Determining Riemannian Manifolds From Nonlinear Wave Observations at a Single Point

TL;DR

This paper demonstrates that the topological, differential, and geometric structure of an unknown Riemannian manifold can be uniquely determined by measurements of nonlinear wave observations at a single point.

Contribution

It introduces a method to recover the entire structure of a Riemannian manifold from localized nonlinear wave data, advancing inverse problems in geometric analysis.

Findings

Unique determination of manifold structure from single-point data

Recovery of topological, differential, and geometric information

Applicability to inverse problems in geometric analysis

Abstract

We show that on an a-priori unknown Riemannian manifold $(M,g)$, measuring the source-to-solution map for the semilinear wave equation at a single point determines the topological, differential, and geometric structure.