Integrable Geodesic Flows on Cones over Riemannian Manifolds

2025/11/03 by Andrey E. Mironov, Mironov, Andrey E., Siyao Yin +1
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2511.01566

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/31

Abstract

In this paper we study the behavior of geodesics on cones over arbitrary C³-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for radial geodesics; thus, the geodesic flow is superintegrable. Moreover, we prove that the geodesic flow restricted to the open dense subset of the cotangent bundle corresponding to all non-radial trajectories is Liouville--Arnold integrable. This investigation is inspired by our recent results on Birkhoff billiards inside cones over convex manifolds where similar results hold true.

Citations

• Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures
• Integrable Birkhoff Billiards inside Cones
• Rich quasi-linear system for integrable geodesic flows on 2-torus

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• Integrable geodesic flows on 2-torus: formal solutions and variational principle
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