ON TIMELIKE HYPERSURFACES OF THE MINKOWSKI 4-SPACE WITH 1-PROPER SECOND MEAN CURVATURE VECTOR

The mean curvature vector field of a submanifold in the Euclidean $n$-space is said to be $proper$ if it is an eigenvector of the Laplace operator $\Delta$. It is proven that every hypersurface with proper mean curvature vector field in the Euclidean 4-space ${\Bbb E}^4$ has constant mean curvature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4-space ${\Bbb E}^4_1$. Let ${\textbf x}:M_1^3\rightarrow{\Bbb E}_1^4$ be the isometric immersion of a timelike hypersurface $M^3_1$ in ${\Bbb E}_1^4$. The second mean curvature vector field ${\textbf H}_2$ of $M_1^3$ is called {\it 1-proper} if it is an eigenvector of the Cheng-Yau operator $\mathcal{C}$ (which is the natural extension of $\Delta$). We show that each $M^3_1$ with 1-proper ${\textbf H}_2$ has constant scalar curvature. By a classification theorem, we show that such a hypersurface is $\mathcal{C}$-biharmonic, $\mathcal{C}$-1-type or  null-$\mathcal{C}$-2-type. Since the shape operator of $M^3_1$ has four possible matrix forms, the results will be considered in four different cases.