Determination of a Riemannian manifold from the distance difference functions

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 ${g \vert}_M$ on it when we are given the distance difference data, that is, the set $F$, the metric ${g \vert}_F$, and the collection $\mathcal{D}(M) = \lbrace D_x ; x \in M \rbrace$. 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.

Keywords

inverse problems, distance functions, embeddings of manifolds, wave equation

2010 Mathematics Subject Classification

35R30, 53C22

Full Text (PDF format)

Received 2 May 2016

Accepted 11 August 2017

Published 28 June 2019