Francisco J. Palomo

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.

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to derive a rigidity result. Specifically, we obtain a complete characterization of all compact spacelike submanifolds with parallel mean curvature vector field that lie in the light cone of the Lorentz-Minkowski spacetime: they must be totally umbilical spheres contained in a spacelike hyperplane in Lorentz-Minkowski spacetime.

Starting from a Riemannian conformal structure on a manifold M, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth 1-parameter family of self-adjoint tensor fields. Then, every metric in the conformal class corresponds to the induced metric on M seen as a codimension two spacelike submanifold into these Lorentzian manifolds. Under suitable choices of the 1-parameter family of tensor fields, there exists a lightlike normal vector field along such spacelike submanifolds whose Weingarten endomorphism provide a Mobius structure on the Riemannian conformal structure. Conversely, every Mobius structure on a Riemannian conformal structure arises in this way. Flat Mobius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.