On tori without conjugate points

Summary In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.Oblatum 22-IX-1993 & 25-IV-1994Supported in part by NSF grant #DMS90-01707 and #DMS85-05550 while at MSRISupported by an NSF Postdoctoral Fellowship