Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

Abstract

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

“…Finally the vanishing of K ∇ to infinite order on G b × [0, η 0 ) follows as in the proof of [UV16, Proposition 3.3] (also see [Ept20,p.45]), where it is shown that e − σR X 1+xR X χ P decays exponentially (or vanishes identically) as R → ∞, and upon taking into account that all other factors of the kernel grow at most polynomially fast as R → ∞, uniformly in η. Lemma 6.4. Let the hypotheses and notations of Lemma 6.3 be in effect.…”

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

“…The geodesic ray transform is closely related to the boundary rigidity problem [60,68] and the spectral rigidity of closed Riemannian manifolds [22,23,56]. Other recent considerations include generalizations of many existing results to some classes of open Riemannian manifolds [18,21,24,44] and to the matrix weighted ray transforms [38,59] as well as their statistical analysis [45,46].…”