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# Communications in Analysis and Geometry

## Volume 11 (2003)

### Number 2

### The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary

Pages: 207 – 221

DOI: http://dx.doi.org/10.4310/CAG.2003.v11.n2.a2

#### Authors

#### Abstract

We study the problem of determining a complete Riemannian manifold with boundary from the Cauchy data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to find an unknown conductivity inside a given body from measurements done on the boundary of the body. Here, we show that one can reconstruct a complete, real-analytic, Riemannian manifold *M* with compact boundary from the set of Cauchy data, given on a non-empty open subset Γ of the boundary, of all harmonic functions with Dirichlet data supported in Γ, provided dim *M ≥ 3*. We note that for this result we need no assumption on the topology of the manifold other than connectedness, nor do we need a priori knowledge of all of δ*M*.