Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 080, 50 pages      arXiv:1704.02542      https://doi.org/10.3842/SIGMA.2018.080

Differential Geometric Aspects of Causal Structures

Omid Makhmali
Institute of Mathematics, Polish Academy of Sciences, 8 Śniadeckich Str., 00-656 Warszawa, Poland

Received April 25, 2017, in final form July 23, 2018; Published online August 02, 2018

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.

Key words: causal geometry; conformal geometry; equivalence method; Cartan connection; parabolic geometry.

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