Rigidity in fixed angle inverse scattering for Riemannian metrics

TL;DR

This paper addresses the inverse scattering problem for Riemannian metrics, demonstrating that smooth metrics differing from Euclidean outside a ball can be uniquely identified using geometric and topological methods.

Contribution

It proves a rigidity result showing that certain smooth Riemannian metrics can be distinguished from Euclidean metrics using fixed angle inverse scattering techniques.

Findings

Smooth metrics outside a ball are distinguishable from Euclidean metrics.

Uses distorted plane waves and unique continuation methods.

Establishes a rigidity theorem for inverse scattering in Riemannian geometry.

Abstract

The fixed angle inverse scattering problem for a velocity consists in determining a sound speed, or a Riemannian metric up to diffeomorphism, from measurements obtained by probing the medium with a single plane wave. This is a formally determined inverse problem that is open in general. In this article we consider the rigidity question of distinguishing a sound speed or a Riemannian metric from the Euclidean metric. We prove that a general smooth metric that is Euclidean outside a ball can be distinguished from the Euclidean metric. The methods involve distorted plane waves and a combination of geometric, topological and unique continuation arguments.