Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

Details Riemannian manifolds not quasi-isometric to leaves in codimension one foliations Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

2009-11-24

arxiv.org/abs/0911.4665v1

Mathematics

Geometric Topology

36 pages, 10 figures

Scientific paper

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.

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