Uniqueness of shortest closed geodesics for generic Finsler metrics

TL;DR

This paper proves that for a generic set of Finsler metrics on a closed manifold, each non-trivial homotopy class contains exactly one shortest geodesic loop, highlighting a generic uniqueness property.

Contribution

It establishes the generic uniqueness of shortest geodesics within each non-trivial homotopy class for Finsler metrics on closed manifolds.

Findings

Residual subset of Finsler metrics with unique shortest geodesics

Uniqueness holds in every non-trivial homotopy class

Results extend to conformal classes of metrics

Abstract

In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.