A local Lorentzian Ferrand-Obata theorem for conformal vector fields

TL;DR

This paper proves that conformal vector fields on closed real-analytic Lorentzian manifolds are either locally isometric or the manifold is conformally flat, advancing understanding of Lorentzian conformal geometry.

Contribution

It establishes a local Lorentzian version of the Lichnerowicz conjecture for real-analytic manifolds, improving normal form results for conformal vector fields.

Findings

Flow is locally isometric or manifold is conformally flat

Improved normal forms for conformal vector fields

Global arguments rely on compactness assumption

Abstract

For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere conformally flat. The main theorem can be viewed as a local version of the Lorentzian Lichnerowicz conjecture in the real-analytic setting. The key result is an optimal improvement of the local normal forms for conformal vector fields of [FM13], which focused on non-linearizable singularities. This article is primarily concerned with essential linearizable singularities, and the proofs include global arguments which rely on the compactness assumption.