Extensions maximales et classification des tores lorentziens munis d'un champ de Killing

TL;DR

This paper classifies Lorentzian tori and Klein bottles with a Killing vector field by constructing universal extensions of Lorentzian surfaces, analyzing their symmetries, completeness, and minimal quotients.

Contribution

It introduces the concept of universal extensions for Lorentzian surfaces with a Killing field and provides a classification of such surfaces, including tori and Klein bottles.

Findings

Construction of universal extensions characterized by reflexivity and weak completeness.

Uniformization results for compact and analytical Lorentzian surfaces.

Classification of Lorentzian tori and Klein bottles with a Killing vector field.

Abstract

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call "universal extensions". They are characterized by a condition of symmetry, the "reflexivity", and a by a rather weak completeness assumption, the absence of "saddles at infinity". Considering these surfaces as model spaces, we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, which is uniquely determined) in the following cases: compact surfaces and analytical surfaces. It allows us to give a classification of Lorentzian tori and Klein bottles with a Killing vector field.