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Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $ 3$-manifolds


Author: William M. Goldman
Journal: Trans. Amer. Math. Soc. 278 (1983), 573-583
MSC: Primary 53C20; Secondary 57R99
DOI: https://doi.org/10.1090/S0002-9947-1983-0701512-8
MathSciNet review: 701512
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Abstract: A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point $ x$, a metric which is flat in a neighborhood of $ x$. In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank $ 3$. Specifically, we show that such conformally flat manifolds are covered by either the sphere, a flat torus, or a Hopf manifold--in particular, their fundamental groups contain abelian subgroups of finite index. These results are applied to show that certain $ {T^2}$-bundles over $ {S^1}$ (namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of $ 3$-manifolds known not to admit conformally flat structures.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701512-8
Article copyright: © Copyright 1983 American Mathematical Society

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