Minimal geodesics and integrable behavior in geodesic flows

2016/01/25 by Jan Philipp Schröder, Schröder, Jan Philipp
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1601.06691

openalex publication_date 2016/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Lecture notes on Mather's theory for Lagrangian systems
• Topological entropy of minimal geodesics and volume growth on surfaces
• Minimal geodesic foliation on T² in case of vanishing topological entropy
• Uniqueness of shortest closed geodesics for generic Finsler metrics
• An entropic characterization of the flat metrics on the two torus

Related

• Integrable geodesic flows on Riemannian manifolds: Construction and Obstructions
• Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
• Integrable geodesic flow in 3D and webs of maximal rank
• New examples of manifolds with completely integrable geodesic flows
• Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity