Uniqueness in an Integral Geometry Problem and an Inverse Problem for the Kinetic Equation

TL;DR

This paper proves the uniqueness of a Riemannian metric determination from boundary distances by reducing the problem to an inverse source problem for a kinetic equation and applying Fourier analysis in semi-geodesic coordinates.

Contribution

It introduces a novel approach linking integral geometry and inverse kinetic problems to establish uniqueness in metric reconstruction.

Findings

Uniqueness theorem for Riemannian metric recovery from boundary data

Reduction of the geometric problem to an inverse source problem

Application of Fourier analysis in semi-geodesic coordinates

Abstract

In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation on a Riemannian manifold and then the uniqueness theorem is proved in semi-geodesic coordinates by using the tools of Fourier analysis.