[Feuilletages transversalement homogènes non unimodulaires]
En calculant sa cohomologie basique, on donne des conditions suffisantes pour qu’un feuilletage transversalement homogène defini sur une variété compacte soit minimalisable. Comme application, on démontre que si le feuilletage est non unimodulaire alors soit la variété ambiante, soit l’adhérence des feuilles, soit un fibré principal associé au feuilletage, fibrent sur .
We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over .
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Keywords: Transversely homogeneous foliation, Lie foliation, base-like cohomology, unimodular foliation
Mot clés : Feuilletage transversalement homogène, feuilletage de Lie, cohomologie basique, feuilletage unimodulaire
Macías-Virgós, Enrique 1 ; Martín-Méndez, Pedro L. 1
CC-BY-ND 4.0
@article{AIF_2021__71_2_849_0,
author = {Mac{\'\i}as-Virg\'os, Enrique and Mart{\'\i}n-M\'endez, Pedro L.},
title = {Non-unimodular transversely homogeneous foliations},
journal = {Annales de l'Institut Fourier},
pages = {849--887},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {71},
number = {2},
year = {2021},
doi = {10.5802/aif.3412},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3412/}
}
TY - JOUR AU - Macías-Virgós, Enrique AU - Martín-Méndez, Pedro L. TI - Non-unimodular transversely homogeneous foliations JO - Annales de l'Institut Fourier PY - 2021 SP - 849 EP - 887 VL - 71 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3412/ DO - 10.5802/aif.3412 LA - en ID - AIF_2021__71_2_849_0 ER -
%0 Journal Article %A Macías-Virgós, Enrique %A Martín-Méndez, Pedro L. %T Non-unimodular transversely homogeneous foliations %J Annales de l'Institut Fourier %D 2021 %P 849-887 %V 71 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3412/ %R 10.5802/aif.3412 %G en %F AIF_2021__71_2_849_0
Macías-Virgós, Enrique; Martín-Méndez, Pedro L. Non-unimodular transversely homogeneous foliations. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 849-887. doi : 10.5802/aif.3412. https://aif.centre-mersenne.org/articles/10.5802/aif.3412/
[1] Secondary characteristic classes of transversely homogeneous foliations, CRM Preprint Series (2012) no. 1103, pp. 1-57
[2] Transversely Homogeneous Foliations, Ph. D. Thesis, Washington University in St. Louis (USA) (1978) | MR
[3] Transversely homogeneous foliations, Ann. Inst. Fourier, Volume 29 (1979) no. 4, pp. 143-158 | DOI | Numdam | MR | Zbl
[4] The base-like cohomology of a class of transversely homogeneous foliations, Bull. Sci. Math., Volume 104 (1980) no. 3, pp. 301-303 | MR | Zbl
[5] Flots riemanniens, Transversal structure of foliations (Toulouse, 1982) (Astérisque), Société Mathématique de France, 1984 no. 116, pp. 31-52 | Numdam | MR | Zbl
[6] Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, 9, North-Holland; Elsevier, 1975, viii+174 pages
[7] On deformations of transversely homogeneous foliations, Topology, Volume 40 (2001) no. 6, pp. 1363-1393 | DOI | MR | Zbl
[8] Structures géométriques invariantes et feuilletages de Lie, Indag. Math., New Ser., Volume 1 (1990) no. 3, pp. 323-333 | DOI | Zbl
[9] Analysis on Lie groups. An introduction, Cambridge Studies in Advanced Mathematics, 110, Cambridge University Press, 2008, x+302 pages | DOI | MR
[10] Feuilletages du plan, feuilletages de Lie (France), Ph. D. Thesis, Université Louis Pasteur (1973)
[11] Connections, curvature, and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, Academic Press Inc., 1976, xxi+593 pages
[12] A duality theorem for cohomology of Lie algebras, Math. USSR, Sb., Volume 12 (1970), pp. 638-644 | DOI | Zbl
[13] Differential geometry and symmetric spaces, Pure and Applied Mathematics, 12, Academic Press Inc., 1962, xiv+486 pages
[14] A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Am. Math. Soc., Volume 11 (1960), pp. 236-242 | DOI | MR | Zbl
[15] Lie groups, Lie algebras, and cohomology, Mathematical Notes, 34, Princeton University Press, 1988, xii+510 pages | MR
[16] Homotopy groups in Lie foliations, Trans. Am. Math. Soc., Volume 344 (1994) no. 2, pp. 701-711 | DOI | MR | Zbl
[17] Non-unimodular Lie foliations, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 5, pp. 359-362 | DOI | MR | Zbl
[18] Duality and minimality in Riemannian foliations, Comment. Math. Helv., Volume 67 (1992) no. 1, pp. 17-27 | DOI | MR | Zbl
[19] Riemannian foliations, Progress in Mathematics, 73, Birkhäuser, 1988, xii+339 pages (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | DOI
[20] Harmonic integrals on foliated manifolds, Am. J. Math., Volume 81 (1959), pp. 529-536 | DOI | MR | Zbl
[21] Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys., Volume 16 (2009) no. 3, pp. 450-466 | DOI | MR | Zbl
[22] Group theory, Dover Publications, 1987, xiv+479 pages | MR
[23] On fibering certain foliated manifolds over , Topology, Volume 9 (1970), pp. 153-154 | DOI | MR | Zbl
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