On surfaces with no conjugate points

Abstract

A complete Riemannian manifold M has no conjugate points if any two points in its universal cover are joined by a unique geodesic. If the sectional curvature of M is nonpositive, then M has no conjugate points; the converse is not true even for compact surfaces. A natural question is to what extent properties of manifolds of nonpositive sectional curvature are valid for manifolds with no conjugate points. For example, by the Gauss-Bonnet theorem, any metric of nonpositive curvature on the torus T is flat. In 1943, E. Hopf [4] proved Theorem. Any metric onT with no conjugate points is flat. The best way to explain the purpose of our paper and to introduce the necessary notations is to give an outline of Hopf s argument. He considers the Riccati equation