An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere

Abstract

Let M be an n-dimensional closed submanifold with parallel mean curvature in S^n+p, $\tilde{h}$ the trace free part of the second fundamental form, and $\tilde{\sigma}$(u) = ||$\tilde{h}$(u, u)||² for any unit vector u ∈ TM. We prove that there exists a positive constant C(n, p, H) (≥ 1/3) such that if $\tilde{\sigma}$(u) ≤ C(n, p, H), then either $\tilde{\sigma}$(u) ≡ 0 and M is a totally umbilical sphere, or $\tilde{\sigma}$(u) ≡ C(n, p, H). A geometrical classification of closed submanifolds with parallel mean curvature satisfying $\tilde{\sigma}$(u) ≡ C(n, p, H) is also given. Our main result is an extension of the Gauchman theorem [4].