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 , 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 .
@cite_1
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Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed.