Simple Riemannian surfaces are scattering rigid

TL;DR

This paper proves that simple two-dimensional Riemannian surfaces are scattering rigid, meaning their metrics can be uniquely determined from boundary geodesic directions, establishing a key equivalence with lens rigidity.

Contribution

It demonstrates the equivalence of scattering and lens rigidity for 2D simple manifolds, extending understanding of boundary rigidity in Riemannian geometry.

Findings

Simple 2D manifolds are scattering rigid.

Scattering rigidity is equivalent to lens rigidity in this setting.

Includes the flat disk as a key example.

Abstract

Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this implies that two-dimensional simple manifolds (such as the flat disk) are scattering rigid since they are lens/boundary rigid (Pestov--Uhlmann, 2005).