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

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.