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.
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.
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Mot clés : Surfaces lorentziennes, champs de Killing
Keywords: Lorentzian surfaces, Killing fields
@article{AIF_2020__70_1_67_0,
author = {Bavard, Christophe and Mounoud, Pierre},
title = {Extensions maximales et classification des tores lorentziens munis d{\textquoteright}un champ de {Killing}},
journal = {Annales de l'Institut Fourier},
pages = {67--168},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {70},
number = {1},
year = {2020},
doi = {10.5802/aif.3309},
language = {fr},
url = {http://archive.numdam.org/articles/10.5802/aif.3309/}
}
TY - JOUR AU - Bavard, Christophe AU - Mounoud, Pierre TI - Extensions maximales et classification des tores lorentziens munis d’un champ de Killing JO - Annales de l'Institut Fourier PY - 2020 SP - 67 EP - 168 VL - 70 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3309/ DO - 10.5802/aif.3309 LA - fr ID - AIF_2020__70_1_67_0 ER -
%0 Journal Article %A Bavard, Christophe %A Mounoud, Pierre %T Extensions maximales et classification des tores lorentziens munis d’un champ de Killing %J Annales de l'Institut Fourier %D 2020 %P 67-168 %V 70 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3309/ %R 10.5802/aif.3309 %G fr %F AIF_2020__70_1_67_0
Bavard, Christophe; Mounoud, Pierre. Extensions maximales et classification des tores lorentziens munis d’un champ de Killing. Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 67-168. doi : 10.5802/aif.3309. http://archive.numdam.org/articles/10.5802/aif.3309/
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