Abstract
In this paper, we are interested in extending the study of spherical curves in R-3 to the submanifolds in the Euclidean space Rn+p. More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space Rn+p lies on the hypersphere Sn+p-1 (c) (standardly imbedded sphere in Rn+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Delta of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S-n(c) (cf Theorem 4.1).