Submanifolds for which a lower bound of the Ricci curvature is achieved

Abstract

Sharp estimates for the Ricci curvature of a submanifold Mn of an arbitrary Riemannian manifold Nn+p are established. It is shown that the equality in the lower estimate of the Ricci curvature of Mn in a space form Nn+p(c) is achieved only when Mn is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in Nn+p(c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.