Compact Submanifolds in a Euclidean Space

ABSTRACT

In this paper we use bounds on the Ricci curvature of an n-dimensional compact submanifold M of the Euclidean space Rn+p to obtain a characterization of a sphere (cf. Theorem 1). Also we obtain a lower bound on the integral of the square of the length of mean curvature vector field H of this submanifold and show that the lower bound is attended if and only if the submanifold is a sphere giving another characterization of a sphere (cf. Theorem 2).