Umbilicity of space-like submanifolds of Minkowski space

Abstract

We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.