Lorentzian manifolds with a conformal action of SL(2,R)

TL;DR

This paper investigates conformal actions of simple Lie groups on compact Lorentzian manifolds, establishing conditions under which such actions are isometric or the manifold is conformally flat, and extending classification results for these groups.

Contribution

It proves an alternative for conformal group actions on compact Lorentzian manifolds and extends classification results to certain simple Lie groups of real-rank 1.

Findings

Either the group acts isometrically or the manifold is conformally flat.

Extension of Frances and Zeghib's theorem to some simple Lie groups of real-rank 1.

Provides a step towards classifying conformal groups of compact Lorentz manifolds.

Abstract

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywhere locally conformally diffeomorphic to Minkowski space-time. When the group is non-compact and not locally isomorphic to SO(1,n), n>1, we derive global conclusions, extending a theorem of Frances and Zeghib to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of the conformal groups of compact Lorentz manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib at the end of the 1990's.