Umbilicity of (Space-Like) Submanifolds of Pseudo-Riem-annian Space Forms

We study (v-)umbilic (space-like) submanifolds of pseudo-riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo euclidean space and relate this notion to umbilicity. finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.a pseudo-riemannian submanifold min m (a pseudo riemannian manifold) is called v -umbilic if the shape operator of m along v epsilon x^up tack(m) is s =lambda id(tm) for some lambda epsilon c(infinity) (m). a totally semi-umbilical space-like submanifold of a pseudo euclidean space is an space-like submanifold for which the curvature ellipse degenerates into a segmentat every point except perhaps at isolated points. one of our main results says that if m n is an space-like (immersed) submanifold of g^n+m (p);+m ,n-> 2 , then m is totally semi-umbilical and c^iotax il ,b^iotax (some special normal vectors) are in the same direction for every ii e t; m and all (except isolated) points of m, if and only if there exist linearly independent normal fields vn +1'... 'vn +m -1 locally defined at every non-umbilical point of m, such that m is va-umbilical, n +1~ alpha<-:n+m-l.