On two theorems about local automorphisms of geometric structures
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 175-208.

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 A-rigid structures, and give a statement and a simpler proof of this result in the setting of Cartan geometries.

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 A-structures rigides. Nous donnons un énoncé et une preuve élémentaire de ce résultat dans le cadre des géométries de Cartan.

Published online:
DOI: 10.5802/aif.3009
Classification: 53A40,  53B15,  53C24
Keywords: Cartan geometries, local homogeneity, orbits of local automorphisms
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     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},
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     doi = {10.5802/aif.3009},
     language = {en},
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Pecastaing, Vincent. On two theorems about local automorphisms of geometric structures. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 175-208. doi : 10.5802/aif.3009. https://aif.centre-mersenne.org/articles/10.5802/aif.3009/

[1] Benoist, Yves Orbites des structures rigides (d’après M. Gromov), Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995) (Progr. Math.) Tome 145, Birkhäuser Boston, Boston, MA, 1997, pp. 1-17 | DOI

[2] Čap, Andreas; Schichl, Hermann Parabolic geometries and canonical Cartan connections, Hokkaido Math. J., Tome 29 (2000) no. 3, pp. 453-505 | DOI

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

[4] D’Ambra, G.; Gromov, M. Lectures on transformation groups: geometry and dynamics, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 19-111

[5] Feres, Renato Rigid geometric structures and actions of semisimple Lie groups, Rigidité, groupe fondamental et dynamique (Panor. Synthèses) Tome 13, Soc. Math. France, Paris, 2002, pp. 121-167

[6] García-Río, E.; Gilkey, P.; Nikcevic, S. Homothety curvature homogeneity (2013) (http://arxiv.org/abs/1309.5332)

[7] Gromov, Michael Rigid transformations groups, Géométrie différentielle (Paris, 1986) (Travaux en Cours) Tome 33, Hermann, Paris, 1988, pp. 65-139

[8] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, xiv+247 pages (Graduate Texts in Mathematics, No. 21)

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

[10] Nomizu, Katsumi On local and global existence of Killing vector fields, Ann. of Math. (2), Tome 72 (1960), pp. 105-120

[11] Opozda, Barbara Affine versions of Singer’s theorem on locally homogeneous spaces, Ann. Global Anal. Geom., Tome 15 (1997) no. 2, pp. 187-199 | DOI

[12] Opozda, Barbara On locally homogeneous G-structures, Geom. Dedicata, Tome 73 (1998) no. 2, pp. 215-223 | DOI

[13] Palais, Richard S. A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No., Tome 22 (1957), iii+123 pages

[14] Podestà, F; Spiro, A 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] Sharpe, R.W. Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Foreword by S. S. Chern., Berlin: Springer, 1997, xix + 421 pages

[16] Singer, I. M. Infinitesimally homogeneous spaces, Comm. Pure Appl. Math., Tome 13 (1960), pp. 685-697

[17] Tanaka, Noboru On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.), Tome 2 (1976) no. 1, pp. 131-190

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