6533b860fe1ef96bd12c3b70

RESEARCH PRODUCT

The X-Ray Transform for Connections in Negative Curvature

Mikko SaloGabriel P. PaternainGunther UhlmannGunther UhlmannGunther UhlmannColin Guillarmou

subject

Mathematics - Differential GeometryPure mathematicsHermitian bundlesGeodesic[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]Connection (vector bundle)Boundary (topology)Dynamical Systems (math.DS)X-ray transforms01 natural sciencesinversio-ongelmatHiggs fieldsTensor fieldMathematics - Analysis of PDEsFOS: MathematicsSectional curvatureMathematics - Dynamical Systems0101 mathematicsmath.APMathematical PhysicsPhysicsX-ray transformParallel transport010102 general mathematicsStatistical and Nonlinear Physicsconnections010101 applied mathematicsHiggs fieldmath.DGDifferential Geometry (math.DG)[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Mathematics::Differential Geometrymath.DSAnalysis of PDEs (math.AP)[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]

description

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.

yearjournalcountryeditionlanguage
2016-04-01Communications in Mathematical Physics
https://doi.org/10.1007/s00220-015-2510-x