The Maslov cocycle, smooth structures, and real-analytic complete integrability

This paper proves two main results. First, it is shown that if $\Sigma$ is a smooth manifold homeomorphic to the standard $n$-torus ${\bf T}^n = {\bf R}^n/{\bf Z}^n$ and $H$ is a real-analytically completely integrable convex hamiltonian on $T^* \Sigma$, then $\Sigma$ is diffeomorphic to ${\bf T}^n$. 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.