Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T19:07:41.556Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamique du pseudo-groupe des isométries locales sur une variété Lorentzienne analytique de dimension 3

Published online by Cambridge University Press:  01 August 2008

SORIN DUMITRESCU
SORIN DUMITRESCU*
Affiliation:
Département de Mathématiques d’Orsay, Équipe de Topologie et Dynamique, Bat. 425, U.M.R. 8628 C.N.R.S., Univ. Paris-Sud (11), 91405 Orsay Cedex, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Résumé

Soit (M,g) une variété lorentzienne analytique réelle de dimension 3 compacte et connexe. Nous démontrons que l’existence d’une orbite ouverte (non vide) du pseudo-groupe des isométries locales implique que la métrique lorentzienne est localement homogène (i.e. le pseudo-groupe des isométries locales de g agit transitivement sur M).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1091 - 1116
Copyright
Copyright © 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, S. and Stuck, G.. The isometry group of a compact Lorentz manifold, I. Invent. Math. 129 (1997), 239–261.CrossRefGoogle Scholar
[2]Amores, A. M.. Vector fields of a finite type G-structure. J. Differential Geom. 14(1) (1979), 1–6.CrossRefGoogle Scholar
[3]Babillot, M., Feres, R. and Zeghib, A.. Rigidité, Groupe Fondamental et Dynamique (Panoramas et Synthèses, 13). Ed. P. Foulon. Soc. Math. France, Paris, 2002.Google Scholar
[4]Benoist, Y.. Orbites des Structures Rigides (d’après M. Gromov) (Feuilletages et Systèmes Intégrables, Montpellier, 1995). Birkhäuser, Boston, 1997, pp. 1–17.Google Scholar
[5]Benoist, Y., Foulon, P. and Labourie, F.. Flots d’Anosov à distributions stables et instables différentiables. J. Amer. Math. Soc. 5 (1992), 33–74.Google Scholar
[6]D’Ambra, G.. Isometry groups of Lorentz manifolds. Invent. Math. 92 (1988), 555–565.CrossRefGoogle Scholar
[7]D’Ambra, G. and Gromov, M.. Lectures on Transformations Groups: Geometry and Dynamics (Surveys in Differential Geometry). Cambridge University Press, Cambridge, 1990, pp. 19–111.Google Scholar
[8]Friedbert, P., Tricerri, F. and Vanhecke, L.. Curvature invariants, differential operators and local homogeneity. Trans. Amer. Math. Soc. 348 (1996), 4643–4652.Google Scholar
[9]Gromov, M.. Rigid transformation groups. Géométrie Différentielle (Travaux en Cours, 33). Ed. D. Bernard et Y. Choquet-Bruhat. Hermann, Paris, 1988, pp. 65–141.Google Scholar
[10]Humphreys, J.. Linear Algebraic Groups (Graduate Texts in Mathematics, 21). Springer, Berlin, 1975.CrossRefGoogle Scholar
[11]Lie, S.. Theorie der Transformationsgruppen. Math. Ann. 16 (1880), 441–528.CrossRefGoogle Scholar
[12]Milnor, J.. Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293–329.CrossRefGoogle Scholar
[13]Molino, P.. Riemannian Foliations. Birkhauser, Basle, 1988.CrossRefGoogle Scholar
[14]Mumford, D.. Introduction to Algebraic Geometry. Harvard University, Cambridge, MA, 1966.Google Scholar
[15]Nomizu, K.. On local and global existence of Killing vector fields. Ann. of Math. (2) 72 (1960), 105–120.CrossRefGoogle Scholar
[16]Rosenlicht, M.. On quotient varieties and the affine embedding of certain homogeneous spaces. Trans. Amer. Math. Soc. 101 (1961), 211–223.CrossRefGoogle Scholar
[17]Rosenlicht, M.. A remark on quotient spaces. An. Acad. Brasil. Ciênc. 35(4) (1963), 487–489.Google Scholar
[18]Singer, I.. Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13 (1960), 685–697.CrossRefGoogle Scholar
[19]Wolf, J.. Spaces of constant curvature (McGraw-Hill Series in Higher Mathematics). McGraw-Hill, New York, 1967.Google Scholar
[20]Zeghib, A.. Killing fields in compact Lorentz 3-manifolds. J. Differential Geom. 43 (1996), 859–894.CrossRefGoogle Scholar
[21]Zeghib, A.. On affine actions of Lie groups. Math. Z. 227 (1998), 245–262.CrossRefGoogle Scholar
[22]Zeghib, A.. Geodesic foliations in Lorentz 3-manifolds. Comment. Math. Helv. 74 (1999), 1–21.CrossRefGoogle Scholar
[23]Zimmer, R.. Ergodic Theory and Semi-Simple Groups. Birkhäuser, Boston, 1984.CrossRefGoogle Scholar