Lightlike manifolds and Cartan geometries

Abstract

Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold $N$ provides a lightlike metric $h$ with radical distribution globally spanned by a vector field $Z$. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold $(N,h)$