DISTANCE DIFFERENCE REPRESENTATIONS OF SUBSETS OF COMPLETE RIEMANNIAN MANIFOLDS (Spectral and Scattering Theory and Related Topics)

Abstract

Let (N, g) be a complete smooth Riemannian manifold with the distance function d(x, y), Usubset N be relatively compact, open subset with smooth boundary. We also assume that overline{U} is geodesically convex. Also, let Msubset U be an open subset with smooth boundary such that overline{M}subset U. We assume that the topology of M and metric g|_{M} are unknown. Let F=overline{U}backslash M be the observa tion domain. For xin M we denote by D_{x} the distance difference function D_{x} : Ftimes Frightarrow mathbb{R}, given by D_{x}(z_{1}, z_{2})=d(x, z_{1})-d(x, z_{2}), z_{1}, z_{2}in F. We show that the manifold M and the metric g|_{M} on it can be determined uniquely, up to an isometry, when we are given the set F, the metric g|_{F}, and the collection D(M)={D_{x};xin M} of distance difference functions. The embedded image mathcal{D}(M) of the manifold M, in the vector space C(Ftimes F), is the distance difference representation of manifold M. The inverse problem of determining (M, g) from prime D(M) arises for example 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. The results presented in this paper generalize the earlier results where N is assumed to be compact that the observation domain F is assumed to be the whole complement of M in N.