On submanifolds with parallel mean curvature vector

Abstract

We consider Mⁿ, n ≥ 3, a complete, connected submanifold of a space form $\tilde{M}^{n+p}(\tilde{c})$, whose non vanishing mean curvature vector H is parallel in the normal bundle. Assuming the second fundamental form h of M satisfies the inequality <h>² ≤ n² |H|²/(n - 1), we show that for $\tilde{c}$ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mⁿ is totally umbilic. For the case $\tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $\tilde{c}$ ≥ 0. When M is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when $\tilde{c}$ < 0, reduces the codimension to 3.