Inverse problems for differential forms on Riemannian manifolds with boundary

TL;DR

This paper demonstrates that the geometry and metric of certain Riemannian manifolds with boundary can be uniquely reconstructed from boundary measurements of harmonic forms, extending previous results to higher forms and non-compact cases.

Contribution

It generalizes inverse boundary value problem results for harmonic functions to harmonic differential forms of arbitrary degree on Riemannian manifolds, including non-compact cases.

Findings

Manifold and metric reconstructed from Cauchy data for harmonic k-forms

Extension of previous results from functions to differential forms

Reconstruction possible for both compact and certain non-compact manifolds

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.