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 $g{\mathrm{\mid }}_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g{\mathrm{\mid }}_F$, 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 $\mathbb R\times N$whenweobservein$F$ 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.