Commentarii Mathematici Helvetici

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Volume 93, Issue 2, 2018, pp. 401–439
DOI: 10.4171/CMH/439

Published online: 2018-05-31

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

Vincent Pecastaing[1]

(1) Université Paris-Sud, Paris-Saclay, Orsay, France

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$(1,n), n \geq 2$, we derive global conclusions, extending a theorem of [18] to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of conformal groups of compact Lorentzian manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib [1, 2, 32].

Keywords: Lorentzian geometry, conformal geometry, dynamical systems, actions of Lie groups

Pecastaing Vincent: Lorentzian manifolds with a conformal action of SL(2,R). Comment. Math. Helv. 93 (2018), 401-439. doi: 10.4171/CMH/439