Intrinsic geometry of oriented congruences in three dimensions

TL;DR

This paper develops a geometric framework for oriented congruences in three dimensions, identifying invariants and constructing Lorentzian metrics with special curvature properties, including Ricci-flat and Einstein metrics.

Contribution

It introduces a new invariant classification of 3D oriented congruence structures and constructs explicit Lorentzian metrics with notable curvature characteristics.

Findings

Classified local invariants of oriented congruence structures.

Constructed explicit Ricci-flat and Einstein Lorentzian metrics.

Connected 3D CR structures to 4D Lorentzian geometries.

Abstract

Starting from the classical notion of an oriented congruence (i.e. a foliation by oriented curves) in $R^3$, we abstract the notion of an oriented congruence structure. This is a 3-dimensional CR manifold $(M,H, J)$ with a preferred splitting of the tangent space $TM=V\oplus H$. We find all local invariants of such structures using Cartan's equivalence method refining Cartan's classification of 3-dimensional CR structures. We use these invariants and perform Fefferman like constructions, to obtain interesting Lorentzian metrics in four dimensions, which include explicit Ricci-flat and Einstein metrics, as well as not conformally Einstein Bach-flat metrics.