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 $(D_n,P_{1,2})$ and $(B_{n-1},P_{1,2})$, when $n\ge 4$, 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.

50 pages; Typos are corrected; Section 4 is removed and will be expanded in another article. Subsection 3.2 is removed due to an unsatisfactory assumption for Theorem 3.5 to be true

References in corpus (7)

• Raychaudhuri equation and singularity theorems in Finsler spacetimes
• On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type
• Geometry of third-order ODEs
• Differential Geometric Aspects of Causal Structures
• Contact Path Geometries
• Lie contact structures and chains
• Causal geometries, null geodesics, and gravity

Cited by in corpus (7)

• On the definition and examples of cones and Finsler spacetimes
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Differential Geometric Aspects of Causal Structures
• GL(2)-geometry and complex structures
• The Cayley cubic and differential equations
• Lightlike hypersurfaces and time-minimizing geodesics in cone structures
• Twistor Theory of Dancing Paths