Section: New Results

Stability properties of geodesic flows on Riemannian manifolds

Participants : Ludovic Rifford, Rafael Ruggiero [PUC, Rio de Janeiro, Brazil] .

In a paper by Rifford and Ruggiero [25], the $C^2$-structural stability conjecture from Mañé's viewpoint for geodesics flows of compact manifolds without conjugate points is investigated. The structural stability conjecture is an open problem in the category of geodesic flows because the $C^1$ closing lemma is not known in this context. Without the $C^1$ closing lemma, we combine the geometry of manifolds without conjugate points and a recent version of Franks' Lemma from Mañé's viewpoint to prove the conjecture for compact surfaces, for compact three dimensional manifolds with quasi-convex universal coverings where geodesic rays diverge, and for $n$-dimensional, generalized rank one manifolds.