Pseudo-Conformal actions of semisimple Lie groups

TL;DR

This work resolves the homogeneous case of the pseudo-Riemannian Lichnerowicz conjecture by showing that a compact, connected pseudo-Riemannian manifold $M$ admitting a conformal, essentially transitive action of a semisimple group $G$ is conformally flat, with ${ m Ein}^{p,q}$ arising in the maximal-structure scenario. The authors translate the geometric problem into a Lie-algebraic framework using root decompositions, parabolic subalgebras, and a Gauss map, then introduce the distortion map and the pairing condition to constrain the possible algebras. A central theme is the heredity principle, which produces Möbius-type subalgebras and enables a rank-two analysis that narrows ${rak g}$ to ${rak sp}(p,q)$; subsequent case analyses either prove flatness directly or identify the Einstein-universe model as the homogeneous quotient. The key conclusion is that any such homogeneous conformal action forces conformal flatness, with the extremal case realized by the standard action on ${ m Ein}^{4p-1,4q-1}$. These results extend the understanding of conformal geometry in the pseudo-Riemannian setting and connect to broader classification efforts of essential conformal actions.

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.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof