Eisenhart Problem to Submanifolds in Non-Flat Real Space Form

Abstract

We apply the Eisenhart problem to study the geometric properties of<br />submanifold $M$ of non-flat real space form. It is shown that $M$<br /> is a hypersphere $S^{3}$ when $sigma$ is parallel. When $sigma$ is either semi-parallel or recurrent,<br /> then $M$ is either an extrinsic sphere and normal flat or mean curvature vector is parallel in the normal space, respectively.