On timelike hypersurfaces of the Minkowski ۴-space with ۱-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 ۴-space {\Bbb E}^۴ has constant mean curvature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski ۴-space {\Bbb E}^۴_۱. Let {\textbf x}:M_۱^۳\rightarrow{\Bbb E}_۱^۴ be the isometric immersion of a timelike hypersurface M^۳_۱ in {\Bbb E}_۱^۴. The second mean curvature vector field {\textbf H}_۲ of M_۱^۳ is called {\it ۱-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^۳_۱ with ۱-proper {\textbf H}_۲ has constant scalar curvature. By a classification theorem, we show that such a hypersurface is \mathcal{C}-biharmonic, \mathcal{C}-۱-type or  null-\mathcal{C}-۲-type. Since the shape operator of M^۳_۱ has four possible matrix forms, the results will be considered in four different cases.

کلیدواژه ها:

Weak convex ، Lorentz hypersurface ، Biharmonic ، mathcal{C}-harmonic