Feuilletages transversalement projectifs sur les variétés de Seifert
Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1551-1613.

Soit M une variété de Seifert de groupe fondamental non virtuellement résoluble. Soit Φ un feuilletage de dimension 1 sur M, muni d’une structure projective réelle transverse. On suppose que Φ satisfait la propriété de relèvement des chemins, i.e., que l’espace des feuilles du relèvement de Φ dans le revêtement universel de M est séparé au sens de Hausdorff. On montre qu’à revêtements finis près, Φ est soit une fibration projective, soit un feuilletage géodésique convexe, soit un feuilletage horocyclique projectif.

Let M be a Seifert manifold with non-solvable fundamental group. Let Φ be a one- dimensional foliation on M, equipped with a transverse real projective structure. We assume moreover that Φ satisfies the Homotopy Lifting Property, i.e., that the leaf space of the lifting of Φ in the universal covering of M satisfies the Hausdorff separation property. Then, up to finite coverings, Φ belongs to one of the following three families of transversely projective foliations: the family of projective fibrations, the family of convex geodesic foliations, or the family of projective horocyclic foliations.

DOI : 10.5802/aif.1988
Classification : 57M50, 57R30, 37D40
Mot clés : feuilletage transversalement projectif, variété de Seifert
Keywords: transversely projective foliations, Seifert manifolds
Barbot, Thierry 1

1 École Normale Supérieure de Lyon, UMPA, UMR 5669, 46 allée d'Italie, 69364 Lyon Cedex 7 (France)
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Barbot, Thierry. Feuilletages transversalement projectifs sur les variétés de Seifert. Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1551-1613. doi : 10.5802/aif.1988. http://www.numdam.org/articles/10.5802/aif.1988/

[1] T. Barbot Actions de groupes sur les 1-variétés non séparées et feuilletages de codimension un, Ann. Fac. Sci. Toulouse Math. (6), Volume 7 (1998) no. 4, pp. 559-597 | DOI | Numdam | MR | Zbl

[2] T. Barbot Variétés affines radiales de dimension trois, Bull. Soc. Math. France, Volume 128 (2000), pp. 347-389 | Numdam | MR | Zbl

[3] T. Barbot Flag structures on Seifert manifolds, Geom. Topol., Volume 5 (2001), pp. 227-266 | DOI | MR | Zbl

[4] T. Barbot Plane affine geometry of Anosov flows, Ann. Sci. École Norm. Sup., Volume 34 (2001) no. 6, pp. 871-889 | Numdam | MR | Zbl

[5] Y. Benoist; F. Labourie Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables, Invent. Math., Volume 111 (1993) no. 2, pp. 285-308 | DOI | MR | Zbl

[6] Y. Benoist Nilvariétés projectives, Comment. Math. Helv., Volume 69 (1994) no. 3, pp. 447-473 | DOI | MR | Zbl

[7] Y. Benoist Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000), pp. 149-193 | DOI | MR | Zbl

[8] Y. Benoist Tores affines, Crystallographic groups and their generalizations (Kortrijk, 1999) (Contemp. Math.), Volume 262 (2000), pp. 1-37 | Zbl

[9] Y. Benoist Convexes divisibles, C. R. Acad. Sci. Paris, Sér. I Math., Volume 332 (2001) no. 5, pp. 387-390 | DOI | MR | Zbl

[10] M. Brunella On transversely holomorphic flows. I, Invent. Math., Volume 126 (1996) no. 2, pp. 265-279 | DOI | MR | Zbl

[11] H. Busemann; P. Kelly Projective geometry and projective metrics, Academic Press, 1953 | MR | Zbl

[12] J. Cantwell; L. Conlon Endsets of exceptionnal leaves; a theorem of G. Duminy, Foliations: geometry and dynamics (Warsaw, 2000) (2002), pp. 225-261 | Zbl

[13] Y. Carrière; F. Dal'bo; G. Meigniez Inexistence de structures affines sur les fibrés de Seifert, Math. Ann., Volume 296 (1993), pp. 743-753 | DOI | MR | Zbl

[14] Y. Carrière Flots riemanniens, Transversal structure of foliations (Toulouse, 1982) (Astérisque), Volume 116 (1984), pp. 31-52 | Numdam | Zbl

[15] S. Choi Convex decomposition of real projective surfaces. I: π-annuli and convexity, J. Diff. Geom., Volume 40 (1994), pp. 165-208 | MR | Zbl

[16] S. Choi; W.M. Goldman The classification of real projective structures on compact surfaces, Bull. Amer. Math. Soc., Volume 34 (1997), pp. 161-171 | DOI | MR | Zbl

[17] S. Choi The decomposition and classification of radiant affine 3-manifolds, avec un appendice par T. Barbot, Mem. Amer. Math. Soc., Volume 154 (2001) no. 730 | MR | Zbl

[18] S. Dupont Solvariétés projectives de dimension trois (1999) (Thèse Université Paris VII)

[19] D.B.A. Epstein Foliations with all leaves compact, Ann. Inst. Fourier, Volume 26 (1976) no. 1, pp. 265-282 | DOI | Numdam | MR | Zbl

[20] D. Fried Affine 3-manifolds that fiber by circles (1992) (preprint I.H.E.S)

[21] D. Fried Transitive Anosov flows and pseudo-Anosov maps, Topology, Volume 22 (1983) no. 3, pp. 299-303 | DOI | MR | Zbl

[22] D. Fried; W.M. Goldman Three-dimensional affine crystallographic groups, Adv. in Math., Volume 47 (1983) no. 1, pp. 1-49 | DOI | MR | Zbl

[23] E. Ghys Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Sci. École Norm. Sup. (4), Volume 20 (1987) no. 2, pp. 251-270 | Numdam | MR | Zbl

[24] E. Ghys On transversely holomorphic flows. II, Invent. Math., Volume 126 (1996) no. 2, pp. 281-286 | DOI | MR | Zbl

[25] W.M. Goldman Convex real projective structures on compact surfaces, J. Diff. Geom., Volume 31 (1990), pp. 791-845 | MR | Zbl

[26] W.M. Goldman Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987) (Contemp. Math.), Volume 74 (1988), pp. 169-198 | Zbl

[27] A. Haefliger Groupoïde d'holonomie et classifiants, Astérisque, Volume 116 (1984), pp. 70-97 | Numdam | MR | Zbl

[28] G. Hector Feuilletages en cylindres, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) (Lecture Notes in Math.), Volume 597 (1977), pp. 252-270 | Zbl

[29] W. Jaco Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, Volume 43 (1980) | Zbl

[30] S. Matsumoto Affine flows on 3-manifolds, Mem. Amer. Math. Soc., Volume 162 (2003) no. 771 | MR | Zbl

[31] S. Matsumoto; T. Tsuboi Transverse intersections of foliations in three-manifolds, Monogr. Enseign. Math., Volume 38 (2001) | MR | Zbl

[32] W. Thurston The geometry and topology of 3-manifolds, Princeton Lect. Notes, Chapitre 13, 1977

[33] W. Thurston Foliations on 3-manifolds which are circle bundles (1972) (Thesis, Berkeley)

[34] F. Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math., Volume 3 (1967), pp. 308-333 | DOI | MR | Zbl

[34] F. Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. ibid, Volume 4 (1967), pp. 87-117 | MR | Zbl

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