Open problems and questions about geodesics

Abstract

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.

This is the final version of the paper

References in corpus (17)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Diastolic and isoperimetric inequalities on surfaces
• Sharp systolic inequalities for Reeb flows on the three-sphere
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
• Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level
• Metrisability of three-dimensional projective structures
• The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles
• Chains in CR geometry as geodesics of a Kropina metric
• The contact property for nowhere vanishing magnetic fields on the two-sphere
• Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$
• Finsler surfaces with prescribed geodesics
• Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics
• The Palais-Smale condition for contact type energy levels for convex lagrangian systems
• Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics
• Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
• On the geodesic flow of surfaces of nonpositive curvature
• Curvature and partial hyperbolicity

Cited by in corpus (19)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
• Connecting and closed geodesics of a Kropina metric
• The Stratification of Singular Locus and Closed Geodesics on Orbifolds
• Almost every path structure is not variational
• On closed geodesics in Lorentz manifolds
• Homologically visible closed geodesics on complete surfaces
• On the ergodicity of geodesic flows on surfaces of nonpositive curvature
• The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
• On the ergodicity of geodesic flows on surfaces without focal points
• The Length of the Shortest Closed Geodesic on a Surface of Finite Area
• Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures
• Deformational rigidity of integrable metrics on the torus
• A systolic inequality for geodesic flows on the two-sphere
• An inequality for length and volume in the complex projective plane
• Finding Geodesics on Surfaces Using Taylor Expansion of Exponential Map
• On the existence of geodesic vector fields on closed surfaces
• Converting homotopies to isotopies and dividing homotopies in half in an effective way
• Geodesics as deformations of one-parameter subgroups in homogeneous manifolds