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We consider the nonlinear problem of determining a connection and a Higgsfield from the corresponding parallel transport along geodesics on a Riemannianmanifold with boundary, in any dimension. The problem can be reduced to anintegral geometry question of some attenuated geodesic ray transform through apseudolinearization argument. We show injectivity (up to natural obstructions)and stability estimates for both the linear and nonlinear problems for genericsimple metrics and generic connections and Higgs fields, including thereal-analytic ones. We consider the problems on simple manifolds in order tomake the exposition of the main ideas clear and concise, many results of thispaper are still true under much weaker geometric assumptions, in particularconjugate points and trapped geodesics are allowed and the boundary is notnecessarily convex.

Generic injectivity and stability of inverse problems for connections