Calder\'on problem for systems via complex parallel transport

TL;DR

This paper advances the Calderón problem for systems with unknown lower order terms by establishing uniqueness results using complex parallel transport and ray transform concepts on certain Riemannian manifolds.

Contribution

It introduces new methods involving complex ray transform and parallel transport to prove uniqueness in Calderón problems for systems with connections and potentials.

Findings

Unique determination of connection and potential from boundary data

Development of complex ray transform and parallel transport concepts

Extension of results to manifolds embedded in Euclidean space

Abstract

We consider the Calder\'on problem for systems with unknown zeroth and first order terms, and improve on previously known results. More precisely, let $(M, g)$ be a compact Riemannian manifold with boundary, let $A$ be a connection matrix on $E = M \times \mathbb{C}^r$ and let $Q$ be a matrix potential. Let $\Lambda_{A, Q}$ be the Dirichlet-to-Neumann map of the associated connection Laplacian with a potential. Under the assumption that $(M, g)$ is isometrically contained in the interior of $(\mathbb{R}^2 \times M_0, c(e \oplus g_0))$, where $(M_0, g_0)$ is an arbitrary compact Riemannian manifold with boundary, $e$ is the Euclidean metric on $\mathbb{R}^2$, and $c > 0$, we show that $\Lambda_{A, Q}$ uniquely determines $(A, Q)$ up to natural gauge invariances. Moreover, we introduce new concepts of complex ray transform and complex parallel transport problem, and study their fundamental properties and relations to the Calder\'on problem.