Essential conformal actions of PSL(2,R) on real-analytic compact Lorentz manifolds

TL;DR

This paper proves that real-analytic compact Lorentz manifolds of dimension at least three with a conformal essential PSL(2,R) action are conformally flat, extending to actions of semi-simple Lie groups, and advances understanding of Lorentzian geometry.

Contribution

It establishes conformal flatness for such manifolds under PSL(2,R) actions using Gromov's rigidity, and extends results to semi-simple Lie group actions, addressing a Lorentzian analogue of Lichnerowicz's question.

Findings

Conformal flatness of manifolds with PSL(2,R) actions.

Extension of results to semi-simple Lie group actions.

Application of Gromov's rigidity to Lorentzian geometry.

Abstract

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least $3$ admitting a conformal essential (i.e. conformal, but not isometric) action of a Lie group locally isomorphic to PSL(2,R). It is established by using a general result of M. Gromov on local isometries of real-analytic $A$-rigid geometric structures. As corollary, we deduce the same conclusion for conformal essential actions of connected semi-simple Lie groups on real-analytic compact Lorentz manifolds. This work is a contribution to the understanding of the Lorentzian version of a question asked by A. Lichnerowicz.