The Dirichlet to Neumann mapping for harmonic differential forms

We show that the full symbol of the Dirichlet to Neumann map of the k-form Laplace’s equation on a Riemannian manifold (of dimension greater than 2) with boundary determines the full Taylor series of the metric at the boundary. This extends the result of Lee and Uhlmann for the case k = 0. The proof avoids the computation of the full symbol by using the calculus of pseudo-differential operators parametrized by a boundary normal coordinate and recursively calculating the principal symbol of the difference of boundary operators. It is hoped that this will inspire the further use of this technique which simplifies the proof of such uniqueness results for inverse boundary value problems for systems of partial differential equations.