On space-like submanifolds of a sphere

Abstract

Y. B. Shen [5] studied curvature pinching for 3-dimensional minimal submanifold in a sphere. In addition, Shen showed that if the scalar curvature of M3 is larger than 4, then M3 is totally geodesic. The purpose of this paper is to study geometry of an n- dimensional compact maximal spacelike submanifold in a (n + pj-dimensional unit sphere of constant curvature 1and index p by 'studying the Ricci and scalar curvatures of M. We do this by proving the following: 'l. Let M be an n-dimensional compact maximal space-like submanifold of S;+p. If the: (i) Ricci curvature R is less than (n -1) 3p -1 ,then Mis totally geodesic. 2p-l " J' Scalar curvature p is less than (n- 1)(n + -._P-J'then M is totally g~Odesic. 2p-1 Ricci curvature of M is bounded from above by 3, then M3 is isomorphic to S3.