The anisotropic conductivity inverse boundary value problem (or recons
truction problem for anisotropic electrical impedance tomography) is p
resented in a geometric formulation and a uniqueness result is proved,
under two different hypotheses, for the case where the conductivity i
s known up to a multiplicative scalar field. The first of these result
s 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 Le
e and Uhlmann and for C-3 conductivities close to constant by Sylveste
r. The apparatus of G-structures is then used to show that a conformal
mapping of a Riemannian manifold which fixes all points on the bounda
ry must be the identity. A second approach, which proves the result in
the piecewise analytic category, is a straightforward extension of th
e work of Kohn and Vogelius on the isotropic problem.