Almost Robinson geometries

TL;DR

This paper studies the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, providing a classification based on intrinsic torsion and exploring their conformal and optical properties.

Contribution

It offers a comprehensive classification of almost Robinson geometries via intrinsic torsion and investigates their conformal invariants and optical geometry analogues.

Findings

Classification of almost Robinson manifolds based on intrinsic torsion

Relation between Robinson structures and leaf space geometry

Conformally invariant properties of these structures

Abstract

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy-Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.