Conformally flat structures on three-dimensional manifolds: The existence problem. I
type: Item
instance of: scholarly article
title: Conformally flat structures on three-dimensional manifolds: The existence problem. I
zbMATH Open document ID: 0694.53029
DOI: 10.1007/BF00971261
author: M. Eh. Kapovich
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: A well-known result of \textit{W. Thurston} [Ann. Math., II. Ser. 124, 203- 246 (1986; Zbl 0668.57015)] asserts that on any closed atoroidal Haken manifold, there exists a hyperbolic structure, i.e., a metric of constant sectional curvature -1. Consequently, on such a manifold there exists a conformally flat structure [for this notion, see \textit{N. H. Kuiper}, ibid. 50, 916-924 (1949; Zbl 0041.093)]. On the other hand, \textit{W. Goldman}'s results [Trans. Am. Math. Soc. 278, 573-583 (1983; Zbl 0518.53041)] imply that on solv- or nil-manifolds there are no conformally-flat structures. In the paper under review, the author makes obvious a larger class of closed Haken 3-manifolds for which there are conformally flat structures. Essentially, he proves two results which represent the first two steps in proving the general existence result for this class. The complete proof should appear in a forthcoming paper. More precisely, let M be a Seifert fiber space or a closed oriented 3-manifold obtained by glueing a finite number of Seifert fiber spaces along their boundary tori, whose fundamental group \(\pi_ 1(M)\) is non-solvable. Then there is a ``finite'' covering \(M_ 0\) over M which admits the introduction of a uniformizable conformally flat structure. Some examples are given.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 4136910
zbMATH Keywords: closed Haken 3-manifolds
zbMATH Keywords: conformally flat structures
zbMATH Keywords: Seifert fiber space
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00971261
OpenAlex ID: W2062030098
description: scientific article; zbMATH DE number 4136910