Conformally homogeneous Lorentzian spaces

TL;DR

This paper classifies 1-connected Lorentzian manifolds with transitive essential conformal groups, showing they are complete homogeneous plane waves, and details their conformal transformation groups.

Contribution

It proves that such manifolds are necessarily complete homogeneous plane waves and characterizes their conformal groups, completing the classification of these Lorentzian spaces.

Findings

Manifolds are complete homogeneous plane waves.

Conformal transformation groups are 1-dimensional extensions of isometry groups.

Classification of 1-connected Lorentzian manifolds with transitive essential conformal groups.

Abstract

We prove that if a 1-connected non-conformally flat conformal Lorentzian manifold $(M,c)$ admits a connected essential transitive group of conformal transformations, then there exists a metric $g\in c$ such that $(M,g)$ is a complete homogeneous plane wave. This finishes the classification of 1-connected Lorentzian manifolds, which admit transitive essential conformal group. We also prove that the group of conformal transformations of a non-conformally flat 1-connected homogeneous plane wave $(M,g)$ consists of homotheties, and it is a 1-dimensional extension of the group of isometries.