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Compact hypersurfaces in a unit sphere

Published online by Cambridge University Press:  12 July 2007

Qing-Ming Cheng ,
Shichang Shu  and
Young Jin Suh
Qing-Ming Cheng
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan ([email protected])
Shichang Shu
Affiliation:
Department of Mathematics, Weinan Teachers' College, Weinan 714000, Shaanxi, People's Republic of China ([email protected])
Young Jin Suh
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea ([email protected])
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Abstract

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 135 , Issue 6 , December 2005 , pp. 1129 - 1137
Copyright
Copyright © Royal Society of Edinburgh 2005

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