no. 4
Riemannian manifolds not quasi-isometric to leaves in codimension one foliations
[Variétés riemanniennes qui ne sont pas quasi-iométriques à feuille d’un feuilletage de codimension un]
Schweitzer, Paul A.1
1 Depto. de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, RJ 22453-900, Brasil
Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1599-1631.

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.

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.

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, Tome 61 (2011) no. 4, pp. 1599-1631. doi : 10.5802/aif.2653. https://aif.centre-mersenne.org/articles/10.5802/aif.2653/

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