On submanifolds with parallel mean curvature vector

Abstract

We consider Mn, n ≥ 3, a complete, connected submanifold of a space form M̃n+p(c̃), whose non vanishing mean curvature vector H is parallel in the normal bundle. Assuming the second fundamental form h of M satisfies the inequality 2 ≤ n2|H|2/(n - 1), we show that for c̃ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mn is totally umbilic. For the case c̃ < 0, one imposes an additional condition that is trivially satisfied when c̃ ≥ 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 c̃ < 0, reduces the codimension to 3.