On the Integrability of the Geodesic Flow on a Friedmann-Robertson-Walker Spacetime

We study the geodesic flow on the cotangent bundle of a Friedman-Robertson-Walker spacetime (M, g). On this bundle, the HamiltonJacobi equation is completely separable and this separability leads us to construct four linearly independent integrals in involution i.e. Poisson commuting amongst themselves and pointwise linearly independent. These integrals involve the six linearly independent Killing fields of the background metric g. As a consequence, the geodesic flow on an FRW background is completely integrable in the Liouville-Arnold sense. For the case of a spatially closed universe we construct families of invariant by the flow sub manifolds.