Symplectic Rigidity for Anosov Hypersurfaces

There is a canonical exact symplectic structure on the unit tangent bundle of a Riemannian manifold M given by pulling-back the symplectic two form ω and Liouville one form λ from the cotangent bundle T ∗M using the Riemannian metric. The pull-back of λ gives a contact form on level-sets of the length function on TM . The geodesic flow of M is given by the Reeb vectorfield of this contact structure, and the invariants of this flow are very important invariants of the symplectic manifold with boundary, or even, in some cases, of the open symplectic manifold. In such favorable circumstances, symplectic equivalence can apply much stronger rigidity results. For example, the following result is a straightforward application of the symplectic homology theory, see [4], and a theorem of J. Otal [18] and C. Croke [5].