The second closed geodesic, the fundamental group, and generic Finsler metrics

Abstract

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.

Multiplicity of non-contractible closed geodesics on Finsler compact space forms

“…Parallel transport along c as well as the index and nullity of the closed geodesic of the Finsler metrics agree with the parallel transport and the index and nullity of the osculating Riemannian metric. In a similar way given in section 3 of [38] to extend the bumpy metric theorem to osculating Riemannian metrics, F is a residual set in strong C 4 topology due to a local perturbative argument from [22], which was used by Rademacher to prove a similar conclusion with Riemannian metrics in C 2 topology, cf. [34].…”