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

## TL;DR

This paper classifies conformal actions of rank 1 simple Lie groups on compact pseudo-Riemannian manifolds, establishing conditions for conformal flatness and identifying groups acting on Lorentzian manifolds.

## Contribution

It determines the minimal index for such actions, proves conformal flatness in optimal cases, and completes classification of semi-simple Lie groups acting on compact Lorentzian manifolds.

## Key findings

- Minimal index for conformal actions identified
- Conformal flatness proven for optimal index cases
- List of groups acting on compact Lorentzian manifolds provided

## 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.