text: 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.
  @cite_1: We establish a generic sufficient condition for a compact n-dimensional manifold bearing an integrable geodesic flow to be the n-torus. As a complementary result, we show that in the case of domains of possible motions with boundary, the first Betti number of the domain of possible motions may be arbitrarily large.
summary: @cite_1 , Rudnev and Ten assume that a geodesic flow is completely integrable with a non-degenerate first-integral map on an @math -dimensional compact manifold with first Betti number equal to @math . Non-degeneracy means, amongst other things, that the singular set is stratified by the rank of the first integral map and each stratum is a symplectic submanifold on which the system is completely integrable. From these hypotheses, they deduce that there is a lagrangian torus @math such that the natural map @math (figure ) is a homeomorphism . Theorem 2 of @cite_1 states that @math is a diffeomorphism, but this is mistaken. It is shown only that @math is a @math smooth map, hence by invariance of domain, a homeomorphism. To prove that @math is a diffeomorphism one must prove that the Maslov cocycle of @math vanishes, or something equivalent. This is the first difficulty in proving theorem .