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

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

2007-08-23

arxiv.org/abs/0708.3157v3

American Journal of Mathematics. 131(5) : 1311--1336, October 2009

Mathematics

Symplectic Geometry

19 pages; v2: Proposition 4.1 is corrected. Main results are unchanged

Scientific paper

10.1353/ajm.0.0069

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.

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