The Geodesic X-ray Transform with a GL(n,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 all such transforms in a neighborhood 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.