Integrable Geodesic Flows on Cones over Riemannian Manifolds

Andrey E. Mironov; Siyao Yin

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arXiv · 2511.01566

Integrable Geodesic Flows on Cones over Riemannian Manifolds

Andrey E. Mironov·Siyao Yin

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Abstract

In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-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.

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BibTeX RIS

Andrey E. Mironov, Siyao Yin. 2026-02-06. Integrable Geodesic Flows on Cones over Riemannian Manifolds. https://arxiv.org/abs/2511.01566

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