Integrable Geodesic Flows on Cones over Riemannian Manifolds

TL;DR

This paper demonstrates that geodesic flows on cones over smooth Riemannian manifolds are superintegrable and Liouville--Arnold integrable on non-radial trajectories, extending previous billiard results.

Contribution

It establishes superintegrability and Liouville--Arnold integrability of geodesic flows on cones over arbitrary smooth Riemannian manifolds, generalizing prior billiard findings.

Findings

Geodesic flow admits first integrals determining almost all geodesics.

Flow is superintegrable except for radial geodesics.

Flow is Liouville--Arnold integrable on non-radial trajectories.

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.