Reconstruction of a compact Riemannian manifold from the scattering data of internal sources

TL;DR

This paper demonstrates that, under generic conditions, a compact Riemannian manifold with boundary can be reconstructed up to isometry solely from scattering data obtained from internal sources, specifically the exit directions of geodesics.

Contribution

It introduces a method to reconstruct a manifold from internal scattering data, extending inverse problems in geometric analysis.

Findings

Reconstruction of the manifold is possible from boundary scattering data.

The method applies under certain generic metric assumptions.

An isometric copy of the original manifold can be recovered.

Abstract

Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary.