Conformal uniqueness results in anisotropic electrical impedance imaging

The anisotropic conductivity inverse boundary value problem (or reconstruction problem for anisotropic electrical impedance tomography) is presented in a geometric formulation and a uniqueness result is proved, under two different hypotheses, for the case where the conductivity is known up to a multiplicative scalar field. The first of these results relies on the conductivity being determined by boundary measurements up to a diffeomorphism fixing points on the boundary, which has been shown for analytic conductivities in three and higher dimensions by Lee and Uhlmann and for C-3 conductivities close to constant by Sylvester. The apparatus of G-structures is then used to show that a conformal mapping of a Riemannian manifold which fixes all points on the boundary must be the identity. A second approach, which proves the result in the piecewise analytic category, is a straightforward extension of the work of Kohn and Vogelius on the isotropic problem.