Scattering rigidity for analytic metrics

TL;DR

This paper proves that for certain analytic negatively curved Riemannian manifolds with convex boundaries, the scattering map uniquely determines the manifold's topology and metric, extending to more general settings under specific conditions.

Contribution

It establishes scattering rigidity results for analytic negatively curved manifolds with convex boundary, including broader cases with no conjugate points and hyperbolic trapped sets.

Findings

Scattering map determines the manifold up to isometry.

Recovery of topology and metric from scattering data.

Results extend to more general analytic settings.

Abstract

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the metric. More generally, our result holds in the analytic category under the no conjugate point and hyperbolic trapped sets assumptions.