Let M be a compact flat Riemannian manifold of dimension n, and Γ its fundamental group. Then we have the following exact sequence (see )
where Zn is a maximal abelian subgroup of Γ and G is a finite group isomorphic to the holonomy group of M. We shall call Γ a Bieberbach group. Let T be a flat torus, and let Ggr act via isometries on T; then ┌ acts isometrically on × T where is the universal covering of M and yields a flat Riemannian structure on ( × T)/Γ. A flat-toral extension (see [9, p. 371]) of the Riemannian manifold M is any Riemannian manifold isometric to ( × T)/Γ where T is a flat torus on which Γ acts via isometries. It is convenient to adopt the convention that a single point is a 0-dimensional flat torus. If this is done, M is itself among the flat toral extensions of M. Roughly speaking, this is a way of putting together a compact flat manifold and a flat torus to make a new flat manifold the dimension of which is the sum of the dimensions of its constituents. It is, more precisely, a fibre bundle over the flat manifold with a flat torus as fibre.