The scattering relation and the Dirichlet-to-Neumann map

We describe a connection between the scattering relation, the Hilbert transform and the geodesic X-ray transform for non-trapping Riemannian manifolds with strictly convex boundary. This connection was used to solve the boundary rigidity problem in two dimensions for simple manifolds [PU1]. The key point in this development is the proof that the scattering relation determines the Dirichlet-to-Neumann map for the Laplace-Beltrami operator. We give another application of the above mentioned connection: an inversion procedure to reconstruct the conformal factor of a metric from its boundary distance function for two dimensional simple manifolds [PU2]. We also use this connection to give a characterization of the range of the geodesic X-ray transform in terms of the scattering relation for non-trapping manifolds with strictly convex boundary. and inversion formulas, on two dimensional simple manifolds, for the geodesic X-ray transform acting on scalar functions and vector fields for metrics with constant curvature and Fredholm type inversion formulas in the general case [PU3].