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

Received : 2014-02-24
Accepted : 2014-09-03
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3009
Classification:  53A40,  53B15,  53C24
Keywords: Cartan geometries, local homogeneity, orbits of local automorphisms
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {175-208},
     doi = {10.5802/aif.3009},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_1_175_0}
}
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. http://www.numdam.org/item/AIF_2016__66_1_175_0/

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