The Non-Abelian X-Ray Transform on Asymptotically Hyperbolic Spaces
Haim Grebnev
TL;DR
This work analyzes the non-abelian X-ray transform on nontrapping asymptotically hyperbolic spaces with negative curvature. By extending Pestov-type energy methods to vector bundles and developing a robust regularity framework on the unit sphere bundle, it proves a gauge equivalence: identical X-ray data for two pairs $( abla^ ext{E}, ext{Φ})$ and $( ilde abla^ ext{E}, ilde{ ext{Φ}})$ implies a unitary gauge $Q$ with $ ilde{ abla}^ ext{E}=Q^{-1} abla^ ext{E}Q$ and $ ilde{ ext{Φ}}=Q^{-1} ext{Φ}Q$, plus a decay condition $Q- ext{id} o 0$ on $SM$. If the connection curvature vanishes, the Higgs field is injectively determined, yielding a precise nonabelian inverse problem result. The paper also establishes the geometric and analytic machinery—AH geometry, b/0-tangent and cosphere bundles, Sasaki metric, curvature operators, and a Pestov identity for bundles—that underpins the gauge theory and regularity results. These contributions advance inverse problems on noncompact geometries and have potential applications in coherent tomography and related areas. The work situates the non-abelian X-ray transform within a broader variational and geometric framework, enabling stability and uniqueness results in curved, noncompact settings.
Abstract
In this paper we formulate and prove a gauge equivalence for unitary connections and skew-Hermitian Higgs fields of suitable regularity that are mapped to the same function under the non-abelian X-ray transform on nontrapping asymptotically hyperbolic spaces with negative curvature and no nontrivial twisted conformal Killing tensor fields with certain regularity. If one furthermore fixes such a connection with zero curvature, a corollary provides an injectivity result for the non-abelian X-ray transform over skew-Hermitian Higgs fields.