[Variations sur le théorème de l’orbite dense-ouverte de M. Gromov]
We investigate under which conditions a geometric structure which is locally homogeneous on a dense open set is locally homogeneous everywhere. In the case of a 3-dimensional Lorentz metric, this allows us to sharpen the conclusions in Gromov’s open-dense orbit theorem.
Nous étudions sous quelles conditions une structure géométrique qui est localement homogène sur un ouvert dense est localement homogène partout. Dans la cadre des variétés lorentziennes de dimension 3, cela conduit à un renforcement des conclusions dans le théorème de l’orbite dense-ouverte de Gromov.
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DOI : 10.24033/bsmf.2773
Keywords: Differential geometry, rigid structures, pseudo-Riemannian geometry
Mots-clés : Géométrie différentielle, structures rigides, géométrie pseudo-Riemannienne
Frances, Charles 1
@article{BSMF_2018__146_4_713_0,
author = {Frances, Charles},
title = {Variations on {Gromov{\textquoteright}s} open-dense orbit theorem},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {713--744},
year = {2018},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {146},
number = {4},
doi = {10.24033/bsmf.2773},
mrnumber = {3936541},
zbl = {1421.53073},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2773/}
}
TY - JOUR AU - Frances, Charles TI - Variations on Gromov’s open-dense orbit theorem JO - Bulletin de la Société Mathématique de France PY - 2018 SP - 713 EP - 744 VL - 146 IS - 4 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2773/ DO - 10.24033/bsmf.2773 LA - en ID - BSMF_2018__146_4_713_0 ER -
%0 Journal Article %A Frances, Charles %T Variations on Gromov’s open-dense orbit theorem %J Bulletin de la Société Mathématique de France %D 2018 %P 713-744 %V 146 %N 4 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2773/ %R 10.24033/bsmf.2773 %G en %F BSMF_2018__146_4_713_0
Frances, Charles. Variations on Gromov’s open-dense orbit theorem. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 4, pp. 713-744. doi: 10.24033/bsmf.2773
[1] Y. Benoist, “Orbites des structures géométriques rigides (d’après M. Gromov)”, Progress in Math (1997), p. 1–17 | MR | Zbl
[2] E. J. Benveniste & D. Fisher, “Nonexistence of invariant rigid structures and invariant almost rigid structure”, Comm. Anal. Geom. 13 (2005), p. 89–111 | MR | Zbl
[3] Y. Benoist, P. Foulon & F. Labourie, “Flots d’Anosov à distributions stable et instable différentiables”, J. Amer. Math. Soc. 5 (1992), 1, p. 33–74 | MR | Zbl
[4] A. Čap & J. Slovák, “Parabolic geometries. I. Background and general theory”, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, 2009 | MR | Zbl
[5] S. Dumitrescu, An invitation to quasihomogeneous rigid geometric structures. Bridging algebra, geometry, and topology, p. 107–123 | MR
[6] S. Dumitrescu, “Dynamique du pseudo-groupe des isométries locales sur une variété lorentzienne analytique de dimension 3”, Ergodic Theory Dynam. Systems , 28 (2008), 4, p. 1091–1116 | MR | Zbl
[7] S. Dumitrescu & A. Guillot, “Quasihomogeneous analytic affine connections on surfaces”, J. Topol. Anal. 5 (2013), 4, p. 491–532 | MR | Zbl
[8] S. Dumitrescu & K. Melnick, “Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist”. Geom. Dedicata 179 (2015), p. 229–253 | MR
[9] S. Dumitrescu & A. Zeghib, “Géométries Lorentziennes de dimension 3: classification et complétude”, Geom. Dedicata 149 (2010), p. 243–273 | MR | Zbl
[10] G. d’Ambra & M. Gromov, “Lectures on transformations groups: geometry and dynamics”, Surveys in Differential Geometry (Cambridge), (1990), p. 19–111 | MR | Zbl
[11] D. Fisher, “Groups acting on manifolds: around the Zimmer program”, in Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, 2011, p. 72–157 | MR | Zbl
[12] R. Feres & A. Katok, “Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena)”, Handbook of dynamical systems, vol. 1A, North-Holland, 2002, p. 665–763 | MR | Zbl
[13] M. Gromov, “Rigid transformation groups, Geometrie Differentielle, (D. Bernard et Choquet-Bruhat Ed.), Travaux en cours, Hermann, 33, (1988), p. 65–141 | MR | Zbl
[14] A. Iozzi, “Invariant geometric structures: A non-linear extension of the Borel density theorem”, Am. Jour. Math., 114 (1992), p. 627–648 | MR | Zbl
[15] S. Kobayashi & K. Nomizu, “Foundations of differential geometry I”, Interscience Publishers, 1963 | MR
[16] K. Melnick, “A Frobenius theorem for Cartan geometries, with applications”, L’Enseignement Mathématique (Série II) 57 (2011), 1–2, p. 57–89 | MR | Zbl
[17] K. Nomizu, “On local and global existence of Killing fields”, Ann. of Math. 72 (1960), 2, p. 105–112 | MR | Zbl
[18] V. Pécastaing, “On two theorems about local automorphisms of geometric structures”. arXiv:1402.5048 [math.DG]. To appear in Annales de l’institut Fourier | Numdam | MR
[19] R. W. Sharpe, Differential Geometry: Cartan’s generalization of Klein’s Erlangen Program, Springer, 1997 | MR | Zbl
[20] I. Singer, “Infinitesimally homogeneous spaces”, Comm. Pure Appl. Math. 13 (1960), p. 685–697 | MR | Zbl
[21] A. Zeghib, “Killing fields in compact Lorentz 3-manifolds”, J. Differential Geom., 43, (1996), p. 859–894 | MR | Zbl
[22] A. Zeghib, “On Gromovs theory of rigid transformation groups: a dual approach”, Ergodic Theory Dynam. Systems 20 (2000), 3, p. 935–946 | MR | Zbl
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