Scattering Rigidity with Trapped Geodesics

We prove that the flat product metric on D × S is scattering rigid where D is the unit ball in R and n ≥ 2. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map S : U∂M → U−∂M from unit vectors V at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes V to γ′ V (T0) where γV is the unit speed geodesic determined by V and T0 is the first positive value of t (when it exists) such that γV (t) again lies in the boundary. We show that any other Riemannian manifold (M,∂M, g) with boundary ∂M isometric to ∂(D×S) and with the same scattering data must be isometric to D × S. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in (M,∂M, g) have measure 0 in the unit tangent bundle.