3-manifolds Admitting Toric Integrable Geodesic Flows

A toric integrable geodesic flow on a manifold M is a completely integrable geodesic flow such that the integrals generate a homogeneous torus action on the punctured cotangent bundle T ∗M \ M . A toric integrable manifold is a manifold which has a toric integrable geodesic flow. Toric integrable manifolds can be characterized as those whose cosphere bundle (the sphere bundle in the cotangent bundle) have the structure of a contact toric manifold. In this paper, we identify all 3-dimensional toric integrable manifolds using a classification of contact toric manifolds and a method of detecting when a manifold is a cosphere bundle. There turns out to be only three diffeomorphism classes: the 3-sphere, the 3-torus and quotients of the 3-sphere by finite cyclic groups. All of these manifolds then possess toric integrable geodesic flows.