The boundary distance function and the Dirichlet-to-Neumann map
type: Item
instance of: scholarly article
title: The boundary distance function and the Dirichlet-to-Neumann map
zbMATH Open document ID: 1067.53027
author: Leonid Pestov
author: Gunther Uhlmann
published in: Mathematical Research Letters
publication date: 2004-12-29T00:00:00Z
review text: This paper consists in a detailed exposition of some main results on the following inverse problem: given a compact Riemannian manifold \((M, g)\), it is possible to determine \(g\) knowing the action of the distance function \(d_g\) on \(\partial M\times\partial M\). The authors announce the proof, in two dimensions, of a conjecture due to Michel, stating that on a simple manifold the function \(d_g\) determines \(g\) uniquely up to an isometry which is the identity map on the boundary. We recall that \((M, g)\) is called simple if \(\partial M\) is strictly convex and any two points of \(M\) are joined by a unique geodesic. The proof of the conjecture involves the scattering relation and the Dirichlet-Neumann map \(\Lambda_g\). In fact, \((\Lambda_g,\partial M)\) uniquely determines the conformal class of \(g\), when \(M\) is a compact surface. The authors prove that two simple metrics on a compact surface with the same scattering relation have the same Dirichlet-Neumann map. Finally, given two simple conformal metrics \(g\), \(\overline g= \rho g\) on the compact surface \(M\), the authors explain a procedure which allows to recover \(\rho\) from the distance function \(d_{\overline g}(x, y)\), \(x,y\in\partial M\).
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 2123929
zbMATH Keywords: simple manifold
MaRDI profile type: Publication
full work available at URL: mrl.2004.v11.n3.a2
OpenAlex ID: W2087770593
DOI: 10.4310/MRL.2004.V11.N3.A2
description: scientific article; zbMATH DE number 2123929