# Differential Geometric Aspects of Causal Structures

## TL;DR

This paper explores the differential geometric properties of causal structures, solving their local equivalence problem using Cartan's method, and relating them to parabolic geometries with applications to special classes and twistorial constructions.

## Contribution

It provides a classification of causal structures via Cartan's method, linking them to parabolic geometries and introducing a twistorial approach for structures with vanishing Wsf curvature.

## Key findings

- Solved local equivalence problem for causal structures in dimensions ≥4.
- Identified correspondence with specific parabolic geometries.
- Developed a twistorial construction for structures with vanishing Wsf curvature.

## 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 $(D_n,P_{1,2})$ and $(B_{n-1},P_{1,2})$, when $n\geq 4$, and $(D_3,P_{1,2,3})$. 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.