Quadratically Integrable Geodesic Ows on the Torus and on the Klein Bottle

1. If the geodesic ow of a metric G on the torus T 2 is quadrati-cally integrable then the torus T 2 isometrically covers a torus with a Liouville metric on it. 2. The set of quadratically integrable geodesic ows on the Klein bottle is described. x1. Introduction Let M 2 be a smooth close surface with a Riemannian metric G on it. The metric allows to canonically identify the tangent and the co-tangent bundles of the surface M 2. Therefore we have a scalar product and a norm in every co-tangential plane. Deenition 1 Hamiltonian system on the co-tangent plane with the Hamil-tonian H def = jpj 2 is called the geodesic ow of the metric G. It is known that the trajectories of the geodesic ow project (under the natural projection , (x; p) def = x) in the geodesics. Deenition 2 A geodesic ow is called integrable if it is integrable as the Hamiltonian system. That is there exists a function F : T M 2 ! R such that: F is constant on the trajectories, F is functionally independent with H.