Tangentially affine foliations and leafwise affine functions on the torus
type: Item
instance of: scholarly article
title: Tangentially affine foliations and leafwise affine functions on the torus
zbMATH Open document ID: 0782.57018
author: Kazuo Masuda
author: Takashi Inaba
published in: Kodai Mathematical Journal
publication date: 1993-09-07T00:00:00Z
review text: The authors study tangentially affine foliations \({\mathcal F}\) of the torus \(T^ 2\) by curves. That is, there is a distinguished atlas in which the coordinate transformations are affine in the \({\mathcal F}\)-direction. They prove that such foliations exist and that any leafwise affine function \(F:T^ 2\to\mathbb{R}\) for \({\mathcal F}\) is uniquely determined by its values on the union of the compact leaves. Furthermore, if \(L\) is a compact leaf with contracting holonomy on one side and if \(U\) is a small neighborhood of \(L\) on that side, they show that the vector space \({\mathcal A}\) of all leafwise affine functions on \(U\) has dimension 1, 2, or \(\infty\). Finally, in a closing remark they show that the Reeb foliation of \(S^ 3\) admits no tangentially affine structure.
reviewed by: Lawrence Conlon
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 409640
zbMATH Keywords: tangentially affine foliations
zbMATH Keywords: leafwise affine function
zbMATH Keywords: Reeb foliation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039702
OpenAlex ID: W2100888829
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DOI: 10.2996/KMJ/1138039702
description: scientific article; zbMATH DE number 409640