Determination of a Riemannian manifold from the distance difference functions

TL;DR

This paper investigates how to uniquely determine the structure and metric of a Riemannian manifold from distance difference functions, with applications in wave-based inverse problems and geophysical imaging.

Contribution

It establishes conditions under which the manifold's topology, differentiable structure, and metric can be reconstructed from distance difference data.

Findings

Reconstruction of manifold structure from distance difference functions.

Application of results to wave equation inverse problems.

Insights into geophysical imaging techniques.

Abstract

Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\in F=N\setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric $g|_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g|_F$, and the collection $\mathcal D(M)=\{D_x;\ x\in M\}$. Moreover, we consider the embedded image $\mathcal D(M)$ of the manifold $M$, in the vector space $C(F\times F)$, as a representation of manifold $M$. The inverse problem of determining $(M,g)$ from $\mathcal D(M)$ arises e.g. in the study of the wave equation on $\mathbb R\times N$ when we observe in $F$ the waves produced by spontaneous point sources at unknown points $(t,x)\in \mathbb R\times M$. Then $D_x(z_1,z_2)$ is the difference of the times when one observes at points $z_1$ and $z_2$ the wave produced by a point source at $x$ that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.