The following article aims at presenting classical aspects of dynamical systems preserving a geometric structure, focusing on -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 -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 .
@article{TSG_2016-2017__34__1_0, author = {Frances, Charles}, title = {Lorentzian $3$-manifolds and dynamical systems}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {1--32}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {34}, year = {2016-2017}, doi = {10.5802/tsg.353}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.353/} }
TY - JOUR AU - Frances, Charles TI - Lorentzian $3$-manifolds and dynamical systems JO - Séminaire de théorie spectrale et géométrie PY - 2016-2017 SP - 1 EP - 32 VL - 34 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.353/ DO - 10.5802/tsg.353 LA - en ID - TSG_2016-2017__34__1_0 ER -
Frances, Charles. Lorentzian $3$-manifolds and dynamical systems. Séminaire de théorie spectrale et géométrie, Tome 34 (2016-2017), pp. 1-32. doi : 10.5802/tsg.353. http://www.numdam.org/articles/10.5802/tsg.353/
[1] Flots d’Anosov à distributions stable et instable différentiables, J. Am. Math. Soc., Volume 5 (1992) no. 1, pp. 33-74 | DOI | MR
[2] Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables, Invent. Math., Volume 111 (1993) no. 2, pp. 285-308 | DOI | MR
[3] Parabolic geometries. I Background and general theory, Mathematical Surveys and Monographs, 154, American Mathematical Society, 2009, x+628 pages | DOI | MR
[4] Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., Volume 95 (1989) no. 3, pp. 615-628 | DOI | MR
[5] 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] Géométries lorentziennes de dimension 3: classification et complétude, Geom. Dedicata, Volume 149 (2010), pp. 243-273 | DOI | MR
[7] Lorentz dynamics on closed -dimensional manifolds (2016) (https://arxiv.org/abs/1605.05755)
[8] Variations on Gromov’s open-dense orbit theorem, Bull. Soc. Math. Fr., Volume 146 (2018) no. 4, pp. 713-744 | Zbl
[9] Three-dimensional affine crystallographic groups, Adv. Math., Volume 47 (1983) no. 1, pp. 1-49 | DOI | MR
[10] Flots d’Anosov dont les feuilletages stables sont différentiables, Ann. Sci. Éc. Norm. Supér., Volume 20 (1987) no. 2, pp. 251-270 | MR
[11] 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
[12] Rigid transformations groups, Géométrie différentielle (Paris, 1986) (Travaux en Cours), Volume 33, Hermann, 1988, pp. 65-139 | MR
[13] 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
[14] Complétude des variétés lorentziennes à courbure constante, Math. Ann., Volume 306 (1996) no. 2, pp. 353-370 | DOI | MR
[15] Foundations of differential geometry. Vol I, Interscience Publishers, 1963, xi+329 pages | MR | Zbl
[16] -dimensional Lorentz space-forms and Seifert fiber spaces, J. Differ. Geom., Volume 21 (1985) no. 2, pp. 231-268 | MR
[17] A Frobenius theorem for Cartan geometries, with applications, Enseign. Math., Volume 57 (2011) no. 1-2, pp. 57-89 | DOI | MR
[18] On local and global existence of Killing vector fields, Ann. Math., Volume 72 (1960), pp. 105-120 | DOI | MR
[19] On two theorems about local automorphisms of geometric structures, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 175-208 | MR
[20] Variétés anti-de Sitter de dimension 3, ENS Lyon (France) (1999) (Ph. D. Thesis http://www.umpa.ens-lyon.fr/~zeghib/these.salein.pdf)
[21] Differential geometry. Cartan’s generalization of Klein’s Erlangen program, Graduate Texts in Mathematics, 166, Springer, 1997, xx+421 pages | MR | Zbl
[22] Uniformisation des variétés pseudo-riemanniennes localement homogènes, Université de Nice Sophia-Antipolis (France) (2014) (Ph. D. Thesis https://tel.archives-ouvertes.fr/tel-01127086)
[23] Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997, x+311 pages | MR
[24] Killing fields in compact Lorentz -manifolds, J. Differ. Geom., Volume 43 (1996) no. 4, pp. 859-894 | MR
Cité par Sources :