An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝm+2), where G = ℝm+2 ⋊ (O(m+1, 1)×ℝ+). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.
@article{bwmeta1.element.doi-10_2478_s11533-012-0076-9,
author = {Yoshinobu Kamishima},
title = {Lorentzian similarity manifolds},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1771-1788},
zbl = {1267.53073},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0076-9}
}
Yoshinobu Kamishima. Lorentzian similarity manifolds. Open Mathematics, Tome 10 (2012) pp. 1771-1788. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0076-9/
[1] Aristide T., Closed similarity Lorentzian affine manifolds, Proc. Amer. Math. Soc., 2004, 132(12), 3697–3702 http://dx.doi.org/10.1090/S0002-9939-04-07560-4 | Zbl 1056.53017
[2] Barbot T., Charette V., Drumm T., Goldman W., Melnick K., A primer on the (2+1) Einstein universe, In: Recent Developments in Pseudo-Riemannian Geometry, ESI Lect. Math. Phys., European Mathematical Society, Zürich, 2008, 179–229 | Zbl 1154.53047
[3] Baues O., Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 2004, 43(4), 903–924 http://dx.doi.org/10.1016/S0040-9383(03)00083-1 | Zbl 1059.57022
[4] Carrière Y., Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., 1989, 95(3), 615–628 http://dx.doi.org/10.1007/BF01393894 | Zbl 0682.53051
[5] Carrière Y., Dal’bo F., Généralisations du premier théorème de Bieberbach sur les groupes cristallographiques, Enseign. Math., 1989, 35(3–4), 245–262
[6] Charette V., Drumm T.A., Goldman W.M., Affine deformations of a three-holed sphere, Geom. Topol., 2010, 14(3), 1355–1382 http://dx.doi.org/10.2140/gt.2010.14.1355 | Zbl 1202.57001
[7] Chen S.S., Greenberg L., Hyperbolic spaces, In: Contributions to Analysis, Academic Press, New York-London, 1974, 49–87
[8] Fefferman C., Parabolic invariant theory in complex analysis, Adv. in Math., 1979, 31(2), 131–262 http://dx.doi.org/10.1016/0001-8708(79)90025-2
[9] Fried D., Closed similarity manifolds, Comment. Math. Helv., 1980, 55(4), 576–582 http://dx.doi.org/10.1007/BF02566707 | Zbl 0455.57005
[10] Fried D., Flat spacetimes, J. Differential Geom., 1987, 26(3), 385–396 | Zbl 0643.53047
[11] Fried D., Goldman W.M., Three-dimensional affine crystallographic groups, Adv. in Math., 1983, 47(1), 1–49 http://dx.doi.org/10.1016/0001-8708(83)90053-1 | Zbl 0571.57030
[12] Goldman W.M., Kamishima Y., The fundamental group of a compact flat Lorentz space form is virtually polycyclic, J. Differential Geom., 1984, 19(1), 233–240 | Zbl 0546.53039
[13] Goldman W.M., Labourie F., Margulis G., Proper affine actions and geodesic flows of hyperbolic surfaces, Ann. of Math., 2009, 170(3), 1051–1083 http://dx.doi.org/10.4007/annals.2009.170.1051 | Zbl 1193.57001
[14] Grunewald F., Segal D., On affine crystallographic groups, J. Differential Geom., 1994, 40(3), 563–594 | Zbl 0822.20050
[15] Kamishima Y., Conformally flat manifolds whose development maps are not surjective. I, Trans. Amer. Math. Soc., 1986, 294(2), 607–623 http://dx.doi.org/10.1090/S0002-9947-1986-0825725-2 | Zbl 0608.53036
[16] Kamishima Y., Completeness of Lorentz manifolds of constant curvature admitting Killing vector fields, J. Differential Geom., 1993, 37(3), 569–601 | Zbl 0792.53061
[17] Kamishima Y., Nondegenerate conformal structures, CR structures and quaternionic CR structures on manifolds, In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theor. Phys., 16, European Mathematical Society, Zürich, 2010, 863–893 http://dx.doi.org/10.4171/079-1/24 | Zbl 1208.53039
[18] Kamishima Y., Conformally Lorentz parabolic structure and Fefferman Lorentz metrics, preprint available at http://arxiv.org/abs/0911.0867
[19] Koszul J.-L., Lectures on Groups of Transformations, Tata Inst. Fundam. Res. Lect. Math., 32, Tata Institute of Fundamental Research, Bombay, 1965
[20] Kulkarni R.S., Groups with domains of discontinuity, Math. Ann., 1978, 237(3), 253–272 http://dx.doi.org/10.1007/BF01420180 | Zbl 0369.20028
[21] Kulkarni R.S., Pinkall U., Uniformization of geometric structures with applications to conformal geometry, In: Differential Geomtery, Peñíscola, June 2–9, 1985, Lecture Notes in Math., 1209, Springer, Berlin, 1986, 190–209 http://dx.doi.org/10.1007/BFb0076632
[22] Lee K.B., Raymond F., Seifert Fiberings, Math. Surveys Monogr., 166, American Mathematical Society, Providence, 2010
[23] Mess G., Lorentz spacetimes of constant curvature, Geom. Dedicata, 2007, 126, 3–45 http://dx.doi.org/10.1007/s10711-007-9155-7
[24] Milnor J., On fundamental groups of complete affinely flat manifolds, Adv. in Math., 1977, 25(2), 178–187 http://dx.doi.org/10.1016/0001-8708(77)90004-4
[25] Raghunathan M.S., Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb., 68, Springer, New York-Heidelberg, 1972 | Zbl 0254.22005
[26] Wall C.T.C., Geometric structures on compact complex analytic surfaces, Topology, 1986, 25(2), 119–153 http://dx.doi.org/10.1016/0040-9383(86)90035-2