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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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 manifoldβ€”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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Article copyright: © Copyright 1983 American Mathematical Society