Conformally flat structures on three-dimensional manifolds: The existence problem. I
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.
scientific article; zbMATH DE number 4136910
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