The geodesic X-ray transform with a $GL(n,\mathbb{C})$-connection

TL;DR

This paper develops reconstruction formulas for geodesic X-ray transforms with complex connections on simple Riemannian surfaces, establishing injectivity near constant curvature metrics and for small curvature connections, with range characterizations and numerical examples.

Contribution

It introduces new injectivity results for geodesic X-ray transforms with complex connections, extending previous work to non-unitary cases and providing explicit formulas and range descriptions.

Findings

Injectivity established near constant curvature metrics.

Injectivity for connections with small curvature.

Numerical illustrations demonstrating the theory.

Abstract

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absence of connection, then for any smooth enough connection multiplied by a complex parameter, the corresponding transform is injective for all values of that parameter outside a discrete set. Range characterizations are also provided, as well as numerical illustrations.