Given an affine (i.e. connection-preserving) diffeomorphismf of a Riemannian manifoldM, we consider itscenter foliation, N, comprised by the directions that neither expand nor contract exponentially under the action generated byf. The main remarks made here (Corollary 3 and Theorem 7) are: There exists a metric compatible with the Levi-Civita connection for which the universal cover ofM decomposes isometrically as the Riemannian product of the universal cover of a leaf ofN (these covers are all isometric) and the Euclidean space; and ifN is one-dimensional,M is flat and the foliation is (up to finite cover) the fiber foliation of a Riemannian submersion onto a flat torus.