The Rigidity Theorems Of A Class Of Submanifolds In Hyperbolic Spaces

In this article, we mainly study the rigidity problems of the compact or complete submanifolds in hyperbolic spaces Hn+P(-1) with the parallel normalized mean curvature, and obtain some sufficient conditions for the submanifolds to be totally umbilical or pseudo-umbilical.The article divided into three sections:In section one, some basic concepts and associated formulas on submanifolds in hyperbolic spaces are presented.In section two, the compact or complete submanifolds satisfying the parallel nor-malized mean curvature in hyperbolic spaces are studied, some sufficient conditions for the compact or complete submanifolds to be totally umbilical or pseudo-umbilical ones are obtained(see Theorems2.1,2.2,2.3).In section three, by studying the eigenvalue of the two Schrodinger operators LH,LR defined on the compact submanifolds in hyperbolic spaces, we obtain some sufficient conditions for the compact submanifolds to be totally umbilical or pseudo-umbilical(see Theorems3.1,3.2).