Scattering rigidity for analytic metrics (Q6191989)

The scattering data of a compact connected Riemannian manifold \((M,g)\) with boundary \(\partial M\) is the set of endpoints together with the normalized tangent vectors at both endpoints of the geodesics with endpoints in \(\partial M\). The geometric inverse problem considered here is to determine properties of \(M\) from this scattering data. The main result here is that for analytic negatively curved Riemannian manifolds with analytic strictly convex boundary the scattering data determines the manifold up to isometry, and in particular determines both the topology and the metric on the manifold. The result also holds for analytic manifolds satisfying the no conjugate point and hyperbolic trapped set assumptions.