The diffeomorphism group of a Lie foliation

Gilbert Hector[1]; Enrique Macías-Virgós[2]; Antonio Sotelo-Armesto[2]

  • [1] Université Claude Bernard Lyon 1 Institut Camille Jordan 69622 Villeurbanne (France)
  • [2] Universidade de Santiago de Compostela Facultade de Matemáticas Dpto. Xeometria e Topoloxia 15782-Santiago de Compostela (Spain)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 1, page 365-378
  • ISSN: 0373-0956

Abstract

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We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus T n , n ≥ 2 . We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are ± Id and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for T 2 , P. Iglesias and G. Lachaud for codimension one foliations on T n , n ≥ 2 , and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau’s diffeological spaces.

How to cite

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Hector, Gilbert, Macías-Virgós, Enrique, and Sotelo-Armesto, Antonio. "The diffeomorphism group of a Lie foliation." Annales de l’institut Fourier 61.1 (2011): 365-378. <http://eudml.org/doc/219837>.

@article{Hector2011,
abstract = {We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus $T^n$, $n\ge 2$. We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are $\pm \mathrm\{Id\}$ and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and G. Lachaud for codimension one foliations on $T^n$, $n\ge 2$, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau’s diffeological spaces.},
affiliation = {Université Claude Bernard Lyon 1 Institut Camille Jordan 69622 Villeurbanne (France); Universidade de Santiago de Compostela Facultade de Matemáticas Dpto. Xeometria e Topoloxia 15782-Santiago de Compostela (Spain); Universidade de Santiago de Compostela Facultade de Matemáticas Dpto. Xeometria e Topoloxia 15782-Santiago de Compostela (Spain)},
author = {Hector, Gilbert, Macías-Virgós, Enrique, Sotelo-Armesto, Antonio},
journal = {Annales de l’institut Fourier},
keywords = {Diffeological space; diffeomorphism group; Lie foliation; linear flow; diffeological space},
language = {eng},
number = {1},
pages = {365-378},
publisher = {Association des Annales de l’institut Fourier},
title = {The diffeomorphism group of a Lie foliation},
url = {http://eudml.org/doc/219837},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Hector, Gilbert
AU - Macías-Virgós, Enrique
AU - Sotelo-Armesto, Antonio
TI - The diffeomorphism group of a Lie foliation
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 1
SP - 365
EP - 378
AB - We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus $T^n$, $n\ge 2$. We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are $\pm \mathrm{Id}$ and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and G. Lachaud for codimension one foliations on $T^n$, $n\ge 2$, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau’s diffeological spaces.
LA - eng
KW - Diffeological space; diffeomorphism group; Lie foliation; linear flow; diffeological space
UR - http://eudml.org/doc/219837
ER -

References

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  2. P. Donato, P. Iglesias, Exemples de groupes difféologiques: flots irrationels sur le tore, C. R. Acad. Sci., Paris, Sér. I 301 (1985), 127-130 Zbl0596.58010MR799609
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  7. G. Hector, E. Macías, Diffeological groups, Recent advances in Lie theory. Selected contributions to the 1st colloquium on Lie theory and applications, Vigo, Spain, July 17-22, 2000 25 (2002), 247-260, Lemgo: Heldermann Verlag Zbl1018.58001MR1937968
  8. B. Herrera, M. Llabrés, A. Reventós, Transverse structure of Lie foliations, J. Math. Soc. Japan 48 (1996), 769-795 Zbl0881.57025MR1404823
  9. B. Herrera Gómez, Sobre la estructura transversa de las foliaciones de Lie, (1994) MR2715023
  10. P. Iglesias, Fibrations difféologiques et homotopie, (1985) 
  11. P. Iglesias, G. Lachaud, Espaces différentiables singuliers et corps de nombres algébriques, Ann. Inst. Fourier Grenoble 40 (1990), 723-737 Zbl0703.57017MR1091840
  12. P. Iglesias-Zemmour, Diffeology, (2009) 
  13. E. Macías-Virgós, Homotopy groups in Lie foliations, Trans. Am. Math. Soc. 344 (1994), 701-711 Zbl0818.57016MR1260205
  14. P. Molino, Riemannian foliations, (1988), Birkhauser Zbl0633.53001MR932463
  15. J.-M. Souriau, Groupes différentiels, Differential geometrical methods in mathematical physics (Proc. Conf. Aix-en-Provence Salamanca, 1979). Lecture Notes in Math. 836 (1980), 91-128, Springer Zbl0501.58010MR607688

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