On determining a Riemannian manifold from the Dirichlet-to-Neumann map

We study the inverse problem of determining a Riemannian manifold from the boundary data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to find an unknown conductivity inside a given body from voltage and current measurements made at the boundary of the body. We show that one can reconstruct the conformal class of a smooth, compact Riemannian surface with boundary from the set of Cauchy data. given on a non-empty open subset of the boundary. of all harmonic functions. Also, we show that one can reconstruct in dimension n greater than or equal to 3 compact real-analytic manifolds with boundary from the same information. We make no assumptions on the topology of the manifold other than connectedness. (C) 2001 editions scientiliques et medicales Elsevier SAS.