Conformal actions of solvable Lie groups on closed Lorentzian manifolds

TL;DR

This paper investigates the conformal actions of solvable Lie groups on closed Lorentzian manifolds, establishing conditions for inessentiality and conformal flatness, and relating the structure of the conformal group to the Lorentzian Lichnerowicz conjecture.

Contribution

It characterizes when the conformal group is inessential and proves conformal flatness under certain algebraic conditions, advancing understanding of Lorentzian conformal geometry.

Findings

The conformal group is inessential iff its nilradical is inessential.

Conformal flatness is established on open subsets under algebraic hypotheses.

The radical admits a local embedding into O(2,n) for essential manifolds.

Abstract

We consider conformal actions of solvable Lie groups on closed Lorentzian manifolds. With anterior results in which we addressed similar questions for semi-simple Lie group actions, this work contributes to the understanding of the identity component $G$ of the conformal group of closed Lorentzian manifolds. In the first part of the article, we prove that $G$ is inessential if and only if its nilradical is inessential. In the second, we assume the nilradical essential and establish conformal flatness of the metric on an open subset, under certain algebraic hypothesis on the solvable radical. This is related to the Lorentzian Lichnerowicz conjecture. Finally, we consider the remaining situations where our methods do not apply to prove conformal flatness, and conclude that for an essential closed Lorentzian $n$-manifold, $n \geq 3$, the radical of its conformal group admits a local embedding into $O(2,n)$.