Sur la complétude de certaines variétés pseudo-riemanniennes localement symétriques

Tholozan, Nicolas

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1921-1952.

Nous prouvons que certains espaces pseudo-riemanniens symétriques n’admettent pas d’ouvert strict divisible par l’action d’un groupe discret d’isométries. Autrement dit, si une variété pseudo-riemannienne compacte est localement isométrique à un tel espace, et si son application développante est injective, alors la variété est géodésiquement complète, et donc isométrique à un quotient de l’espace modèle tout entier. Ces résultats étendent, sous une hypothèse supplémentaire (l’injectivité de l’application développante), les théorèmes de Carrière et Klingler selon lesquels les variétés lorentziennes compactes de courbure constante sont géodésiquement complètes.

We prove that certain pseudo-Riemannian symmetric spaces do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed pseudo-Riemannian manifold is locally isometric to such a model, and if its developing map is injective, then the manifold is actually geodesically complete, and therefore isometric to a quotient of the whole model space. Those results extend, under an additional assumption (the injectivity of the developing map), the theorems of Carrière and Klingler stating that closed Lorentz manifolds of constant curvature are geodesically complete.

Classification : 53C50, 53C35, 22E40
Mot clés : Variété pseudo-riemannienne, $(G,X)$–structure, action proprement discontinue
Keywords: Pseudo-Riemannian manifold, $(G,X)$–structure, properly discontinuous action

[1] Benoist, Yves Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., Volume 164 (2006) no. 2, pp. 249-278 | DOI | Zbl

[2] Borel, Armand Compact Clifford-Klein forms of symmetric spaces, Topology, Volume 2 (1963), pp. 111-122 | Zbl

[3] Borel, Armand; Tits, Jacques Groupes réductifs, Publ. Math. Inst. Hautes Études Sci., Volume 27 (1965), pp. 55-150 | Numdam | Zbl

[4] Carrière, Yves Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., Volume 95 (1989) no. 3, pp. 615-628 | DOI | Zbl

[5] Dumitrescu, Sorin; Zeghib, Abdelghani Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds, Math. Ann., Volume 345 (2009) no. 1, pp. 53-81 | DOI | Zbl

[6] Ehresmann, Charles Sur les Espaces localement homogènes, Enseign. Math., Volume 35 (1936), pp. 317-333 | Zbl

[7] Frances, Charles Sur le groupe d’automorphismes des géométries paraboliques de rang 1, Ann. Sci. École Norm. Sup. (4), Volume 40 (2007) no. 5, pp. 741-764 | DOI | Numdam | Zbl

[8] Fried, David; Goldman, William; Hirsch, Morris W. Affine manifolds with nilpotent holonomy, Comment. Math. Helv., Volume 56 (1981) no. 4, pp. 487-523 | DOI | Zbl

[9] Gallo, Daniel; Kapovich, Michael; Marden, Albert The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 625-704 | DOI | Zbl

[10] Ghys, Étienne Déformations des structures complexes sur les espaces homogènes de $\mathrm{SL}(2,ℂ)$, J. Reine Angew. Math., Volume 468 (1995), pp. 113-138 | DOI | Zbl

[11] Goldman, William; Hirsch, Morris W. The radiance obstruction and parallel forms on affine manifolds, Trans. Amer. Math. Soc., Volume 286 (1984) no. 2, pp. 629-649 | DOI | Zbl

[12] Goldman, William M. Nonstandard Lorentz space forms, J. Differential Geom., Volume 21 (1985) no. 2, pp. 301-308 http://projecteuclid.org/getRecord?id=euclid.jdg/1214439567 | Zbl

[13] Goldman, William M. Locally homogeneous geometric manifolds, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi (2010), pp. 717-744 | Zbl

[14] Gromov, Michael Rigid transformation groups, Géométrie différentielle (Paris, 1986) (Travaux en Cours), Volume 33, Hermann, Paris, 1988, pp. 65-139 | Zbl

[15] Guediri, Mohammed; Lafontaine, Jacques Sur la complétude des variétés pseudo-riemanniennes, J. Geom. Phys., Volume 15 (1995) no. 2, pp. 150-158 | DOI | Zbl

[16] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna Anosov representations and proper actions (http://arxiv.org/abs/1502.03811)

[17] Guéritaud, François; Kassel, Fanny Maximally stretched laminations on geometrically finite hyperbolic manifolds (http://arxiv.org/abs/1307.0250, à paraître à Geometry & Topology)

[18] Jo, Kyeonghee; Kim, Inkang Convex affine domains and Markus conjecture, Math. Z., Volume 248 (2004) no. 1, pp. 173-182 | DOI | Zbl

[19] Kassel, Fanny Proper actions on corank-one reductive homogeneous spaces, J. Lie Theory, Volume 18 (2008) no. 4, pp. 961-978 | Zbl

[20] Kassel, Fanny Quotients compacts d’espaces homogènes réels ou $p$-adiques, Université de Paris-Sud 11 (2009) (Ph. D. Thesis)

[21] Klingler, Bruno Complétude des variétés lorentziennes à courbure constante, Math. Ann., Volume 306 (1996) no. 2, pp. 353-370 | DOI | Zbl

[22] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser Boston Inc., Boston, MA, 2002, pp. xviii+812 | Zbl

[23] Kobayashi, Toshiyuki On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys., Volume 12 (1993) no. 2, pp. 133-144 | DOI | Zbl

[24] Kobayashi, Toshiyuki Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann., Volume 310 (1998), pp. 394-408 | Zbl

[25] Kulkarni, Ravi S.; Pinkall, Ulrich Uniformization of geometric structures with applications to conformal geometry, Differential geometry, Peñíscola 1985 (Lecture Notes in Math.), Volume 1209, Springer, Berlin, 1986, pp. 190-209 | DOI | Zbl

[26] Kulkarni, Ravi S.; Raymond, Frank $3$-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom., Volume 21 (1985) no. 2, pp. 231-268 http://projecteuclid.org/getRecord?id=euclid.jdg/1214439564 | Zbl

[27] Markus, Lawrence Cosmological models in differential geometry, University of Minnesota Press, 1963

[28] Salein, François Variétés anti-de Sitter de dimension 3 exotiques, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 1, pp. 257-284 | Numdam | Zbl

[29] Smillie, John David Affinely flat manifolds, ProQuest LLC, Ann Arbor, MI, 1977 (PhD Thesis–University of Chicago)

[30] Thurston, William The Geometry and topology of 3-manifolds, Princeton University Press, 1980

[31] Wolf, Joseph A. Spaces of constant curvature, Publish or Perish Inc., Boston, Mass., 1974, pp. xv+408 | Zbl

[32] Zeghib, Abdelghani On closed anti-de Sitter spacetimes, Math. Ann., Volume 310 (1998) no. 4, pp. 695-716 | Zbl