The X-ray transform for connections in negative curvature

TL;DR

This paper investigates the injectivity and uniqueness of the X-ray transform for connections and Higgs fields on negatively curved manifolds, including cases with boundary and trapped geodesics, using energy identities and singularity analysis.

Contribution

It extends integral geometry results to higher-dimensional negatively curved manifolds with boundary and trapped geodesics, providing new injectivity and uniqueness theorems for connections and Higgs fields.

Findings

Injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle.

Connections and Higgs fields are determined up to gauge by boundary parallel transport.

Results hold for manifolds with or without boundary, including trapped geodesics.

Abstract

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.