Geom Dedicata (2008) 134:131–138 DOI 10.1007/s10711-008-9249-x ORIGINAL PAPER A note on the divergence of geodesic rays in manifolds without conjugate points Rafael Oswaldo Ruggiero Received: 25 April 2006 / Accepted: 17 March 2008 / Published online: 2 April 2008 © Springer Science+Business Media B.V. 2008 Abstract Let ( M, g) be a compact Riemannian manifold without conjugate points and let ( ˜ M, ˜ g) be its universal covering endowed with the pullback of the metric g by the cover- ing map. We show that geodesic rays in ( ˜ M, ˜ g) which meet an axis of a covering isometry diverge from this axis. This result generalizes well known results by Morse and Hedlund in the context of globally minimizing geodesics in the universal covering of compact surfaces. Keywords Conjugate points · Divergence of rays Mathematics Subject Classification (2000) 53C70 1 Introduction The purpose of this article is to study the problem of the divergence of geodesic rays in mani- folds without conjugate points with no further assumptions on either the sectional curvatures or the Jacobi fields. Recall that a C ∞ Riemannian, n-dimensional manifold ( M, g) has no conjugate points if the exponential map is nonsingular at every point. The universal cover- ing ˜ M of a manifold ( M, g) is diffeomorphic to R n , and the metric spheres in ( ˜ M, ˜ g)—the universal covering endowed with the pullback of g—are diffeomorphic to the standard sphere in R n . Given a point p ∈ ˜ M, and two geodesics γ , β in ( ˜ M, g) parametrized by arclength such that p = γ(0) = β(0), we say that these geodesics diverge if lim t →+∞ d (γ (t ),β(t )) =∞. Although two different geodesic rays starting from a point in ˜ M diverge in all well-known examples of manifolds without conjugate points (e.g., nonpositive curvature, no focal points, bounded asymptote), there is no general proof of this fact so far. Let us make a brief account of the literature concerning this problem. Hopf in a cel- ebrated paper [8] showed that two dimensional tori without conjugate points are flat. So R. O. Ruggiero (B ) Dep. de Matemática, Pontificia Universidade Católica do Rio de Janeiro, PUC-Rio, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brazil e-mail: rorr@mat.puc-rio.br 123