Cartan geometries with model the future lightlike cone of Lorentz-Minkowski spacetime

Abstract

This paper develops the theory of Cartan geometries modeled on the future lightlike cone of Lorentz Minkowski spacetime, which we refer to as lightlike Cartan geometries. We show that such geometries naturally induce on the base manifold a lightlike metric, a globally defined radical vector field, and two additional compatible structures. Within this framework, we construct the standard tractor bundle associated with a lightlike Cartan geometry, showing that it extends the tangent bundle of the base manifold and carries a canonical metric linear connection. This construction provides an alternative characterization of lightlike Cartan geometries purely in terms of vector bundle data. Using this alternative description, we analyze the additional geometric information encoded in the Cartan connection beyond the metric and radical data, and we show how it can be extracted via natural decompositions of the standard tractor bundle. Our results underscore the intrinsic significance of Cartan connection methods in the study of lightlike geometry and open new avenues for the analysis of geometric structures that are neither parabolic nor reductive.