Leo Butler - A New Class of Homogeneous Manifolds with Liouville-Integrable Geodesic Flows

A family of nilmanifolds possessing Riemannian metrics whose geodesic flow is Liouville-integrable is demonstrated. These homogeneous spaces are of the form , where H is a connected, simply-connected and two-step nilpotent Lie group and D is a discrete, cocompact subgroup of H. The metric on these homogeneous spaces is obtained from a left-invariant metric on H. These nilmanifolds provide the first example of manifolds whose fundamental group possesses no commutative subgroup of finite index, yet they admit a Liouville-integrable geodesic flow. The conclusions of Taimanov's theorem do not obtain in the category of Liouville-integrable geodesic flows with smooth first integrals.