On a compact Riemannian manifold with boundary, the Dirichlet-to-Neumann map for harmonic differential k-forms sends prescribed boundary data of a form annihilated by the Hodge Laplacian to the induced normal derivative at the boundary. It is a classical pseudodifferential operator of order one, so it carries a full symbol expanding in terms homogeneous in the boundary cotangent variable. The central question is what geometry that symbol encodes: for a manifold of dimension greater than two and for every degree k, the full symbol of this operator determines the complete Taylor series of the metric at the boundary in boundary normal coordinates, extending the scalar (k=0) result of Lee and Uhlmann to forms. The proof factorizes the Hodge Laplacian near the boundary into first-order pseudodifferential factors and solves the resulting operator-valued Riccati equation symbol by symbol, with the factorization carried out in a parameter-dependent calculus so the recursion is uniform. A corollary shows the same conclusion follows from the natural Neumann-type boundary data. The problem is the differential-form analogue of the Calderon inverse conductivity problem, and it is the linear model behind inverse boundary determination for the vector Helmholtz system of electromagnetism, where the electric and magnetic fields are naturally one-forms, and for the system of linear elasticity.
A differential k-form on a compact Riemannian manifold with boundary is harmonic when it is annihilated by the Hodge Laplacian, the operator d*delta + delta*d built from the exterior derivative d and…