Determination of a Riemannian manifold from the distance difference functions

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Lassas , M & Saksala , T 2019 , ' Determination of a Riemannian manifold from the distance difference functions ' , Asian Journal of Mathematics , vol. 23 , no. 2 , pp. 173-199 . https://doi.org/10.4310/AJM.2019.v23.n2.a1

Title: Determination of a Riemannian manifold from the distance difference functions
Author: Lassas, Matti; Saksala, Teemu
Other contributor: University of Helsinki, Department of Mathematics and Statistics
University of Helsinki, Department of Mathematics and Statistics


Date: 2019-04
Language: eng
Number of pages: 27
Belongs to series: Asian Journal of Mathematics
ISSN: 1093-6106
DOI: https://doi.org/10.4310/AJM.2019.v23.n2.a1
URI: http://hdl.handle.net/10138/326781
Abstract: Let (N, g) be a Riemannian manifold with the distance function d(x, y) and an open subset M subset of N. For x is an element of M we denote by D-x the distance difference function D-x:F x F -> R, given by D-x(z(1), z(2)) = d(x, z(1)) - d(x, z(2)), z(1), z(2) is an element of F = N \ M. We consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric g vertical bar M on it when we are given the distance difference data, that is, the set F, the metric g vertical bar F, and the collection D(M) = {D-x; x is an element of M}. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F x 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 x N when we observe in F the waves produced by spontaneous point sources at unknown points (t, x) is an element of R x 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.
Subject: Inverse problems
distance functions
embeddings of manifolds
wave equation
INVERSE PROBLEM
PHOTOACOUSTIC TOMOGRAPHY
WAVE-EQUATION
ALGORITHM
RECONSTRUCTION
INTEGRABILITY
EQUIVALENCE
SCATTERING
RECOVERY
INDEX
111 Mathematics
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