Locally conformally homogeneous Lorentzian spaces

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.