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Comptes Rendus Mathématique
Volume 341, n° 6
pages 361-364 (septembre 2005)
Doi : 10.1016/j.crma.2005.08.005
Received : 15 August 2004 ;  accepted : 24 July 2005
PseudoRiemannian geometry and actions of simple Lie groups
Géométrie pseudoRiemannienne et actions des groupes de Lie simples
 

Raul Quiroga-Barranco 1
Centro de Investigaciones en Matemáticas, A.P. 402, Guanajuato, Gto., C.P. 36000, México 

Abstract

Let G be a connected noncompact simple Lie group acting isometrically on a connected compact pseudoRiemannian manifold M . Denote with   and   the dimension of the maximal null subspaces tangent to G and M , respectively. Then we always have  . Our main result states that, if  , then the G -action is, up to a finite covering, an algebraic action. We use this to obtain a complete characterization of a large family of G -actions, thus providing a partial positive answer to the conjecture proposed in Zimmerʼs program for pseudoRiemannian manifolds. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Soit G un groupe de Lie simple non compact connexe agissant isométriquement sur une variété pseudoRiemannienne compacte connexe M . Dénotez avec   et   la dimension des sous-espaces nuls maximales tangents á G et M , respectivement. Alors nous avons toujours  . Notre résultat principal déclare que, si  , alors le action de G est, jusquʼà une revêtement finie, une action algébrique. Nous employons ceci pour obtenir une caractérisation complète dʼune famille nombreuse de actions de G , de ce fait fournissant une réponse positive partielle à la conjecture proposé dans le programme de Zimmer pour le variété pseudoRiemannienne. Pour citer cet article : R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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1  Research supported by SNI-México and Conacyt Grant 44620.


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