Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds

Abstract

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we obtain as corollary the list of semi-simple Lie groups without compact factor that can act on compact Lorentzian manifolds. We also derive consequences in CR geometry via the Fefferman fibration.

23 pages

References in corpus (2)

• Lorentzian manifolds with a conformal action of SL(2,R)
• Essential conformal actions of PSL(2,R) on real-analytic compact Lorentz manifolds

Cited by in corpus (1)

• Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds