Transversely projective foliations on Seifert manifolds.
type: Item
instance of: scholarly article
title: Transversely projective foliations on Seifert manifolds.
zbMATH Open document ID: 1036.57006
author: Thierry Barbot
publication date: 2003-12-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116081
review text: In this paper, a projective structure means a \((PGL(3,\mathbb{R}),\mathbb{R} P^2)\) structure. In the Memoir [Affine flows on 3-manifolds, Mem. Am. Math. Soc. 771 (2003; Zbl 1026.57022)], \textit{S. Matsumoto} introduced a property which he called the ``homotopy lifting property'', which says that the space of leaves of the lift of the foliation to the universal cover of the ambient manifold is Hausdorff separated. In the paper under review, the author studies transversely projective foliations on three-manifolds, and the main result is the following Theorem: Let \(M\) be a Seifert fibered three-manifold whose base is an orbifold of nonpositive Euler characteristic and let \(\Phi\) be a transversely projective foliation on \(M\) satisfying the homotopy lifting property. Then, up to passing to a finite cover, \(\Phi\) is conjugate to a foliation which is of one of the following types: -- a projective fibration, that is, a foliation whose leaves are the fibres of a Seifert fibration over an orbifold equipped with a real projective structure; -- a ``convex geodesic foliation'', which is typically the foliation of a geodesic flow of a surface equipped with a metric of constant negative curvature; -- a ``horocyclic projective foliation'' associated to the geodesic foliation above. In each case, the conjugacy respects the projective structures.
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 2014686
zbMATH Keywords: real projective structure
zbMATH Keywords: transversely projective foliation
zbMATH Keywords: Seifert fibre space
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
OpenAlex ID: W2460801968
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DOI: 10.5802/AIF.1988
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2014686