Locally conformally homogeneous Lorentzian spaces

TL;DR

This paper classifies locally conformally homogeneous Lorentzian manifolds of dimension at least 3, showing they are either conformally flat or equivalent to homogeneous plane waves, and explores their geometric structures using Gromov's theory.

Contribution

It generalizes previous results by demonstrating the structure of such manifolds and establishing their relation to plane waves and Penrose limits.

Findings

Manifolds are either conformally flat or locally conformally equivalent to homogeneous plane waves.

Existence of a codimension-one lightlike foliation of Heisenberg type in non-conformally flat cases.

The plane wave metric coincides with the Penrose limit along some null geodesic.

Abstract

We study locally conformally homogeneous Lorentzian manifolds of dimension at least $3$, admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such manifold $(M,g)$ is either conformally flat, or locally conformally equivalent to a homogeneous plane wave. When the manifold is non-conformally flat, we show the existence of a codimension-one lightlike foliation of Heisenberg type, which leads to the plane wave structure. Our approach relies on tools from Gromov's theory of rigid transformations. Finally, we observe that the plane wave metric in the conformal class coincides with the Penrose limit of $(M,g)$ along some null geodesic.