Splittings and Cr-structures for manifolds with nonpositive sectional curvature

Let (M) over tilde (n) denote the universal covering space of a compact Rie mannian manifold, M-n, with sectional curvature, -1 less than or equal to K -Mn less than or equal to 0. We show that a collection of deck transformati ons of (M) over tilde (n), satisfying certain (metric dependent) conditions , determines an open dense subset of Mn, at every paint of which, there exi sts a local isometric splitting with nontrivial flat factor. Such a collect ion, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injectiv e F-structure, for which additional conditions hold. It follows in particul ar that the minimal volume of M-n vanishes. We show that an abelian structu re exists if the injectivity radius at all points of M-n is less than epsil on (n) > 0. This yields a conjecture of Buyalo as well as a strengthened ve rsion of the conclusion of Gromov's "gap conjecture" in our special situati on. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled chan ges of metric.