Umbilicity of (Space-Like) Submanifolds of Pseudo-Riemannian Space Forms

Abstract

We study 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 M in (a pseudo Riemannian manifold) is called -umbilic if the shape operator of M along is for some . 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 segment at every point except perhaps at isolated points. One of our main results says that if is an space-like (immersed) submanifold of , then M is totally semi-umbilical and (some special normal vectors) are in the same direction for every and all (except isolated) points of M, if and only if there exist linearly independent normal fields locally defined at every non-umbilical point of M, such that M is -umbilical, .