The X-ray transform for connections in negative curvature

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.

48 pages, 1 figure, final version to appear in CMP

References in corpus (5)

• Pollicott-Ruelle resonances for open systems
• Invariant distributions and X-ray transform for Anosov flows
• The inverse problem for the local geodesic ray transform
• Lens rigidity for manifolds with hyperbolic trapped set
• Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

Cited by in corpus (13)

• Killing and Conformal Killing tensors
• Tensor tomography on Cartan-Hadamard manifolds
• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• Isometric extensions of Anosov flows via microlocal analysis
• On the ergodicity of the frame flow on even-dimensional manifolds
• On the ergodicity of unitary frame flows on Kähler manifolds
• Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces
• Mathematical Study of Scattering Resonances
• Towards Brin's conjecture on frame flow ergodicity: new progress and perspectives
• Geodesic ray transform with matrix weights for piecewise constant functions
• Correspondence between Pestov and Weitzenböck identities
• The attenuated ray transforms on Gaussian thermostats with negative curvature
• Generic injectivity and stability of inverse problems for connections