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In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset R of metrics on a given compact Riemannian manifold such tha tif g∈R,then its associated geodesic flow φgt is expansive if and only if the closure of the set of periodic orbits of φgt is a uniformly hyperbolic set.
For surfaces, we obtain a stronger statement: there exists a C2-residual R such that if g ∈ R, then its associated geodesic flow φgt is expansive if and only if φgt is an Anosov flow.
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Expansiveness Residual sets Anosov Geodesic flows