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Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection*

Published online by Cambridge University Press:  22 January 2016

Joseph Erbacher
Joseph Erbacher*
Affiliation:
Brown University, University of Southern California
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In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 45 , February 1972 , pp. 139 - 165
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

Footnotes

*

This paper is part of the author’s doctoral dissertation written under the direction of Professor K. Nomizu at Brown University. The research was partially supported by the National Science Foundation.

References

[1] Kobayashi, and Nomizu, , Foundations of Differential Geometry, Vol. I-II, John Wiley and Sons Inc., 1963, 1969.Google Scholar
[2] Nomizu, and Smyth, , A formula of Simon’s type and hypersurfaces with constant mean curvature, J. Differential Geometry 3 (1969), 367–377.Google Scholar
[3] Simons, , Minimal Varieties in Riemannian Manifolds, Ann. of Math. 88 (1968), 62–105.Google Scholar