Vol. 228, No. 1, 2006

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A generalization of Gauchman’s rigidity theorem

Hong-Wei Xu, Wang Fang and Fei Xiang

Vol. 228 (2006), No. 1, 185–199
Abstract

We generalize the well-known Gauchman theorem for closed minimal submanifolds in a unit sphere, and prove that if M is an n-dimensional closed submanifold of parallel mean curvature in Sn+p and if σ(u) 1
3 for any unit vector u TM, where σ(u) = h(u,u)2, and h is the second fundamental form of M, then either σ(u) H2 and M is a totally umbilical sphere, or σ(u) 1
3. Moreover, we give a geometrical classification of closed submanifolds with parallel mean curvature satisfying σ(u) 1
3.

Keywords
closed submanifolds, rigidity theorem, parallel mean curvature
Mathematical Subject Classification 2000
Primary: 53C40, 53C42
Milestones
Received: 12 February 2005
Accepted: 6 July 2005
Published: 1 November 2006
Authors
Hong-Wei Xu
Center of Mathematical Sciences
Zhejiang University
Hangzhou 310027
The People’s Republic of China
Wang Fang
Center of Mathematical Sciences
Zhejiang University
Hangzhou 310027
The People’s Republic of China
Fei Xiang
Center of Mathematical Sciences
Zhejiang University
Hangzhou 310027
The People’s Republic of China