Riemannian manifolds not quasi-isometric to leaves in codimension one foliations
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1599-1631.

Every open manifold L of dimension greater than one has complete Riemannian metrics g with bounded geometry such that (L,g) is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of (L,g) suffice to make it quasi-isometric to a leaf of such a foliation. We introduce the ‘bounded homology property’, a semi-local property of (L,g) that is necessary for it to be a leaf in a compact manifold in codimension one, up to quasi-isometry. An essential step involves a partial generalization of the Novikov closed leaf theorem to higher dimensions.

Chaque variété ouverte L de dimension plus grande que 1 possède des métriques Riemanniennes complètes g avec géométrie bornée telles que (L,g) n’est pas quasi-isométrique à une feuille d’un feuilletage de codimension un d’une variété fermée. Donc il n’y a pas de conditions sur la géométrie locale de (L,g) qui suffisent pour qu’elle soit quasi-isométrique à une feuille de tel feuilletage. Nous introduisons la «  propriété d’homologie bornée  », une propriété semi-locale de (L,g) qui est nécessaire pour qu’elle puisse être feuille d’un feuilletage de codimension 1 d’une variété compacte, à une quasi-isométrie près. Une étape essentielle de la démonstration utilise une généralisation partielle du théorème de la feuille fermée de Novikov aux dimensions plus grandes.

DOI: 10.5802/aif.2653
Classification: 57R30, 53C12, 53B20, 53C40
Keywords: codimension one foliation, Reeb component, non-leaf, geometry of leaves, bounded homology property
Mot clés : feuilletages de codimension un, composante de Reeb, non-feuille, géométrie des feuilles, propriété d’homologie bornée
Schweitzer, Paul A. 1

1 Depto. de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, RJ 22453-900, Brasil
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Schweitzer, Paul A. Riemannian manifolds not quasi-isometric to leaves in codimension one foliations. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1599-1631. doi : 10.5802/aif.2653. http://www.numdam.org/articles/10.5802/aif.2653/

[1] Alcalde, F.; Hector, G.; Schweitzer, P.A. The structure of generalized Reeb components (2009) (Preprint)

[2] Alcalde, F.; Hector, G.; Schweitzer, P.A. A generalization of Novikov’s Theorem on the existence of Reeb components in codimension one foliations (2010) (In preparation)

[3] Attie, O.; Hurder, S. Manifolds which cannot be leaves of foliations, Topology, Volume 35 (1996), pp. 335-353 | DOI | MR | Zbl

[4] Camacho, C.; Lins Neto, A. Geometric Theory of Foliations, Birkhäuser Verlag, 1986 (Translation of Teoria Geométrica das Folheações, Projeto Euclides, IMPA, Rio de Janeiro, 1981) | MR

[5] Cantwell, J.; Conlon, L. Every surface is a leaf, Topology, Volume 26 (1987), pp. 265-285 | DOI | MR | Zbl

[6] do Carmo, M. Riemannian Geometry, Birkhäuser Verlag, 1998 (Translation of Geometria Riemanniana, Projeto Euclides, IMPA, Rio de Janeiro, 1988) | MR

[7] Ghys, E. Une variété qui n’est pas une feuille, Topology, Volume 24 (1985), pp. 67-73 | MR | Zbl

[8] Ghys, E. Topologie des feuilles génériques, Annals of Math., Volume 141 (1995), pp. 387-422 | DOI | MR | Zbl

[9] Haefliger, A. Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, Volume 16 (1962), pp. 367-397 | EuDML | Numdam | MR | Zbl

[10] Haefliger, A. Travaux de Novikov sur les feuilletages, Séminaire Bourbaki, 1968 no. 339 | Numdam | Zbl

[11] Hector, G. Croissance des feuilletages presque sans holonomie, Foliations and Gelfand-Fuks Cohomology, Rio de Janeiro, 1976, Volume 652, Springer Lecture Notes in Mathematics (1978), pp. 141-182 | MR | Zbl

[12] Hector, G.; Hirsch, U. Introduction to the geometry of foliations, B, Vieweg, Braunschweig, 1983 | MR

[13] Inaba, T.; Nishimori, T.; Takamura, M.; Tsuchiya, N. Open manifolds which are non-realizable as leaves, Kodai Math. J., Volume 8 (1985), pp. 112-119 | DOI | MR | Zbl

[14] Januszkiewicz, T. Characteristic invariants of noncompact Riemannian manifolds, Topology, Volume 23 (1984), pp. 299-302 | DOI | MR | Zbl

[15] Novikov, S.P. Topology of foliations, Trans. Moscow Math. Soc., Volume 14 (1965), pp. 268-304 | MR | Zbl

[16] Phillips, A.; Sullivan, D. Geometry of leaves, Topology, Volume 20 (1981), pp. 209-218 | DOI | MR | Zbl

[17] Reeb, G. Sur certaines propriétés topologiques des variétés feuilletées, Actual. Sci. Ind. 1183, Hermann, Paris, 1952 | MR | Zbl

[18] Schweitzer, P.A. Surfaces not quasi-isometric to leaves of foliations of compact 3-manifolds, Analysis and geometry in foliated manifolds, Proceedings of the VII International Colloquium on Differential Geometry, Santiago de Compostela, 1994, World Scientific, Singapore (1995), pp. 223-238 | MR | Zbl

[19] Siebenmann, L.C. Deformation of homeomorphisms on stratified sets, Comment. Math. Helv., Volume 4 (1972), pp. 123-163 | DOI | MR | Zbl

[20] Solodov, V.V. Components of topological foliations (Russian), Mat. Sb. (N.S.), Volume 119 (1982), pp. 340-354 | MR | Zbl

[21] Sondow, J. When is a manifold a leaf of some foliation?, Bull. Amer. Math. Soc., Volume 81 (1975), pp. 622-624 | DOI | MR | Zbl

[22] Sullivan, D. Cycles for the dynamical study of foliated manifolds and complex manifolds, Inventiones Math., Volume 36 (1976), pp. 225-255 | DOI | MR | Zbl

[23] Walczak, P A virtual leaf, Int. J., Bifur. and Chaos, Volume 7 (1996), pp. 1845-1852 | MR | Zbl

[24] Zeghib, A. An example of a 2-dimensional no leaf, Proceedings of the 1993 Tokyo Foliations Symposium, World Scientific, Singapore (1994), pp. 475-477 | MR

[25] Zetti, A. Sturm-Liouville Theory, Amer. Math. Soc., Providence, 2005

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