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 in 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

Primary: 53C40, 53C42

Authors