Localization in the classification of flat manifolds (Q1099819)

This paper deals with smooth compact manifolds carrying a Riemannian connection for which the Levi-Civita connection is flat. Two compact flat Riemannian manifolds are defined to be comparable if each one is a covering space of the other in such a way that the covering maps are affine and compositions of these maps in either order increase distance locally by a constant factor. The study of comparability classes instead of affine-equivalence classes corresponds to the localization of the algebra in calculations. The author also investigates the extent to which the cohomology of the manifold is an invariant of the comparability class. The paper concludes with a calculation of the comparability classes when the holonomy group is the metacyclic group \(D_{pq}\), p, q prime.