Scattering and lens rigidity on some Riemannian manifolds with trapped geodesics

We discuss how to show that the flat solid torus is scattering rigid.
We will consider compact Riemannian manifolds M with boundary N.  We let IN be the unit vectors to M whose base point is on N and point inwards towards M.  Similarly we define OUT.  The scattering data (loosely speaking) of a Riemannian manifold with boundary is map from IN to OUT which assigns to each unit vector V of IN a the unit vector W in OUT.  W will be the tangent vector to the geodesic determined by V when that geodesic first hits the boundary N again.  This may not be defined for all V since the geodesic might be trapped (i.e. never hits the boundary again).  A manifold is said to be scattering rigid if any other Riemannian manifold Q with boundary isometric to N and with the same scattering data must be isometric to M.
In this talk we will discuss the scattering rigidity problem and related inverse problems.  There are a number of manifolds that are known to be scattering rigid and there are examples that are not scattering rigid. All of the known examples of non-rigidity have trapped geodesics in them.
In particular, we will see that the flat solid torus is scattering rigid.  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 any such Q  have measure 0 in the unit tangent bundle of Q.  We will also consider scattering rigidity of a number of two dimensional manifolds (joint work with Pilar Herreros) which have trapped geodesics.