Differential Geometric Aspects of Causal Structures

Omid Makhmali

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

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.

Détail

Keywords: causal geometry; conformal geometry; equivalence method; Cartan connection; parabolic geometry.

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