A generalization of Gauchman's rigidity theorem

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) = {double pipe}h(u, u){double pipe}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.