Normal contact structures on 3-manifolds
type: Item
instance of: scholarly article
title: Normal contact structures on 3-manifolds
zbMATH Open document ID: 0897.53024
author: Hansjörg Geiges
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-07-20T00:00:00Z
review text: A closed 3-manifold \(M\) is called a contact manifold if it carries a global differential 1-form \(\eta\) such that \(\eta \wedge d\eta\) is a volume form. \(M\) admits a global nonvanishing vector field \(\xi\) such that \(\eta(\xi)=1\) and \(d\eta(\xi,X)=0\) for all vector fields \(X\) on \(M\). On a contact distribution \({\mathcal D}=\ker \eta\) one can find an endomorphism \(J\colon{\mathcal D}\to{\mathcal D}\) compatible with \(d\eta\) in the sense that \(d\eta(JX,JY)=d\eta(X,Y)\). \(J\) is defined uniquely up to homotopy. The triple \((J,\xi,\eta)\), in other words, the reduction of the structure group of \(M\) to \(\text{U}(1)\times 1\), is called an almost contact structure on \(M\). If an almost complex structure \(\overline J\) defined on \(M\times \mathbb{R}\) by \(\overline J(X,t)=(JX-t\xi,\eta(X))\) is integrable, then the almost contact structure \((J,\xi,\eta)\) and the contact form \(\eta\) are called normal. In this paper, the author determines manifolds diffeomorphic to closed 3-manifolds that admit normal contact or almost contact forms. In Theorem 1, the author proves that a closed 3-manifold admits a normal contact form if and only if it is diffeomorphic to a manifold of the form \(\Gamma \backslash X\), where \(X\) is either \(S^3\), \(\widetilde{\text{SL}}_2\) (the universal covering of \(\text{SL}_2(\mathbb{R})\)), or \(\text{Nil}^3\) (the Heisenberg group), and \(\Gamma\) is a (torsion free) subgroup of the identity component \(\text{Isom}_0 X\) of \(X\). The manifolds in this theorem are the Seifert fibred 3-manifolds with nonzero Euler number over orientable base orbifolds. \textit{H. Sato} [Tôhoku Math. J., II. Ser. 29, 577-584 (1977; Zbl 0382.53031)] proved that if \(M\) admits a normal almost contact structure, then \(\pi_2(M)=0\) or \(M\) is diffeomorphic to \(S^2 \times S^1\). In Theorem 2, the author completes the investigation begun by Sato and proves that \(M\) admits a normal almost contact structure if and only if \(M\) is diff…
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 1140478
zbMATH Keywords: 3-manifold
zbMATH Keywords: contact structure
zbMATH Keywords: almost contact structure
zbMATH Keywords: normal almost contact structure
zbMATH Keywords: normal contact structure
zbMATH Keywords: compact complex surface
zbMATH Keywords: spherical geometry
zbMATH Keywords: Seifert manifold
reviewed by: Andrew Bucki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225112
OpenAlex ID: W1987427496
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Wikidata QID: Q126022108
DOI: 10.2748/TMJ/1178225112
description: scientific article; zbMATH DE number 1140478