Submanifolds with constant scalar curvature

Abstract

Let Mⁿ be a compact submanifold of S^n+p(c) with constant scalar curvature. In this paper, we prove that if the squared norm S of the second fundamental form satisfies a certain inequality, then Mⁿ is a totally umbilic or eqality holds and we described all Mⁿ that satisfy this equality.