Submersions of CR-submanifolds with totally umbilical fibers

Abstract

The purpose of this paper is to study totally umbilical submersions of a $CR$-submanifolds $M$ of a Kaehler manifolds $\bar{M}$ onto an almost Hermitian manifolds $N$ such that characteristic vector field is vertical or horizontal vector field. We first show the totally geodesicity of fibers of such submersions. In addition, inequality in terms of O'Neill tensor fields is presented and new results regarding the equality sitation are obtained. Finally, the geodesicity of the fibers by submersion with parallel mean curvature vector field was investigated in complex space form.