Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Compact hypersurfaces in a unit sphere

Qing-Ming Chenga1, Shichang Shua2 and Young Jin Suha3

a1 Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan (cheng@ms.saga-u.ac.jp)

a2 Department of Mathematics, Weinan Teachers' College, Weinan 714000, Shaanxi, People's Republic of China (xysxssc@yahoo.com.cn)

a3 Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea (yjsuh@wmail.knu.ac.kr)

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 S0308210500004303inline001 is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies

S0308210500004303disp001

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 S0308210500004303inline002 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
S0308210500004303disp002

This gives a partial answer for the problem proposed by Cheng.

(Received August 31 2004)

(Accepted February 09 2005)