Toric integrable geodesic flows in odd dimensions

TL;DR

This paper proves that odd-dimensional compact Riemannian manifolds with toric integrable geodesic flows are topologically tori, extending understanding of integrable systems and contact geometry in differential topology.

Contribution

It establishes that under certain conditions, such manifolds are topologically equivalent to tori, revealing new classifications of manifolds with integrable geodesic flows.

Findings

Cosphere bundle is equivariantly contactomorphic to that of a torus.

Manifold $Q$ is homeomorphic to an $n$-dimensional torus.

Results apply when $n eq 3$ odd or $ ext{pi}_1(Q)$ is infinite.

Abstract

Let $Q$ be a compact, connected $n$-dimensional Riemannian manifold, and assume that the geodesic flow is toric integrable. If $n \neq 3$ is odd, or if $\pi_1(Q)$ is infinite, we show that the cosphere bundle of $Q$ is equivariantly contactomorphic to the cosphere bundle of the torus $\T^n$. As a consequence, $Q$ is homeomorphic to $\T^n$.