Integrable Geodesic Flows on Cones over Riemannian Manifolds

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.

💡 Research Summary

The paper investigates the dynamics of geodesics on cones built over arbitrary closed $C^3$‑smooth Riemannian manifolds $\Gamma$. By embedding $\Gamma$ isometrically into a Euclidean space $\mathbb R^{N+1}$ (via Nash’s theorem) and defining the cone $K={tp\mid p\in\Gamma,;t\in\mathbb R}$, the authors equip $K$ with the induced metric $g=dt^2+t^2 g_\Sigma$, where $\Sigma=K\cap S^N$ is the intersection with the unit sphere and inherits the metric of $\Gamma$. This metric is a warped product, a setting that allows explicit calculations of geodesic equations.

The first key observation (Lemma 1) is the existence of a conserved quantity
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