Regular projectively Anosov flows with compact leaves

[Flots projectivement Anosov réguliers avec des feuilles compactes]

Noda, Takeo

Annales de l'Institut Fourier, Tome 54 (2004), p. 481-497 / Harvested from Numdam

Résumé

Cet article concerne les flots projectivement Anosov, dont les feuilletages stable et instable ${\mathcal {F}}^s$ et ${\mathcal {F}}^u$ sont lisses, sur une variété de Seifert $M$. Nous prouvons que si l’un des feuilletages ${\mathcal {F}}^s$ ou ${\mathcal {F}}^u$ contient une feuille compacte, alors le flot $\phi ^t$ se décompose en union finie de modèles définis sur ${T^2} \times I$ et ayant pour bord les feuilles compactes. La variété $M$ est donc homeomorphe au tore ${T^3}$. Dans la preuve, nous obtenons également un théorème qui classifie les feuilletages de codimension un sur les variétés de Seifert ayant des feuilles compactes qui sont des tores incompressibles.

Abstract

This paper concerns projectively Anosov flows $\phi ^t$ with smooth stable and unstable foliations ${\mathcal {F}}^s$ and ${\mathcal {F}}^u$ on a Seifert manifold $M$. We show that if the foliation ${\mathcal {F}}^s$ or ${\mathcal {F}}^u$ contains a compact leaf, then the flow $\phi ^t$ is decomposed into a finite union of models which are defined on ${T^2} \times I$ and bounded by compact leaves, and therefore the manifold $M$ is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible tori.

Détail

Publié le : 2004-01-01

Zbl 1058.57021 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2026
Classification:  57R30,  37D30,  53C12,  53C15
Mots clés: flots projectivement Anosov, feuilletages stables, structures de bi-contact

Bibliographie

[A] M. Asaoka Classification of regular and non-degenerate projectively Anosov flows on three manifolds (Preprint) | Zbl 05128991

[Ba] T. Barbot Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier, Tome 46 (1996) no. 5, pp. 1451-1517 | Numdam | MR 1427133 | Zbl 0861.58028

[Br1] M. Brittenham Essential laminations in Seifert-fibered spaces, Topology, Tome 32 (1993) no. 1, pp. 61-85 | MR 1204407 | Zbl 0791.57013

[Br2] M. Brittenham Essential laminations in Seifert-fibered spaces: boundary behavior, Topology Appl., Tome 95 (1999) no. 1, pp. 47-62 | MR 1691931 | Zbl 0937.57016

[D] A. Denjoy Sur les courbes définies par les équation différentielles à la surface du tore, J. de Math. (9), Tome 11 (1932), pp. 333-375 | JFM 58.1124.04

[EHN] D. Eisenbud; U. Hirsch; W. Neumann Transverse foliations of Seifert bundles and self homeomorphism of the circle, Comment. Math. Helv., Tome 56 (1981), pp. 638-660 | MR 656217 | Zbl 0516.57015

[ET] Y. Eliashberg; W.P. Thurston Confoliations, Amer. Math. Soc., University Lecture Series 13 (1998) | Zbl 0893.53001

[F] S. Fenley Anosov flows in 3-manifolds, Ann. of Math. (2), Tome 139 (1994), pp. 79-115 | MR 1259365 | Zbl 0796.58039

[Gh1] E. Ghys Déformations de flots d'Anosov et de groupes fuchsiens, Ann. Inst. Fourier, Tome 42 (1992) no. 1-2, pp. 209-247 | Numdam | MR 1162561 | Zbl 0759.58036

[Gh2] E. Ghys Rigidité différentiable des groupes fuchsiens, I.H.É.S. Publ. Math., Tome 78 (1993), pp. 163-185 | Numdam | MR 1259430 | Zbl 0812.58066

[GO] D. Gabai; U. Oertel Essential laminations in $3$-manifolds, Ann. of Math. (2), Tome 130 (1989) no. 1, pp. 41-73 | MR 1005607 | Zbl 0685.57007

[H] D. Hardorp All compact orientable three manifolds admit total foliations, Memoirs Amer. Math. Soc., Tome 233 (1980) | Zbl 0435.57005

[J] W. Jaco Lectures on three-manifold topology, CBMS, Amer. Math. Soc. (Regional Conference Series in Mathematics 43) (1980) | MR 565450 | Zbl 0433.57001

[L] G. Levitt Feuilletages des variétés de dimension $3$ qui sont des fibrés en cercles, Comment. Math. Helv., Tome 53 (1978) no. 4, pp. 572-594 | MR 511848 | Zbl 0393.57004

[Ma] S. Matsumoto Foliations of Seifert fibered space over $S^2$, Foliations (Tokyo, 1983), North-Holland (Adv. Studies Pure Math.) Tome 5 (1985), pp. 325-339 | MR 877337 | Zbl 0645.57020

[Mi1] Y. Mitsumatsu Anosov flows and non-Stein symplectic manifolds, Ann. Inst. Fourier, Tome 45 (1995) no. 5, pp. 1407-1421 | Numdam | MR 1370752 | Zbl 0834.53031

[Mi2] Y. Mitsumatsu Foliations and contact structures on 3-manifolds, Foliations: geometry and dynamics (Warsaw, 2000), World Sci. Publishing (2002), pp. 75-125 | MR 1882766 | Zbl 1008.57003

[Mi3] Y. Mitsumatsu Projectively Anosov flows and bi-contact structures on (Preprint in preparation)

[MR] R. Moussu; R. Roussarie Relations de conjugaison et de cobordisme entre certains feuilletages, I.H.É.S. Publ. Math., Tome 43 (1974), pp. 142-168 | Numdam | MR 358810 | Zbl 0356.57018

[Nd] T. Noda Projectively Anosov flows with differentiable (un)stable foliations, Ann. Inst. Fourier, Tome 50 (2000) no. 5, pp. 1617-1647 | Numdam | MR 1800129 | Zbl 1023.37014

[NT] T. Noda; T. Tsuboi Regular projectively Anosov flows without compact leaves, Foliations: geometry and dynamics (Warsaw, 2000), World Sci. Publishing (2002), pp. 403-419 | MR 1882782 | Zbl 1002.37016

[Nv] S.P. Novikov Topology of foliations, Trudy Moskov. Mat. Ob., Tome 14 (1965), pp. 248-278 | MR 200938 | Zbl 0247.57006

[Nv] S.P. Novikov Topology of foliations, Amer. Math. Soc. (1967), pp. 286-304 | Zbl 0247.57006

[O] P. Orlik Seifert manifolds, Springer, Lecture Notes in Math., Tome 291 (1972) | MR 426001 | Zbl 0263.57001

[Sc] P. Scott The geometries of $3$-manifolds, Bull. London Math. Soc., Tome 15 (1983), pp. 401-487 | MR 705527 | Zbl 0561.57001

[Sch] A.J. Schwarz A generalization of a Poincaré-Bendixon theorem to closed two dimensional manifolds, Amer. J. Math., Tome 85 (1963), pp. 453-458 | MR 155061 | Zbl 0116.06803

[Ta] I. Tamura Topology of foliations : an introduction. Transl. from the 1976 Japanese edition., Amer. Math. Soc., Translation of Mathematical Monographs, Tome 97 (1992) | MR 1151624 | Zbl 0742.57001

[Th] W.P. Thurston Foliations of 3-manifolds which are circle bundles (1972) (Ph. D. Thesis, UC Berkeley)

[Ts] T. Tsuboi Regular projectively Anosov flows on the Seifert fibered spaces (Preprint)