Conformally flat manifolds with nilpotent holonomy and the uniformization problem for -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 , a metric which is flat in a neighborhood of . In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank . 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 -bundles over (namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of -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