Pseudo-Spherical Submanifolds Of A Pseudo-Euclidean Space

Decomposing the position vectors of a submanifold into the tangential and normal components and by means of moving frame, the present paper studies the characterizations of pesudo-spherical submanifolds of a pseudo-Euclidean space. We firstly study the space-like pesudo-spherical submanifolds and obtain a necessary and sufficient condition: there exists a unit normal vector satisfying the given three conditions. Specially , we also study Chen submanifolds and obtain that they are pseudo-spherical submanifolds if the mean curvature vector is parallel and not isotropic and the support function doesn't chang its sign.We also study the compact submanifolds whose normal vectors are all time-like. It is observed that the tangential components of the position vectors being harmonic , the support function of the submanifold being contant andthe function F=and the Ricci curvature in the direction of Ψ~Tsatisfying the condition: S(Ψ~T,Ψ~T) ≥n~2(1 + F)~2 provides three necessaryand sufficient conditions. At the same time, we obtain another necessary and sufficient condition for the submanifold with parallel mean curvature vector.