On continuous geodesic foliations of hyperbolic manifolds
type: Item
instance of: scholarly article
title: On continuous geodesic foliations of hyperbolic manifolds
zbMATH Open document ID: 0789.57019
DOI: 10.1007/BF01232666
published in: Inventiones Mathematicae
publication date: 1994-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/144145
review text: We prove the following result: Theorem: A compact hyperbolic 3-manifold does not possess a \(\mathbb{C}^ 0\)- geodesic foliation. (Here hyperbolic means of constant curvature \(=- 1\).) The theorem will be proved for foliations of dimension 1, because the codimension 1 case is well known [see for instance: \textit{E. Ghys}, Comment. Math. Helv. 58, 543-572 (1983; Zbl 0534.57015); or the author, Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, No. 2, 171-188 (1991; Zbl 0738.53019)]. The proof is done in 3 steps. We suppose that such a foliation exists. We can assume it oriented by a unit tangent vector field. The first step of the proof is dynamic. We remark that the generated flow is expansive, and then of pseudo-Anosov type. From this, we deduce, in the second step, that the flow is Lipschitz. In the final step, we treat Lipschitz foliations.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 549645
zbMATH Keywords: compact hyperbolic 3-manifold
zbMATH Keywords: \(\mathbb{C}^ 0\)-geodesic foliation
author: Abdelghani Zeghib
MaRDI profile type: Publication
cites work: The Hadamard-Cartan theorem in locally convex metric spaces
cites work: Q4067705
cites work: Q5565773
cites work: Classification des feuilletages totalement géodésiques de codimension un
cites work: Expansive homeomorphisms of compact surfaces are pseudo-Anosov
cites work: Expansive homeomorphisms of surfaces
cites work: Q4710945
description: scientific article; zbMATH DE number 549645