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 γ \gamma of H tends uniformly to zero as the distance from p to γ \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 {T_t} the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to T t {T_t} then T t {T_t} is topologically transitive on SM. We also prove that if M ′ M’ is a normal covering of M then T t {T_t} is topologically transitive on S M ′ SM’ if T t {T_t} is topologically transitive on SM.