Extensions maximales et classification des tores lorentziens munis d’un champ de Killing  [ Maximal extensions and classification of Lorentzian tori with a Killing field ]
Annales de l'Institut Fourier to appear, , 102 p.

We define a family of model spaces for 2-dimensional Lorentzian geometry, consisting of simply connected inextendable Lorentzian surfaces admitting a Killing field. These spaces, called “universal extensions”, are constructed by an extension process and characterized by symmetry and completeness conditions. In general, these surfaces have a rich combinatorics and admit many quotient spaces and many divisible open sets. As applications, we show the existence of plenty (both topologically and geometrically) of Lorentzian surfaces with a Killing field. We also prove uniformisation results for the compact case and for the analytic case, which in particular allows us to give a classification of Lorentzian tori and Klein bottles with a Killing field.

Nous introduisons une famille d’espaces modèles pour la géométrie lorentzienne en dimension 2. Il s’agit de surfaces lorentziennes simplement connexes, inextensibles et possédant un champ de Killing. Désignées par « extensions universelles », elles sont construites par un procédé d’extension et caractérisées par certaines conditions de symétrie et de complétude. Ces surfaces possèdent généralement une combinatoire très riche, de nombreux quotients et de nombreux ouverts divisibles. Comme application, nous obtenons l’existence d’une grande diversité, tant topologique que géométrique, de surfaces lorentziennes munies d’un champ de Killing. Nous établissons aussi des résultats d’uniformisation pour le cas compact et pour le cas analytique, ce qui nous permet notamment de classer les tores lorentziens et les bouteilles de Klein possédant un champ de Killing.

Received : 2017-01-16
Revised : 2018-09-10
Accepted : 2018-09-25
Classification:  53C50
Keywords: Lorentzian surfaces, Killing fields
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     author = {Bavard, Christophe and Mounoud, Pierre},
     title = {Extensions maximales et classification des tores lorentziens munis d'un champ de Killing},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
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Bavard, Christophe; Mounoud, Pierre. Extensions maximales et classification des tores lorentziens munis d’un champ de Killing. Annales de l'Institut Fourier, to appear, 102 p.

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