Let Mn denote the universal covering space of a compact Rie-mannian manifold, Mn, with sectional curvature, -1≤Kmn≤O.We show dependent) conditions, determinesan open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a coolection, which we call an abelian structure, also gives rise factor. Such a collectionk, which we call an abelian structure, also gives rise to an essentially canonical Cr-stucture in the sense of Buyalo, i.e. an atalas to an essentially canonical Cr-stucture in the sense of Buyalo, i.e an atlas for an injective F-STUCTURE, for which additional conditions hold, It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ε(n)>O.This yields a conjecture of Buyalo as well as a strength-ened version of the conclusion of Gromov's gap conjecture in our special ituation. In addition, we observe that abelian stuructures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.

Let Mi dGH −→ X denote a sequence of n-manifolds converging to a compact metric space, X, in the Gromov-Hausdorff topology such that the sectional curvature is bounded in absolute value and dim(X)<n. We prove the following stability result: If the fundamental groups of Miare torsion groups of uniformly bounded exponents and the second twisted Betti numbers of Mi vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {Mi} such that the distance functions of the pullback metrics converge to a pseudo-metric in C0-norm. Furthermore, M admits a foliation with leaves diffeomorphic to flat manifolds (not necessarily compact) such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics. These results lead to a few interesting applications.