A local Lorentzian Ferrand-Obata theorem for conformal vector fields
Sorin Dumitrescu, Charles Frances, Karin Melnick, Vincent Pecastaing, Abdelghani Zeghib
TL;DR
The paper proves a local version of the Lorentzian Lichnerowicz conjecture in the real-analytic category: on a closed Lorentzian manifold of dimension at least 3, any conformal vector field is locally inessential unless the metric is conformally flat. The authors develop a robust Cartan-geometric framework to study conformal structures, employing a refined local linearization result for singularities, a detailed taxonomy of singularities, and global dynamical arguments that rely on compactness. They show that regular and isometry-like singularities yield local inessentiality, while contracting/expanding singularities force conformal flatness; mixed and balanced singularities are handled via a combination of pp-wave polarizations and polarization arguments, ultimately driving the nonflat case to flatness. The work connects local dynamical rigidity with global conformal geometry, providing a pathway toward understanding conformal compactifications and rigidity phenomena in Lorentzian geometry within the Zimmer program context.
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.