This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus T n = R n Z n and H is a real-analytically completely integrable convex hamiltonian on T ∗Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a real-analytically completely integrable riemannian metric hamiltonian.
Taimanov @cite_27 has proven that if a compact manifold @math admits a real-analytically completely integrable geodesic flow, then @math is almost abelian of rank at most @math ; @math ; and there is an injection @math where @math . These constraints are ineffective for exotic tori.
@cite_27
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In this paper, (Liouville) integrability of geodesic flows on non-simply-connected manifolds is studied. In particular, the following result is obtained: A geodesic flow on a real-analytic Riemannian manifold cannot be integrable in terms of analytic functions if either 1) the fundamental group of the manifold contains no commutative subgroup of finite index, or 2) the first Betti number of the manifold over the field of rational numbers is greater than the dimension (the manifold is assumed to be closed). Bibliography: 11 titles.