Submanifolds with parallel mean curvature vector in spheres

Abstract

Let Mn be a compact submanifold of a sphere with parallel mean curvature vector h. We introduce a tensor ϕ, related to h and the second fundamental form, and show that if (j) satisfies a certain inequality, then ϕ = 0 and Mn is totally umbilic or equality holds. The estimate is sharp in the sense that we describe all Mn that satisfy this equality. © 1994 Tohoku University, Mathematical Institute.