Twisting non-shearing congruences of null geodesics, almost CR structures, and Einstein metrics in even dimensions

TL;DR

This paper explores the geometric structures induced by twisting non-shearing null geodesic congruences in even-dimensional Lorentzian manifolds, linking curvature conditions to Einstein metrics and almost CR structures, and presenting new solutions.

Contribution

It establishes necessary and sufficient conditions on the Weyl tensor for twist-induced almost Robinson structures and introduces new Einstein metrics derived from almost Kaehler-Einstein manifolds.

Findings

Characterization of Weyl tensor conditions for almost Robinson structures.

Identification of Einstein metrics including known and new solutions.

Construction of Einstein metrics from almost Kaehler-Einstein manifolds.

Abstract

We investigate the geometry of a twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. We give a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, that is, the screen bundle of the congruence is equipped with a bundle complex structure. In this case, the (local) leaf space of the congruence acquires a partially integrable contact almost CR structure of positive definite signature. We give further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure, and for the flatness of the latter. We show that under a mild natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR-Einstein structure on the leaf space of the congruence. These metrics depend on three parameters, and include the Fefferman-Einstein metric and Taub-NUT-(A)dS metric in the integrable case. In the non-integrable case, we obtain new solutions to the Einstein field equations, which, we show, can be constructed from strictly almost Kaehler-Einstein manifolds.