Topological obstructions to the integrability of geodesic flows on nonsimply connected Riemannian manifolds

Let (M, g) be a Cinfinity Riemannian manifold and &phis;t : T*M → T*M be the geodesic flow of g on the cotangent bundle T*M of M. The flow &phis; t is hamiltonian with respect to the energy H( m,p) = ½g-1( p,p) and the canonical symplectic form on T* M. The flow &phis;t is said to be integrable if there exists a dense submanifold X ⊂ T*M such that: (i) X = ∪a∈A Xalpha; (ii) for each alpha ∈ A, there is a homeomorphism halpha : Xalpha → TkxD 2n-k ; (iii) each Xalpha is &phis; t-invariant; and (iv) &phis;t is conjugated by halpha to a translation-type flow on the invariant tori Tk x {pt.}. The flow &phis;t is Cr integrable if there exists a Cr map F : T* M → R2n-k such that F|X is a proper submersion with fibres homeomorphic to Tk . The most commonly known integrable geodesic flows are those that are Liouville integrable.;This thesis examines the theorems of Kozlov, Tai˘manov and Paternain on the topology of non-simply connected manifolds with integrable geodesics flows, and constructs examples of Co (real-analytic) geodesic flows that are Cinfinity integrable but not Cinfinity integrable. These examples show that the set of Cinfinity manifolds with Cinfinity integrable geodesic flows is much richer than the set of Co manifolds with Co integrable geodesic flows.