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

TL;DR

The paper demonstrates that certain open manifolds with specific Riemannian metrics cannot be quasi-isometric to leaves in codimension one foliations of closed manifolds, introducing a new semi-local property called the bounded homology property.

Contribution

It introduces the bounded homology property as a necessary condition for a manifold to be a leaf in a codimension one foliation, extending the Novikov theorem to higher dimensions.

Findings

Existence of metrics not quasi-isometric to leaves

Introduction of the bounded homology property

Generalization of Novikov's theorem to higher dimensions

Abstract

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.