On the geometry of null hypersurfaces in Minkowski space

Abstract

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space R1n+2, classifying all rotation null hypersurfaces in R1n+2. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the (n+1)-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the 3-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of R14 and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.