Given an affine (i.e. connection-preserving) diffeomorphism f of a Riemannian manifold M, we consider its center foliation, N, comprised by the directions that neither expand nor contract exponentially under the action generated by f. 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 of M decomposes isometrically as the Riemannian product of the universal cover of a leaf of N (these covers are all isometric) and the Euclidean space; and if N 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.462233238