Sur les symétries des structures géométriques rigides
Séminaire de théorie spectrale et géométrie, Volume 28  (2009-2010), p. 29-49

Nous présentons des résultats de classification pour des variétés lorentziennes de dimension trois avec “beaucoup” de symétries locales.

DOI : https://doi.org/10.5802/tsg.277
Classification:  53B21,  53B30,  53C56,  53A55
Keywords:
@article{TSG_2009-2010__28__29_0,
     author = {Dumitrescu, Sorin},
     title = {Sur les sym\'etries des structures g\'eom\'etriques rigides},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     year = {2009-2010},
     pages = {29-49},
     doi = {10.5802/tsg.277},
     language = {fr},
     url = {http://www.numdam.org/item/TSG_2009-2010__28__29_0}
}
Dumitrescu, Sorin. Sur les symétries des structures géométriques rigides. Séminaire de théorie spectrale et géométrie, Volume 28 (2009-2010) , pp. 29-49. doi : 10.5802/tsg.277. http://www.numdam.org/item/TSG_2009-2010__28__29_0/

[1] A. Amores, Vector fields of a finite type G-structure, J. Differential Geom., 14(1), 1979, 1-6. | MR 577874 | Zbl 0414.53030

[2] G. D’Ambra, Isometry groups of Lorentz manifolds, Invent. Math., 92, (1988), 555-565. | MR 939475 | Zbl 0647.53046

[3] G. D’Ambra, M. Gromov, Lectures on transformations groups : geometry and dynamics, Surveys in Differential Geometry (Cambridge), (1990), 19-111. | MR 1144526 | Zbl 0752.57017

[4] M. Babillot, R. Feres, A. Zeghib, Rigidité, Groupe fondamental et Dynamique, (P. Foulon Ed.), Panoramas et Synthèses, 13, (2002). | MR 1993147

[5] Y. Benoist , Orbites des structures rigides (d’après M. Gromov), Feuilletages et systèmes intégrables (Montpellier, 1995), Birkhäuser Boston, (1997), 1-17. | MR 1432904 | Zbl 0880.58031

[6] Y. Benoist, Actions propres sur les espaces homogènes réductifs, Annals of Mathematics, 144, (1996), 315-347. | MR 1418901 | Zbl 0868.22013

[7] Y. Benoist, P. Foulon, F. Labourie, Flots d’Anosov à distributions stables et instables différentiables, Jour. Amer. Math. Soc., 5, (1992), 33-74. | MR 1124979 | Zbl 0759.58035

[8] Y. Benoist, F. Labourie, Sur les espaces homogènes modèles de variétés compactes, Publ. Math. I.H.E.S., 76, (1992), 99-109. | Numdam | MR 1215593 | Zbl 0786.53031

[9] E. Calabi, L. Markus, Relativistic space forms, Ann. of Math., 75, (1962), 63-76. | MR 133789 | Zbl 0101.21804

[10] A. Candel, R. Quiroga-Barranco, Gromov’s centralizer theorem, Geom. Dedicata 100, (2003), 123-155. | MR 2011119 | Zbl 1049.53028

[11] Y. Carrière, Flots riemanniens, dans Structures transverses des feuilletages, Toulouse, Astérisque, 116, (1984), 31-52. | MR 755161 | Zbl 0548.58033

[12] Y. Carrière, Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., 95, (1989), 615-628. | MR 979369 | Zbl 0682.53051

[13] S. Dumitrescu, Dynamique du pseudo-groupe des isométries locales sur une variété lorentzienne analytique de dimension 3, Ergodic Th. Dyn. Systems, 28(4), (2008), 1091-1116. | MR 2437221 | Zbl 1151.53350

[14] S. Dumitrescu, A. Zeghib, Géométries Lorentziennes de dimension 3 : classification et complétude, Geometriæ Dedicata, 149, (2010), 243-273. | MR 2737692 | Zbl pre05818371

[15] C. Ehresmann, Sur les espaces localement homogènes, Enseignement Math., p. 317, (1936).

[16] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles, (1950). | MR 61454 | Zbl 0054.07201

[17] R. Feres, Rigid geometric structures and actions of semisimple Lie groups, Rigidité, groupe fondamental et dynamique, Panorama et synthèses, 13, Soc. Math. France, Paris, (2002). | MR 1993149 | Zbl 1058.53037

[18] J. Ferrand, The action of conformal transformations on a Riemannian manifold, Math. Ann., 304, (1996), 277-291. | MR 1371767 | Zbl 0866.53027

[19] C. Frances, Un théorème de Ferrand-Obata pour les géométries paraboliques de rang un, Ann. Sci. Ens., 40(5),(2007), 741-764. | Numdam | Zbl 1135.53016

[20] C. Frances, Rigidity at the boundary for conformal structures and other Cartan geometries, arXiv.

[21] D. Fried, W. Goldman, Three-dimensional affine crystallographic groups, Adv. Math., 47(1), (1983), 1-49. | MR 689763 | Zbl 0571.57030

[22] P. Friedbert, F. Tricerri, L. Vanhecke, Curvature invariants, differential operators and local homogeneity, Trans. Amer. Math. Soc., 348, (1996), 4643-4652. | MR 1363946 | Zbl 0867.53032

[23] E. Ghys, Flots d’Anosov dont les feuilletages stables sont différentiables, Ann. Sci. Ec. Norm. Sup., 20, (1987), 251-270. | Numdam | MR 911758 | Zbl 0663.58025

[24] W. Goldman, Nonstandard Lorentz space forms, J. Differential Geom.,21(2), (1985), 301-308. | MR 816674 | Zbl 0591.53051

[25] W. Goldman, Y. Kamishima, The fundamental group of a compact flat Lorentz space form is virtually polycyclic, J. Differential Geom., 19(1), (1984), 233-240. | MR 739789 | Zbl 0546.53039

[26] M. Gromov, Rigid transformation groups, Géométrie Différentielle, (D. Bernard et Choquet-Bruhat Ed.), Travaux en cours, Hermann, Paris, 33, (1988), 65-141. | MR 955852 | Zbl 0652.53023

[27] M. Guediri, Sur la complétude des pseudo-métriques invariantes à gauche sur les groupes de Lie nilpotents, Rend. Sem. Mat. Univ. Politec. Torino, 52(4), (1994), 371-376. | MR 1345607 | Zbl 0838.53035

[28] M. Guediri, J. Lafontaine, Sur la complétude des variétés pseudo-riemanniennes, J. Geom. Phys., 15(2), (1995), 150-158. | MR 1310948 | Zbl 0818.53083

[29] M. Guediri, On completeness of left-invariant Lorentz metrics on solvable Lie groups, Rev. Mat. Univ. Complut. Madrid, 9(2), (1996), 337-350. | MR 1430782 | Zbl 0878.53048

[30] O. Kowalski, Counter-example to the second Singer’s theorem, Ann. Global Anal. Geom., 8(2), (1990), 211-214. | Zbl 0736.53047

[31] R. Kulkarni & F. Raymond, 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom., 21(2), (1985), 231-268. | MR 816671 | Zbl 0563.57004

[32] B. Klingler, Complétude des variétés Lorentziennes à courbure sectionnelle constante, Math. Ann., 306, (1996), 353-370. | MR 1411352 | Zbl 0862.53048

[33] S. Kobayashi, Transformation groupes in differential geometry, Springer-Verlag, (1972). | MR 355886 | Zbl 0829.53023

[34] T. Kobayashi, T. Yoshino, Compact Clifford-Klein form of symmetric spaces -revisited, Pure Appl. Math. Q., 1(3), (2005), 591-663. | MR 2201328 | Zbl 1145.22011

[35] A. Koutras, C. McIntosh, A metric with no symmetries or invariants, Class. Quant. Grav., 13(5), (1996), 47-49. | MR 1390083 | Zbl 0851.53063

[36] F. Labourie, Quelques résultats récents sur les espaces localement homogènes compacts, Symposia Mathematica (en l’honneur d’Eugenio Calabi), (1996), 267-283. | MR 1410076 | Zbl 0861.53053

[37] F. Lastaria, F. Tricceri, Curvature-orbits and locally homogeneous Riemannian manifolds, Ann. Mat. Pura Appl., 165(4), (1993), 121-131. | MR 1271415 | Zbl 0804.53072

[38] V. Matveev, Proof of the projective Lichnerowicz-Obata conjecture, J. Diff. Geom., 75(3), (2007), 459-502. | MR 2301453 | Zbl 1115.53029

[39] G. Mess, Lorentz spacetimes of constant curvature, preprint IHES/M/90/28, (1990). | MR 2328921

[40] J. Milnor, Curvatures of Left Invariant Metrics on Lie Groups, Adv. in Math., 21, (1976), 293-329. | MR 425012 | Zbl 0341.53030

[41] P. Molino, Riemannian Foliations, Birkhauser, (1988). | MR 932463 | Zbl 0633.53001

[42] M. Morrill, Nonexistence of compact de Sitter manifolds, PHD, University of California, (1996).

[43] G. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math., 52(2), (1950), 606-636. | MR 48464 | Zbl 0040.15204

[44] K. Nomizu, On local and global existence of Killing vector fields, Ann. of Math. (2), 72, (1960), 105-120. | MR 119172 | Zbl 0093.35103

[45] M. Obata, The conjecture on conformal transformations on riemannian manifolds, J. Diff. Geom., 6, (1971), 247-258. | MR 303464 | Zbl 0236.53042

[46] P. Olver, Equivalence, invariants and symmetry, Cambridge Univ. Press, (1995). | MR 1337276 | Zbl 1156.58002

[47] V. Patrangenaru, Locally homogeneous pseudo-Riemannian manifolds, J. Geom. Phys., 17, (1995), 59-72. | MR 1348747 | Zbl 0832.53017

[48] M. Raghunathan, Discrete subgroups of Lie groups, Springer, (1972). | MR 507234 | Zbl 0254.22005

[49] S. Rahmani, Métriques de Lorentz sur les groupes de Lie unimodulaires de dimension 3, J. Geom. Phys., 9, (1992), 295-302. | MR 1171140 | Zbl 0752.53036

[50] N. Rahmani, S. Rahmani, Lorentzian geometry of the Heisenberg group, Geom. Dedicata, 118, (2006), 133-140. | MR 2239452 | Zbl 1094.53065

[51] F. Salein, Variétés anti-de Sitter de dimension 3 exotiques, Ann. Inst. Fourier, Grenoble, 50(1), (2000), 257-284. | Numdam | MR 1762345 | Zbl 0951.53047

[52] R. Schoen, On the conformal and CR automorphism groups, Geom. Funct. Anal., 5(2), (1995), 464-481. | MR 1334876 | Zbl 0835.53015

[53] P. Scott, The Geometries of 3-manifolds, Bull. London Math. Soc., 15, (1983), 401-487. | MR 705527 | Zbl 0561.57001

[54] R. Sharpe, Differential Geometry, Cartan’s Generalization of Klein’s Erlangen Program, Springer, (1997). | MR 1453120 | Zbl 0876.53001

[55] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. of Amer. Math. Soc., 6(3), (1982), 357-381. | MR 648524 | Zbl 0496.57005

[56] W. Thurston, The geometry and topology of 3-manifolds, Princeton University Press, (1983).

[57] J. Wolf, Spaces of constant curvature, McGraw-Hill Series in Higher Math., (1967). | MR 217740 | Zbl 0162.53304

[58] A. Zeghib, Killing fields in compact Lorentz 3-manifolds, J. Differential Geom., 43, (1996), 859-894. | MR 1412688 | Zbl 0877.53048

[59] R. Zimmer, Ergodic theory and semisimple groups, Birkhäuser, (1984). | MR 776417 | Zbl 0571.58015