no. 6
Differential Geometry
Totally geodesic Riemannian foliations with locally symmetric leaves
[Feuilletages riemanniens totalement géodésiques avec des feuilles localement symétriques]

Présenté par : Étienne Ghys

Quiroga-Barranco, Raul1, 2
1 Centro de Investigación en Matemáticas, A.C. (CIMAT), A.P. 402, Guanajuato, Gto., C.P. 36000, Mexico
2 Departamento de Matemáticas, CINVESTAV, A.P. 14-740, México, D.F., C.P. 07000, Mexico
Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 421-426.

Nous prouvons le caractère arithmétique des feuilletages riemanniens totalement géodésiques, possédant une feuille dense, sur une variété riemannienne complète de volume fini, quand les feuilles sont revêtues de façon isométrique par un espace symétrique irréductible de type noncompact et de rang au moins 2.

We prove the arithmeticity of totally geodesic Riemannian foliations, with a dense leaf, on complete finite volume Riemannian manifolds when the leaves are isometrically covered by an irreducible symmetric space of noncompact type and rank at least 2.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.01.015
Quiroga-Barranco, Raul 1, 2

1 Centro de Investigación en Matemáticas, A.C. (CIMAT), A.P. 402, Guanajuato, Gto., C.P. 36000, Mexico
2 Departamento de Matemáticas, CINVESTAV, A.P. 14-740, México, D.F., C.P. 07000, Mexico 
@article{CRMATH_2006__342_6_421_0,
     author = {Quiroga-Barranco, Raul},
     title = {Totally geodesic {Riemannian} foliations with locally symmetric leaves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--426},
     publisher = {Elsevier},
     volume = {342},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.01.015/}
}
TY  - JOUR
AU  - Quiroga-Barranco, Raul
TI  - Totally geodesic Riemannian foliations with locally symmetric leaves
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 421
EP  - 426
VL  - 342
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.01.015/
DO  - 10.1016/j.crma.2006.01.015
LA  - en
ID  - CRMATH_2006__342_6_421_0
ER  - 
%0 Journal Article
%A Quiroga-Barranco, Raul
%T Totally geodesic Riemannian foliations with locally symmetric leaves
%J Comptes Rendus. Mathématique
%D 2006
%P 421-426
%V 342
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.01.015/
%R 10.1016/j.crma.2006.01.015
%G en
%F CRMATH_2006__342_6_421_0
Quiroga-Barranco, Raul. Totally geodesic Riemannian foliations with locally symmetric leaves. Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 421-426. doi : 10.1016/j.crma.2006.01.015. http://www.numdam.org/articles/10.1016/j.crma.2006.01.015/

[1] Blumenthal, R.A.; Hebda, J.J. Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations, Quart. J. Math. Oxford Ser. (2), Volume 35 (1984) no. 140, pp. 383-392

[2] Candel, A.; Quiroga-Barranco, R. Gromov's centralizer theorem, Geom. Dedicata, Volume 100 (2003), pp. 123-155

[3] Cairns, G. Géométrie globale des feuilletages totalement géodésiques, C. R. Acad. Sci. Paris Sér. I Math., Volume 297 (1983) no. 9, pp. 525-527

[4] Cairns, G. Compact 4-manifolds that admit totally umbilic metric foliations, Brno, 1989, World Sci. Publishing, Teaneck, NJ (1990), pp. 9-16

[5] Cairns, G.; Ghys, É. Totally geodesic foliations on 4-manifolds, J. Differential Geom., Volume 23 (1986) no. 3, pp. 241-254

[6] Gromov, M. Géométrie defférentielle, Colloque Géométrie et Physique de 1986 en l'honneur de André Lichnerowicz (Bernard, D.; Choquet-Bruhat, Y., eds.), 1988, pp. 65-139

[7] Johnson, D.L.; Whitt, L.B. Totally geodesic foliations, J. Differential Geom., Volume 15 (1980) no. 2, pp. 225-235

[8] Molino, P. Riemannian foliations, Prog. Math., vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988 (Translated from the French by Grant Cairns. With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu)

[9] Quiroga-Barranco, R. PseudoRiemannian geometry and actions of simple Lie groups, C.R. Math. Acad. Sci. Paris, Ser. I, Volume 341 (2005) no. 6, pp. 361-364

[10] R. Quiroga-Barranco, Isometric actions of simple Lie groups on pseudoRiemannian manifolds, Ann. of Math., in press

[11] Zimmer, R.J. Arithmeticity of holonomy groups of Lie foliations, J. Amer. Math. Soc., Volume 1 (1988) no. 1, pp. 35-58

Cité par Sources :