Inverse problems for differential forms on Riemannian manifolds with boundary

Abstract

Consider a real-analytic orientable connected complete Riemannian manifold \(M\) with boundary of dimension \(n\ge 2\) and let \(k\) be an integer \(1\le k\le n\). In the case when \(M\) is compact of dimension \(n\ge 3\), we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic \(k\)-forms, given on an open subset of the boundary. This extends a result of [13] when \(k=0\). In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic \(1\)-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when \(M\) is complete non-compact. In the case \(n\ge 3\), this generalizes the results of [12] when \(k=0\). In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic \(1\)-forms.