Minimal geodesics and integrable behavior in geodesic flows

TL;DR

This survey reviews classical and recent results on minimal geodesics in Riemannian and Finsler metrics, focusing on the two-dimensional case, and discusses open problems and potential solutions.

Contribution

It compiles and discusses both classical and recent findings on minimal geodesics, highlighting open problems and initial ideas for solutions in the field.

Findings

Summary of classical and recent results on minimal geodesics

Focus on the two-dimensional case of geodesic flows

Presentation of open problems and preliminary ideas for solutions

Abstract

In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first ideas as to the solutions.