A generalization of Gauchman's rigidity theorem

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 or (u) <= 1/3 for any unit vector u is an element of TM, where sigma (u) = parallel to h(u, u)parallel to(2), and h is the second fundamental form of M, then either sigma (u) equivalent to H-2 and M is a totally umbilical sphere, or sigma (u) equivalent to 1/3. Moreover, we give a geometrical classification of closed submanifolds with parallel mean curvature satisfying sigma (u) equivalent to 1/3.