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

Paul A. Schweitzer[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 (2011)

  • Volume: 61, Issue: 4, page 1599-1631
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Schweitzer, Paul A.. "Riemannian manifolds not quasi-isometric to leaves in codimension one foliations." Annales de l’institut Fourier 61.4 (2011): 1599-1631. <http://eudml.org/doc/219796>.

@article{Schweitzer2011,
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.},
affiliation = {Depto. de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, RJ 22453-900, Brasil},
author = {Schweitzer, Paul A.},
journal = {Annales de l’institut Fourier},
keywords = {codimension one foliation; Reeb component; non-leaf; geometry of leaves; bounded homology property; quasi-isometry; closed leaf theorem},
language = {eng},
number = {4},
pages = {1599-1631},
publisher = {Association des Annales de l’institut Fourier},
title = {Riemannian manifolds not quasi-isometric to leaves in codimension one foliations},
url = {http://eudml.org/doc/219796},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 4
SP - 1599
EP - 1631
AB - 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.
LA - eng
KW - codimension one foliation; Reeb component; non-leaf; geometry of leaves; bounded homology property; quasi-isometry; closed leaf theorem
UR - http://eudml.org/doc/219796
ER -

References

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