  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.
