ON TORI WITHOUT CONJUGATE-POINTS

In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient v ector fields of Busemann functions on the universal cover of such a ma nifold induce a natural foliation (akin to the weak stable foliation f or a Riemannian manifold with negative sectional curvature) on the uni t tangent bundle. The main result in the paper is that the metric is f lat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This co nfirms a conjecture of E. Hopf in this case.