Riemannian manifolds not quasi-isometric to leaves in codimension one foliations
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, p. 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 : https://doi.org/10.5802/aif.2653
Classification:  57R30,  53C12,  53B20,  53C40
Keywords: codimension one foliation, Reeb component, non-leaf, geometry of leaves, bounded homology property
@article{AIF_2011__61_4_1599_0,
     author = {Schweitzer, Paul A.},
     title = {Riemannian manifolds not quasi-isometric to leaves in codimension one foliations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     pages = {1599-1631},
     doi = {10.5802/aif.2653},
     mrnumber = {2951506},
     zbl = {1241.57036},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_4_1599_0}
}
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/item/AIF_2011__61_4_1599_0/

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