Compact Flat Riemannian Manifolds: I

Abstract

We say a riemannian manifold is flat if its curvature vanishes at each point. In this paper we study the classification problem for compact flat manifolds and show that it is related to the problem of finding integral representations for finite groups. The first section contains material that can only be called well known. However, since we are unable to give any references,1 we have elected to include these theorems together with an indication of their proofs. The main result is that compact flat manifolds are classified up to something stronger than diff eomorphism by their fundamental groups, and the fundamental group 7c of a compact flat manifold X satisfies an exact sequence