Pseudo-Conformal actions of semisimple Lie groups

TL;DR

This paper proves that any compact connected pseudo-Riemannian manifold with a transitive, conformal, and essential action by a semisimple group must be conformally flat, advancing understanding of the pseudo-Riemannian Lichnerowicz conjecture.

Contribution

It establishes the conformal flatness of such manifolds in the homogeneous setting, confirming a case of the pseudo-Riemannian Lichnerowicz conjecture.

Findings

Any such manifold is conformally flat.

The result applies to homogeneous pseudo-Riemannian manifolds with semisimple group actions.

Supports the conjecture in the specific setting considered.

Abstract

We consider the pseudo-Riemannian Lichnerowicz conjecture in the homogeneous setting. In particular, we show that any compact connected pseudo-Riemannian manifold $M$ on which a semisimple group $G$ acts conformally, essentially and transitively, is conformally flat.