Some Inequalities For Totally Real Spacelike Submanifolds In An Indefinite Complex Space Form

One of the fundamental problems in the theory of submanifold is to establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds. The basic relations discovered until now are inequalities. The main purpose of this paper is to establish geometric inequalities between the intrinsic invariants and the extrinsic invariants for totally real spacelike submanifolds in an indefinite complex space form. Specially speaking, by using some algebraic inequalities and the optimization method due to T. Oprea,we obtain two inequalities under different conditions for totally real spacelike submnaifolds in an indefinite complex space form. Besides,we give the geometric conditions for the equality cases. Furthermore,we establish two inequalities involving Ricci curvature and the mean curvature under different conditions, which provides an upper bound of Ricci curvature. On the other hand, we obtain the geometric inequal-ity in terms of the Ricci curvature, the mean curvature and the scalar curvature, which provides a lower bound of R,icci curvature. Accord-ingly, two inequalities involving k-Ricci curvature and the T. Oprea’s invariant are obtained, respectively. Finally, by exploring the position of the parallel umbilical normal vectors in the normal bundle of totally real submanifolds, we obtain some respective results in a special case.