DOES THE HYPERBOLICITY OF PERIODIC ORBITS OF A GEODESIC FLOW WITHOUT CONJUGATE POINTS IMPLY THE ANOSOV PROPERTY?

Let (M, g) be a C-infinity compact, connected, boundaryless manifold without conjugate points whose Green bundles are continuous. We show that if the closure of the set of periodic orbits of the geodesic flow is a hyperbolic set then the geodesic flow is Anosov. Combining this fact with recent results relating the structural stability of geodesic flows with the hyperbolicity of the closure of periodic orbits, we conclude that if the geodesic flow of (M, g) is C-2-structurally stable from Mane's viewpoint, then it is an Anosov flow, proving the so-called stability conjecture for the above class of compact manifolds without conjugate points.