A Generalization of Bieberbach's Theorem

In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be separated into two conditions. First, M has an affine structure a distinguished covering by coordinate charts, whose coordinate changes are affine. Second, M has a Riemannian metric whose coefficients in the affine charts are constants. In this paper we relax the second condition. A polynomial Riemannian metric on the affine manifold M is a Riemannian metric whose local expression in affine coordinates has the form ~gi~(x)dxidx ~ where the gij are polynomial functions on Euclidean space E. By letting the gij be rational functions on E, we arrive at the more general notion of a rational Riemannian metric. (It is not assumed that these expressions define Riemannian metrics on all of E.) The object of this paper is to determine which affine manifolds have polynomial Riemannian metrics and to give examples of affine manifold having rational Riemannian metrics.