The stability conjecture for geodesic flows of compact manifolds without conjugate points and quasi-convex universal covering

Let $(M,g)$ be a $C^{\mathrm{\infty }}$ compact, boudaryless connected manifold without conjugate points with quasi-convex universal covering and divergent geodesic rays. We show that the geodesic flow of $(M,g)$ is $C^2$-structurally stable from Mañé’s viewpoint if and only if it is an Anosov flow, proving the so-called $C^1$-stability conjecture.