Reference : Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds
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Physical, chemical, mathematical & earth Sciences : Mathematics
Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds
Pecastaing, Vincent mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] Conformal Geometry ; Pseudo-Riemannian Geometry ; CR Geometry ; Dynamics of Lie groups actions
[en] 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.
To appear in Transformation Groups (Springer)

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