Locally conformally homogeneous Lorentzian spaces

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.

💡 Research Summary

The paper investigates Lorentzian manifolds of dimension ≥ 3 that are locally conformally homogeneous and admit an essential pseudo‑group of local conformal transformations. Extending a recent theorem of Alekseevsky and Galaev, the authors remove the hypotheses of simple connectivity and global conformal homogeneity, replacing them with the weaker condition that the local conformal pseudo‑group acts transitively. Their main result (Theorem 1.1) states that any such manifold (M,g) falls into exactly one of three mutually exclusive categories:

1. Inessential case – the pseudo‑group preserves a genuine Lorentzian metric within the conformal class, i.e. all conformal transformations are actually isometries for some metric g₁∈