Conformal actions of solvable Lie groups on closed Lorentzian manifolds

2023/07/11 by Vincent Pecastaing, Pecastaing, Vincent · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2307.05436

openalex publication_date 2023/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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 ≥ 3, the radical of its conformal group admits a local embedding into O(2,n).

Cited by

• A local Lorentzian Ferrand-Obata theorem for conformal vector fields
• Failure of Lichnerowicz-type result in parabolic geometries of real rank at least 3

Related

• The Lorentzian Lichnerowicz Conjecture for real-analytic, three-dimensional manifolds
• The conformal group of a compact simply connected Lorentzian manifold
• Semi-simple Lie groups acting conformally on compact Lorentz manifolds
• Lorentzian manifolds with transitive conformal group
• Pseudo-Conformal actions of semisimple Lie groups