RICCI CURVATURE OF INTEGRAL SUBMANIFOLDS OF AN S-SPACE FORM

Abstract

Abstract. Involving the Ricci curvature and the squared mean curva-ture, we obtain a basic inequality for an integral submanifold of an S -space form. By polarization, we get a basic inequality for Ricci tensoralso. Equality cases are also discussed. By giving a very simple proof weshow that if an integral submanifold of maximum dimension of an S -spaceform satisfles the equality case, then it must be minimal. These resultsare applied to get corresponding results for C -totally real submanifoldsof a Sasakian space form and for totally real submanifolds of a complexspace form. 1. IntroductionOne of the most fundamental problems in submanifold theory is the follow-ing: Establish simple relationships between the main extrinsic invariants andthe main intrinsic invariants of a submanifold. In [7], B.-Y. Chen established asharp relationship between the Ricci curvature and the squared mean curvaturefor a submanifold in a Riemannian space form with arbitrary codimension. In[8], he gave the corresponding version of this inequality for totally real sub-manifolds in a complex space form. We flnd corresponding results for