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

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Abstract
A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point x x , a metric which is flat in a neighborhood of x x . In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank 3 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 {T^2} -bundles over S 1 {S^1} (namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of 3 3 -manifolds known not to admit conformally flat structures.

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