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 Zimmer's program for pseudoRiemannian manifolds.
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, jusqu'à une revêtement finie, une action algébrique. Nous employons ceci pour obtenir une caractérisation complète d'une 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.
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Raul Quiroga-Barranco 1
@article{CRMATH_2005__341_6_361_0,
author = {Raul Quiroga-Barranco},
title = {PseudoRiemannian geometry and actions of simple {Lie} groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {361--364},
publisher = {Elsevier},
volume = {341},
number = {6},
year = {2005},
doi = {10.1016/j.crma.2005.08.005},
language = {en},
}
Raul Quiroga-Barranco. PseudoRiemannian geometry and actions of simple Lie groups. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 361-364. doi : 10.1016/j.crma.2005.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.08.005/
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