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

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.

38 pages

References in corpus (2)

• Essential conformal actions of PSL(2,R) on real-analytic compact Lorentz manifolds
• Semi-simple Lie groups acting conformally on compact Lorentz manifolds

Cited by in corpus (2)

• Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds
• Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds