CONFORMAL ACTIONS OF REAL-RANK 1 SIMPLE LIE GROUPS ON PSEUDO-RIEMANNIAN MANIFOLDS

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, and 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 g must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes earlier investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank [1, 10, 26]. Combined with these results, we obtain as a corollary the list of semi-simple Lie groups without compact factor that can act conformally on compact Lorentzian manifolds. We also investigate some consequences in CR geometry via the Fefferman fibration. The author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network)