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Calderón problem for systems via complex parallel transport

Mihajlo Cekić

TL;DR

This work advances the Calderón problem for systems by proving uniqueness of a connection $A$ and potential $Q$ on a compact manifold with boundary under a CTA-type embedding into $\mathbb{R}^2 \times M_0$, up to natural gauge. The authors introduce and study the complex ray transform and the complex parallel transport, developing a leafwise Del Bar framework and CGO solutions that reduce the inverse problem to a complex transport problem on a 2-frame leaf bundle $L$. By leveraging limiting Carleman weights, transport reductions, and Gaussian beam techniques, they obtain reconstruction results in both simple and arbitrary transversal geometries, including non-unitary connections and non-skew-Hermitian potentials. The approach uncovers a deep link between complexified geometric optics, transport tomography on leaf spaces, and gauge-invariant recovery, with potential extensions to Lorentzian settings and broader regularity regimes.

Abstract

We consider the Calderón 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 $Λ_{A, Q}$ 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 $Λ_{A, Q}$ 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 Calderón problem.

Calderón problem for systems via complex parallel transport

TL;DR

This work advances the Calderón problem for systems by proving uniqueness of a connection and potential on a compact manifold with boundary under a CTA-type embedding into , up to natural gauge. The authors introduce and study the complex ray transform and the complex parallel transport, developing a leafwise Del Bar framework and CGO solutions that reduce the inverse problem to a complex transport problem on a 2-frame leaf bundle . By leveraging limiting Carleman weights, transport reductions, and Gaussian beam techniques, they obtain reconstruction results in both simple and arbitrary transversal geometries, including non-unitary connections and non-skew-Hermitian potentials. The approach uncovers a deep link between complexified geometric optics, transport tomography on leaf spaces, and gauge-invariant recovery, with potential extensions to Lorentzian settings and broader regularity regimes.

Abstract

We consider the Calderón problem for systems with unknown zeroth and first order terms, and improve on previously known results. More precisely, let be a compact Riemannian manifold with boundary, let be a connection matrix on and let be a matrix potential. Let be the Dirichlet-to-Neumann map of the associated connection Laplacian with a potential. Under the assumption that is isometrically contained in the interior of , where is an arbitrary compact Riemannian manifold with boundary, is the Euclidean metric on , and , we show that uniquely determines 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 Calderón problem.
Paper Structure (24 sections, 27 theorems, 162 equations, 2 figures)

This paper contains 24 sections, 27 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Calderón problem for systems
  3. Complex ray transform and complex parallel transport.
  4. Previous work on Calderón type problems for systems.
  5. Proof ideas.
  6. Further remarks and perspectives.
  7. Organisation of the paper.
  8. Preliminaries
  9. Connection Laplacian
  10. Boundary determination
  11. The Del Bar equation
  12. Extensions.
  13. Topological and geometrical reductions.
  14. Complex Geometric Optics and the Del Bar equation
  15. Limiting Carleman weights
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(M, g)$ be a smooth connected compact Riemannian manifold with boundary, isometrically contained in the interior of $(\mathbb{R}^2 \times M_0, c(e \oplus g_0))$, where $(M_0, g_0)$ is a smooth compact Riemannian manifold with boundary and dimension $\dim M_0 \geq 1$, and $c \in C^\infty(\mathbb Moreover, if $A_1$ and $A_2$ are assumed to be unitary connections, then we may require that $G$ ta

Figures (2)

  • Figure 1: In shaded blue is the bicharacteristic ray $\mathbb{R} \times \gamma$ generated by the geodesic $\gamma$ starting at $x \in \partial M$ and with speed $v$; $N$ is the extension of $M$ constructed in § \ref{['ssec:extensions']}.
  • Figure 2: Visualisation of $\mathcal{D}_{\leq 0}(\mu, p)$ when $(M_0, g_0)$ is the interval with Euclidean metric; in shaded blue is the manifold orthogonal to $\mu$, and in dotted lines is the set of possible vectors in the neighbourhood $U \times V$; $\gamma_\nu$ is the geodesic generated by $\nu$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 50 more