second fundamental tensors’, to appear.Google Scholar
[6]Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Vols. I, II (John Wiley and Sons, Inc., 1963, 1969).Google Scholar
[7]Okumura, M., ‘Submanifolds and a pinching problem on the second fundamental tensors’, Trans. Amer. Math. Soc. 178 (1973), 285–291.CrossRefGoogle Scholar
[8]Omori, H., ‘Isometric immersions of Riemannian manifolds’, J. Math. Soc. Japan 19 (1967), 409–415.Google Scholar