Lorentzian 3-manifolds and dynamical systems
Charles Frances1
1 IRMA 7 rue René Descartes 67000 Strasbourg (France)
Séminaire de théorie spectrale et géométrie, Tome 34 (2016-2017), pp. 1-32.

The following article aims at presenting classical aspects of dynamical systems preserving a geometric structure, focusing on 3-dimensional Lorentzian dynamics. The reader won’t find here any new result, but rather an expository approach of classical ones. Our main goal, in particular, is to introduce part of the techniques and arguments used in [7] to obtain the classification of closed 3-dimensional Lorentzian manifolds admitting a non-compact isometry group. Doing so, we will present a self-contained, and somehow shortened proof of Zeghib’s classification [24] of non-equicontinuous Lorentzian isometric flows in dimension 3.

DOI : 10.5802/tsg.353
Charles Frances 1

1 IRMA 7 rue René Descartes 67000 Strasbourg (France) 
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Charles Frances. Lorentzian $3$-manifolds and dynamical systems. Séminaire de théorie spectrale et géométrie, Tome 34 (2016-2017), pp. 1-32. doi : 10.5802/tsg.353. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.353/

[1] Yves Benoist; Patrick Foulon; François Labourie Flots d’Anosov à distributions stable et instable différentiables, J. Am. Math. Soc., Volume 5 (1992) no. 1, pp. 33-74 | DOI | MR | Zbl

[2] Yves Benoist; François Labourie Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables, Invent. Math., Volume 111 (1993) no. 2, pp. 285-308 | DOI | MR | Zbl

[3] Andreas Čap; Jan Slovák Parabolic geometries. I Background and general theory, Mathematical Surveys and Monographs, 154, American Mathematical Society, 2009, x+628 pages | DOI | MR | Zbl

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

[5] Giuseppina D’Ambra; Mikhael Gromov Lectures on transformation groups: geometry and dynamics, Surveys in differential geometry (Cambridge, MA, 1990), American Mathematical Society; Lehigh University, 1991, pp. 19-111 | MR | Zbl

[6] Sorin Dumitrescu; Abdelghani Zeghib Géométries lorentziennes de dimension 3: classification et complétude, Geom. Dedicata, Volume 149 (2010), pp. 243-273 | DOI | MR | Zbl

[7] Charles Frances Lorentz dynamics on closed 3-dimensional manifolds (2016) (https://arxiv.org/abs/1605.05755)

[8] Charles Frances Variations on Gromov’s open-dense orbit theorem, Bull. Soc. Math. Fr., Volume 146 (2018) no. 4, pp. 713-744 | DOI | MR | Zbl

[9] David Fried; William M. Goldman Three-dimensional affine crystallographic groups, Adv. Math., Volume 47 (1983) no. 1, pp. 1-49 | DOI | MR | Zbl

[10] Étienne Ghys Flots d’Anosov dont les feuilletages stables sont différentiables, Ann. Sci. Éc. Norm. Supér., Volume 20 (1987) no. 2, pp. 251-270 | DOI | MR | Zbl

[11] William M. Goldman; Yoshinobu Kamishima The fundamental group of a compact flat Lorentz space form is virtually polycyclic, J. Differ. Geom., Volume 19 (1984) no. 1, pp. 233-240 | MR | Zbl

[12] Mikhael Gromov Rigid transformations groups, Géométrie différentielle (Paris, 1986) (Travaux en Cours), Volume 33, Hermann, 1988, pp. 65-139 | MR | Zbl

[13] Masahiko Kanai Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dyn. Syst., Volume 8 (1988) no. 2, pp. 215-239 | DOI | MR | Zbl

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

[15] Shoshichi Kobayashi; Katsumi Nomizu Foundations of differential geometry. Vol I, Interscience Publishers, 1963, xi+329 pages | MR | Zbl

[16] Ravi S. Kulkarni; Frank Raymond 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differ. Geom., Volume 21 (1985) no. 2, pp. 231-268 | MR | Zbl

[17] Karin Melnick A Frobenius theorem for Cartan geometries, with applications, Enseign. Math., Volume 57 (2011) no. 1-2, pp. 57-89 | DOI | MR | Zbl

[18] Katsumi Nomizu On local and global existence of Killing vector fields, Ann. Math., Volume 72 (1960), pp. 105-120 | DOI | MR | Zbl

[19] Vincent Pecastaing On two theorems about local automorphisms of geometric structures, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 175-208 | DOI | MR | Zbl

[20] Francçois Salein Variétés anti-de Sitter de dimension 3, ENS Lyon (France) (1999) (Ph. D. Thesis)

[21] Richard W. Sharpe Differential geometry. Cartan’s generalization of Klein’s Erlangen program, Graduate Texts in Mathematics, 166, Springer, 1997, xx+421 pages | MR | Zbl

[22] Nicolas Tholozan Uniformisation des variétés pseudo-riemanniennes localement homogènes, Université de Nice Sophia-Antipolis (France) (2014) (Ph. D. Thesis)

[23] William P. Thurston Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997, x+311 pages | MR | Zbl

[24] Abdelghani Zeghib Killing fields in compact Lorentz 3-manifolds, J. Differ. Geom., Volume 43 (1996) no. 4, pp. 859-894 | DOI | MR | Zbl

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