2015/10/21 by Lassas, Matti, Saksala, Teemu
#35R30 #53C22 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1510.06157
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→ \mathbb 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 \mathcal D(M)=\Dx; x∈ M\. Moreover, we consider the embedded image \mathcal 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 \mathcal D(M) arises e.g. in the study of the wave equation on \mathbb R× N when we observe in F the waves produced by spontaneous point sources at unknown points (t,x)∈ \mathbb 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.