Every open manifold of dimension greater than one has complete Riemannian metrics with bounded geometry such that is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of suffice to make it quasi-isometric to a leaf of such a foliation. We introduce the ‘bounded homology property’, a semi-local property of 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 de dimension plus grande que 1 possède des métriques Riemanniennes complètes g avec géométrie bornée telles que 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 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 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.
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
@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}, pages = {1599--1631}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {4}, year = {2011}, doi = {10.5802/aif.2653}, zbl = {1241.57036}, mrnumber = {2951506}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2653/} }
TY - JOUR AU - Schweitzer, Paul A. TI - Riemannian manifolds not quasi-isometric to leaves in codimension one foliations JO - Annales de l'Institut Fourier PY - 2011 SP - 1599 EP - 1631 VL - 61 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2653/ DO - 10.5802/aif.2653 LA - en ID - AIF_2011__61_4_1599_0 ER -
%0 Journal Article %A Schweitzer, Paul A. %T Riemannian manifolds not quasi-isometric to leaves in codimension one foliations %J Annales de l'Institut Fourier %D 2011 %P 1599-1631 %V 61 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2653/ %R 10.5802/aif.2653 %G en %F 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/articles/10.5802/aif.2653/
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