Transversely homogeneous foliations
Blumenthal, Robert A.
Annales de l'Institut Fourier, Tome 29 (1979), p. 143-158 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Un feuilletage d’une variété s’appelle transversalement homogène s’il peut être défini par des submersions locales prenant leurs valeurs dans un espace homogène G/K telles que les changements des cartes sont des translations. Nous étudions la topologie et la géométrie de ces feuilletages et nous donnons un théorème de structure pour le cas où K est compact. Nous considérons la relation entre les équations de structure de G et l’espace fibré vectoriel transverse au feuilletage, et nous donnons une caractérisation au moyen des formes différentielles pour une grande classe des feuilletages homogènes. Enfin, nous étudions les feuilletages transversalement elliptiques, euclidiens, et hyperboliques.

A foliation of a manifold is transversely homogeneous if it can be defined by local submersions to a homogeneous space G/K which on overlaps differ by translations. We explore the topology and geometry of such foliations and give a structure theorem for the case when K is compact. We investigate the relationship between the structure equations of G and the normal bundle of the foliation and provide a differential forms characterization of a large class of homogeneous foliations. As a special case, we study the transversely elliptic, Euclidean, and hyperbolic foliations.

@article{AIF_1979__29_4_143_0,
     author = {Blumenthal, Robert A.},
     title = {Transversely homogeneous foliations},
     journal = {Annales de l'Institut Fourier},
     volume = {29},
     year = {1979},
     pages = {143-158},
     doi = {10.5802/aif.771},
     mrnumber = {81h:57011},
     zbl = {0405.57016},
     mrnumber = {558593},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1979__29_4_143_0}
}

Blumenthal, Robert A. Transversely homogeneous foliations. Annales de l'Institut Fourier, Tome 29 (1979) pp. 143-158. doi : 10.5802/aif.771. http://gdmltest.u-ga.fr/item/AIF_1979__29_4_143_0/

[1] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc., (3), 25 (1972), 603-614. | MR 52 #577 | Zbl 0259.20045

[2] G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. | MR 54 #1265 | Zbl 0246.57017

[3] A. Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comm. Math. Helv., 32 (1958), 248-329. | MR 20 #6702 | Zbl 0085.17303

[4] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience Tracts in Pure and Appl. Math., 15, Interscience, New York, 1963. | Zbl 0119.37502

[5] R. Palais, A global formulation of the Lie theory of transformation groups, Memoirs of the Amer. Math. Soc., 22 (1957). | MR 22 #12162 | Zbl 0178.26502

[6] J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math., 102 (1975), 327-361. | MR 52 #11947 | Zbl 0314.57018

[7] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (68), Springer-Verlag, Berlin, 1972. | MR 58 #22394a | Zbl 0254.22005

[8] B. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math., (1), 69 (1959), 119-132. | MR 21 #6004 | Zbl 0122.16604

[9] M. Spivak, A comprehensive introduction to differential geometry, vol. I, Publish or Perish, Boston, 1970. | Zbl 0202.52001

[10] J. Tits, Free subgroups in linear groups, J. of Alg., 20 (1972), 250-270. | MR 286898 | MR 44 #4105 | Zbl 0236.20032

[11] J. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. of Diff. Geom., 2 (1968), 421-446. | MR 248688 | MR 40 #1939 | Zbl 0207.51803