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

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.

arXiv comment42 pages. Version accepted to Journal of Geometric Analysis. v1 contains numerical simulations not included in the published version