Geodesic flow in certain manifolds without conjugate points

Abstract:

A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic $\gamma$ of H tends uniformly to zero as the distance from p to $\gamma$ tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and ${T_t}$ the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to ${T_t}$ then ${T_t}$ is topologically transitive on SM. We also prove that if $M’$ is a normal covering of M then ${T_t}$ is topologically transitive on $SM’$ if ${T_t}$ is topologically transitive on SM.