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⊂ N. For x∈ M we denote by Dₓ the distance difference function Dₓ:F× F→ R, given by Dₓ(z₁,z₂)=d(x,z₁)-d(x,z₂), z₁,z₂∈ F=N 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 D(M)=ₓ;\ x∈ M\. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F× F), as a representation of manifold M. The inverse problem of determining $(M,g)$ from D(M) arises e.g. in the study of the wave equation on R× N when we observe in F the waves produced by spontaneous point sources at unknown points (t,x)∈ R× M. Then Dₓ(z₁,z₂) is the difference of the times when one observes at points z₁ and z₂ 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.