Lens rigidity with trapped geodesics in two dimensions

TL;DR

This paper investigates the lens and scattering rigidity of 2D surfaces with boundary, demonstrating that certain flat surfaces are uniquely determined by lens data, while others are not, and establishing rigidity results for negatively curved cylinders.

Contribution

It shows flat cylinders and M"obius strips are determined by lens data, and proves negatively curved cylinders with convex boundary are lens rigid.

Findings

Flat cylinder and M"obius strip are determined by lens data.

Flat M"obius strip is not determined by scattering data.

Negatively curved cylinders with convex boundary are lens rigid.

Abstract

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat M\"obius strip are determined by their lens data. We also see by example that the flat M\"obius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.