Conformal transformations of Cahen–Wallach spaces
[Transformations conformes d’espaces de Cahen–Wallach]
Leistner, Thomas1 ; Teisseire, Stuart2
1 School of Computer and Mathematical Sciences, University of Adelaide, SA 5005, (Australia)
2 Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142 (New Zealand)
Annales de l'Institut Fourier, Online first, 41 p.

Nous étudions les transformations conformes d’espaces symétriques lorentziens indécomposables de courbure sectionnelle non constante, appelés espaces de Cahen–Wallach. Lorsqu’un espace de Cahen–Wallach n’est pas conformement plat, ses transformations conformes sont des homothéties. En utilisant cela, nous montrons qu’une transformation conforme d’un espace de Cahen–Wallach qui n’est pas conformement plat est essentielle si et seulement si il a un point fixe. Nous explorons ensuite la possibilité de groupes proprement discontinus de transformations conformes agissant avec un espace d’orbites compact sur un espace de Cahen–Wallach non conformement plat. Nous montrons qu’un tel groupe ne peut centraliser une homothétie essentielle et que pour les espaces de Cahen–Wallach de type imaginaire, un tel groupe doit être contenu dans les isométries.

We study conformal transformations of indecomposable Lorentzian symmetric spaces of non-constant sectional curvature, the so-called Cahen–Wallach spaces. When a Cahen–Wallach space is conformally curved, its conformal transformations are homotheties. Using this we show that a conformal transformation of a conformally curved Cahen–Wallach space is essential if and only if it has a fixed point. Then we explore the possibility of properly discontinuous groups of conformal transformations acting with a compact orbit space on a conformally curved Cahen–Wallach space. We show that any such group cannot centralise an essential homothety and that for Cahen–Wallach spaces of imaginary type, such a group must be contained within the isometries.

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DOI : 10.5802/aif.3678
Classification : 53C50, 53C18, 53C35, 57S20
Keywords: Lorentzian manifolds, Lorentzian symmetric spaces, conformal geometry, conformal transformations, conformal group actions, cocompact group actions, essential conformal transformations
Mots-clés : Variétés lorentziennes, espaces symétriques lorentziens, géométrie conforme, transformations conformes, actions de groupe conformes, actions de groupe cocompactes, transformations conformes essentielles

Leistner, Thomas 1 ; Teisseire, Stuart 2

1 School of Computer and Mathematical Sciences, University of Adelaide, SA 5005, (Australia)
2 Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142 (New Zealand) 
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Leistner, Thomas; Teisseire, Stuart. Conformal transformations of Cahen–Wallach spaces. Annales de l'Institut Fourier, Online first, 41 p.

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