Recovering Riemannian metrics in monotone families from boundary data

We discuss the inverse problem of determining the anisotropic conductivity of a body described by a compact, orientable, Riemannian manifold M with boundary ∂M, when measurements of electric voltages and currents are taken on all of ∂M. Specifically we consider a one-parameter family of conductivity tensors, extending results obtained in [3] where the simpler Euclidean case is considered. Our problem is equivalent to the geometric one of determining a Riemannian metric in a monotone one-parameter family of metrics from its Dirichlet-to-Neumann map on ∂M.