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

TL;DR

This paper explores how smooth structures influence integrability of Hamiltonian systems, showing that certain topological manifolds admit real-analytic integrable flows, with implications for understanding manifold smoothness and integrability obstructions.

Contribution

It proves that smooth structures can obstruct integrability and provides examples of manifolds with integrable geodesic flows across all smooth structures.

Findings

Smooth topological n-tori with real-analytic integrable Hamiltonians are diffeomorphic to standard tori.

Witten-Kreck-Stolz 7-manifolds admit integrable geodesic flows on all smooth structures.

Additional examples include Eschenburgh and Aloff-Wallach spaces.

Abstract

This paper studies smooth obstructions to integrability and proves two main results. First, it is shown that if a smooth topological n-torus admits a real-analytically completely integrable convex hamiltonian on its cotangent bundle, then the torus is diffeomorphic to the standard n-torus. This is the first known result where the smooth structure of a manifold obstructs complete integrability. Second, it is proven that each one of the Witten-Kreck-Stolz 7-manifolds admit a real-analytically completely integrable geodesic flow on its cotangent bundle. This gives examples of topological manifolds all of whose smooth structures admit a real-analytically completely integrable convex hamiltonian on its cotangent bundle. Additional examples are provided by Eschenburgh and Aloff-Wallach spaces.