Differential Geometric Aspects of Causal Structures

Abstract

This article is concerned with causal structures, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. The local equivalence problem of causal structures on manifolds of dimension at least four is solved using Cartan's method of equivalence, leading to an {e}-structure over some principal bundle. It is shown that these structures correspond to parabolic geometries of type (Dₙ, P₁,₂) and (Bₙ₋₁, P₁,₂), when n ≥ 4, and (D₃, P₁,₂,₃). The essential local invariants are determined and interpreted geometrically. Several special classes of causal structures are considered, including those that are a lift of pseudo-conformal structures and those referred to as causal structures with vanishing Wsf curvature. A twistorial construction for causal structures with vanishing Wsf curvature is given.