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Let $(M,g)$ be a $C^{\infty}$ compact, boudaryless connected manifold withoutconjugate points with quasi-convex universal covering and divergent geodesicrays. We show that the geodesic flow of $(M,g)$ is $C^{2}$-structurally stablefrom Ma\~{n}\'{e}'s viewpoint if and only if it is an Anosov flow, proving theso-called $C^{1}$-stability conjecture.

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