[Sur deux théorèmes portant sur les automorphismes locaux des structures géométriques]
Cet article s’intéresse à des questions autour des orbites des automorphismes locaux de variétés munies de structures géométriques rigides. Nous formulons des conditions suffisantes assurant l’homogénéité locale d’un large spectre de structures géométriques rigides, les géométries de Cartan, étendant ainsi un résultat de Singer sur les variétés riemanniennes localement homogènes. Nous revisitons également un résultat très général de Gromov qui décrit l’agencement des orbites des automorphismes locaux des variétés munies de -structures rigides. Nous donnons un énoncé et une preuve élémentaire de ce résultat dans le cadre des géométries de Cartan.
This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemannian manifolds. We also revisit a strong result of Gromov which describes the structure of the orbits of local automorphisms of manifolds endowed with -rigid structures, and give a statement and a simpler proof of this result in the setting of Cartan geometries.
Accepté le :
Publié le :
DOI : 10.5802/aif.3009
Keywords: Cartan geometries, local homogeneity, orbits of local automorphisms
Mot clés : Géométries de Cartan, homogénéité locale, orbites des automorphismes locaux
@article{AIF_2016__66_1_175_0,
author = {Pecastaing, Vincent},
title = {On two theorems about local automorphisms of geometric structures},
journal = {Annales de l'Institut Fourier},
pages = {175--208},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {66},
number = {1},
year = {2016},
doi = {10.5802/aif.3009},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.3009/}
}
TY - JOUR AU - Pecastaing, Vincent TI - On two theorems about local automorphisms of geometric structures JO - Annales de l'Institut Fourier PY - 2016 SP - 175 EP - 208 VL - 66 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3009/ DO - 10.5802/aif.3009 LA - en ID - AIF_2016__66_1_175_0 ER -
%0 Journal Article %A Pecastaing, Vincent %T On two theorems about local automorphisms of geometric structures %J Annales de l'Institut Fourier %D 2016 %P 175-208 %V 66 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3009/ %R 10.5802/aif.3009 %G en %F AIF_2016__66_1_175_0
Pecastaing, Vincent. On two theorems about local automorphisms of geometric structures. Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 175-208. doi : 10.5802/aif.3009. http://archive.numdam.org/articles/10.5802/aif.3009/
[1] Orbites des structures rigides (d’après M. Gromov), Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995) (Progr. Math.), Volume 145, Birkhäuser Boston, Boston, MA, 1997, pp. 1-17 | DOI
[2] Parabolic geometries and canonical Cartan connections, Hokkaido Math. J., Volume 29 (2000) no. 3, pp. 453-505 | DOI
[3] Parabolic geometries. I, Mathematical Surveys and Monographs, 154, American Mathematical Society, Providence, RI, 2009, x+628 pages (Background and general theory) | DOI
[4] Lectures on transformation groups: geometry and dynamics, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 19-111
[5] Rigid geometric structures and actions of semisimple Lie groups, Rigidité, groupe fondamental et dynamique (Panor. Synthèses), Volume 13, Soc. Math. France, Paris, 2002, pp. 121-167
[6] Homothety curvature homogeneity (2013) (http://arxiv.org/abs/1309.5332)
[7] Rigid transformations groups, Géométrie différentielle (Paris, 1986) (Travaux en Cours), Volume 33, Hermann, Paris, 1988, pp. 65-139
[8] Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, xiv+247 pages (Graduate Texts in Mathematics, No. 21)
[9] A Frobenius theorem for Cartan geometries, with applications, Enseign. Math. (2), Volume 57 (2011) no. 1-2, pp. 57-89 | DOI
[10] On local and global existence of Killing vector fields, Ann. of Math. (2), Volume 72 (1960), pp. 105-120
[11] Affine versions of Singer’s theorem on locally homogeneous spaces, Ann. Global Anal. Geom., Volume 15 (1997) no. 2, pp. 187-199 | DOI
[12] On locally homogeneous -structures, Geom. Dedicata, Volume 73 (1998) no. 2, pp. 215-223 | DOI
[13] A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No., Volume 22 (1957), iii+123 pages
[14] Introduzione ai Gruppi di Transformazione, Volume of the Preprint Series of the Mathematics Department “V. Voleterra” of the University of Ancona, Via delle Brecce Bianche, Ancona, ITALY, 1996
[15] Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Foreword by S. S. Chern., Berlin: Springer, 1997, xix + 421 pages
[16] Infinitesimally homogeneous spaces, Comm. Pure Appl. Math., Volume 13 (1960), pp. 685-697
[17] On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.), Volume 2 (1976) no. 1, pp. 131-190
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