[Géométrie pseudoRiemannienne et actions des groupes de Lie simples]
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.
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.
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@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/
[1] Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds, J. Differential Geom., Volume 60 (2002) no. 3, pp. 355-387
[2] Gromov's centralizer theorem, Geom. Dedicata, Volume 100 (2003), pp. 123-155
[3] Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena), Handbook of Dynamical Systems, vol. 1A, North-Holland, Amsterdam, 2002, pp. 665-763
[4] Rigid transformations groups (D. Bernard; Y. Choquet-Bruhat, eds.), Géométrie defférentielle, Colloque Géométrie et Physique de 1986 en l'honneur de André Lichnerowicz, Hermann, 1988, pp. 65-139
[5] Noncompact simple automorphism groups of Lorentz manifolds and other geometric manifolds, Ann. of Math. (2), Volume 144 (1996) no. 3, pp. 611-640
[6] R. Quiroga-Barranco, Isometric actions of simple Lie groups on pseudoRiemannian manifolds, Ann. of Math., in press
[7] Infinitesimally homogeneous spaces, Comm. Pure Appl. Math., Volume 13 (1960), pp. 685-697
[8] Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), Volume 139 (1994) no. 3, pp. 723-747
[9] Isotropy of semisimple group actions on manifolds with geometric structure, Amer. J. Math., Volume 120 (1998) no. 1, pp. 129-158
[10] On affine actions of Lie groups, Math. Z., Volume 227 (1998) no. 2, pp. 245-262
[11] Actions of semisimple groups and discrete subgroups, Proceedings of the International Congress of Mathematicians, vols. 1, 2, Berkeley, CA, 1986, Amer. Math. Soc., Providence, RI, 1987, pp. 1247-1258
[12] Arithmeticity of holonomy groups of Lie foliations, J. Amer. Math. Soc., Volume 1 (1988) no. 1, pp. 35-58
[13] Automorphism groups and fundamental groups of geometric manifolds, Differential Geometry: Riemannian Geometry, Part 3, Los Angeles, CA, 1990, Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 693-710
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