Hypersurfaces of $E^4_8$ with proper mean curvature vector

Abstract

Submanifolds satisfying $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ are named by B. Y. Chen submanifolds with proper mean curvature vector. We prove that a hypersurface of the pseudo-Euclidean space $E_s^4$ with $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ and diagonalizable shape operator, has constant mean curvature.