Structure of a Morse form foliation on a closed surface in terms of genus
type: Item
instance of: scholarly article
title: Structure of a Morse form foliation on a closed surface in terms of genus
zbMATH Open document ID: 1223.57022
author: Irina Gelbukh
published in: Differential Geometry and its Applications
publication date: 2011-07-19T00:00:00Z
review text: Let \(\omega\) be a closed \(1\)-form with Morse singularities on a genus \(g\) closed surface \(M\). The form \(\omega\) defines a foliation \(\mathcal{F} _{\omega}\) on \(M\backslash Sing\;\omega\) and a singular foliation \(\overline{\mathcal{F}}_{\omega}\) on the whole \(M\). A leaf \(\gamma \in\mathcal{F}_{\omega}\) is compactifiable if \(\gamma\cup Sing\;\omega\) is compact. Non-compactifiable leaves of \(\mathcal{F}_{\omega}\) form minimal components \(\mathcal{C}_{j}^{\min};\) each such leaf is dense in its minimal component. If \(\gamma\) is a compact singular leaf, a small closed tubular neighborhood of \(\gamma\) is denoted by \(V(\gamma)\). The main result of this work is the relation \(c(\omega)+\sum_{\gamma}g(V(\gamma))+g(\bigcup \overline{\mathcal{C}_{j}^{\min}})=g,\) where \(c(\omega)\) is the number of homologically independent compact leaves. This result allows the author to prove a criterion for compactness of \(\overline{\mathcal{F}}_{\omega}\), to estimate the number of its minimal components and to give an upper bound on the rank \(rk\;\omega,\) in terms of genus.
reviewed by: Charalampos Charitos
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 58K65
zbMATH DE Number: 5925815
zbMATH Keywords: Morse form foliation
zbMATH Keywords: minimal component
zbMATH Keywords: compact singular leaves
zbMATH Keywords: genus
zbMATH Keywords: isotropic subgroup
MaRDI profile type: Publication
full work available at URL: j.difgeo.2011.04.029
OpenAlex ID: W1996698310
Wikidata QID: Q115356870
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DOI: 10.1016/J.DIFGEO.2011.04.029
description: scientific article; zbMATH DE number 5925815