Splittings and Cr-structures for manifolds with nonpositive sectional curvature

Abstract

Let M n denote the universal covering space of a compact Riemannian manifold, M n , with sectional curvature, −1≤K Mn ≤0. We show that a collection of deck transformations of M n , satisfying certain (metric dependent) conditions, determines an open dense subset of M n , at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, 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 injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of M n vanishes. We show that an abelian structure exists if the injectivity radius at all points of M n is less than e(n)>0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov’s “gap conjecture” in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.