Generic injectivity and stability of inverse problems for connections

TL;DR

This paper proves injectivity and stability for inverse problems involving connections and Higgs fields on Riemannian manifolds, using integral geometry and pseudolinearization, applicable under broad geometric conditions.

Contribution

It establishes the first generic injectivity and stability results for nonlinear inverse problems for connections and Higgs fields on simple and more general Riemannian manifolds.

Findings

Injectivity up to natural obstructions for generic simple metrics

Stability estimates for both linear and nonlinear problems

Results extend to manifolds with conjugate points and trapped geodesics

Abstract

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinearization argument. We show injectivity (up to natural obstructions) and stability estimates for both the linear and nonlinear problems for generic simple metrics and generic connections and Higgs fields, including the real-analytic ones. We consider the problems on simple manifolds in order to make the exposition of the main ideas clear and concise, many results of this paper are still true under much weaker geometric assumptions, in particular conjugate points and trapped geodesics are allowed and the boundary is not necessarily convex.