MSRI - Inverse Problems for Differential Forms on Riemannian Manifolds with Boundary

Abstract: We are concerned with the problem of recovery of a real-analytic Riemannian complete manifold with boundary from the set of the Cauchy data for harmonic differential forms, given on an open subset of the boundary. In the case when the manifold is compact and the dimension is three or higher, we show that the manifold and the metric can be reconstructed, up to an isometry. In the case of dimension two, it is known that only the conformal class of a compact real-analytic manifold can be determined from the knowledge of the Cauchy data for harmonic functions. Working on the level of
one-forms, in the compact real-analytic case, we are able to reconstruct a two-dimensional manifold up to an isometry. Under additional assumptions on the curvature of the manifold, we carry out the same program when the manifold is complete non-compact. This is joint work with Matti Lassas and Gunther Uhlmann.