Submanifolds Satisfying The Condition ?₂???=?????in Pseudo-Riemannian Space Forms

Let?:M_rⁿ?N_q^n+pbe an isometric immersion of a pseudo-Riemannian mani-fold M_rⁿinto a pseudo-Riemannian manifold N_q^n+p.We say that M_rⁿis a biharmonic submanifold,if its bitension field?₂???vanishes identically.The concept of bihar-monic submanifolds in Euclidean spaces was firstly proposed by B.-Y.Chen during the study of the finite type submanifolds,and for which in general?pseudo?Rieman-nian manifolds was firstly proposed by G.-Y.Jiang as applications of biharmonic maps.In the last decade,biharmonic submanifolds have been a very active research subject in geometry of submanifolds and attracted wide attention.Around Chen's conjecture?i.e.,any biharmonic submanifold in Euclidean space is minimal?,many interesting results were obtained concerning the aspect of existence and classifica-tion problems of nonminimal biharmonic submanifolds.In this paper,we study submanifolds satisfying the condition?₂???=??????where?is a constant,????is the tension field of??in pseudo-Riemannian space forms,and obtain some proper-ties of its mean curvature.Furthermore,we prove some classification theorems of such submanifolds.The main results are the following two parts including that of submanifolds and hypersurfaces.In the first part,we investigate hypersurfaces satisfying the condition ?₂???=?????in pseudo-Riemannian space forms.We give firstly a series of model spaces of such hypersurfaces.Then,using these model spaces,we obtain several classification results of such surfaces in nonflat 3-dimensional Lorentz space forms.For the case of higher dimension,under the assumption that such hypersurfaces have diagnoliz-able shape operator with at most two distinct principal curvatures,we prove a full classification theorem of such hypersurfaces.For the case of three distinct principal curvatures,we prove that such hypersurfaces must have constant mean curvature.In the second part,we study submanifolds satisfying the condition?₂???=?????in pseudo-Riemannian space forms.We investigate such submanifolds with parallel mean curvature vector field,or parallel normalized mean curvature vector field,respectively,under the assumption that submanifolds have diagnolizable shape operator in the direction of the mean curvature vector field,and the mean curvature is nonzero or at most two distinct principal curvatures,obtain an upper bound estimate for their mean curvature.Also,we completely classify such class of pseudo-umbilical submanifolds with parallel normalized mean curvature vector field.