The Lorentzian Lichnerowicz Conjecture for real-analytic, three-dimensional manifolds

TL;DR

This paper proves that for compact, three-dimensional, real-analytic Lorentzian manifolds, the conformal group either preserves a metric in the conformal class or the manifold is conformally flat, advancing understanding of conformal symmetries.

Contribution

It establishes the Lorentzian Lichnerowicz Conjecture for three-dimensional real-analytic manifolds, showing the conformal group has a specific preservation property or flatness.

Findings

Conformal group preserves a metric in the conformal class.

Manifold is conformally flat if the group does not preserve a metric.

Results apply to compact, three-dimensional, real-analytic Lorentzian manifolds.

Abstract

Given a compact, three-dimensional, real-analytic Lorentzian manifold $(M,g)$, we prove that the identity component of the conformal group preserves a metric in the conformal class $[g]$, or $(M,g)$ is conformally flat.