Submanifolds satisfying are named by B. Y. Chen submanifolds with proper mean curvature vector. We prove that a hypersurface of the pseudo-Euclidean space with and diagonalizable shape operator, has constant mean curvature.
Andreas ARVANITOYEORGOS. Filip DEFEVER. George KAIMAKAMIS. "Hypersurfaces of $E^4_8$ with proper mean curvature vector." J. Math. Soc. Japan 59 (3) 797 - 809, July, 2007. https://doi.org/10.2969/jmsj/05930797