A generalization of Gauchman’s rigidity theorem

Hong-Wei Xu, Wang Fang and Fei Xiang

Vol. 228 (2006), No. 1, 185–199

DOI: 10.2140/pjm.2006.228.185

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 S^n+p and if σ(u) ≤ 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) ≡. Moreover, we give a geometrical classification of closed submanifolds with parallel mean curvature satisfying σ(u) ≡.

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