An inequality on submanifolds with parallel mean curvature vector in a space of constant curvature

Let Nn+p(c)be an n+p dimensional Riemannian manifold with constant curvature c and Mn an n dimensional compact submanifold of Nn+p(c). It is known that there is a Simons’inequality when Mn is minimal. Li An Min etc.improved this inequality. Now this paper gives the generalizations of the inequality for the case that the mean curvature vector field of Mn is parallel.