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

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 ∂M .