The tensorial X-ray transform on asymptotically conic spaces

TL;DR

This paper proves the invertibility of the geodesic X-ray transform on one-forms and 2-tensors on asymptotically conic manifolds, extending previous results from functions to tensor fields using advanced pseudodifferential operator techniques.

Contribution

It introduces a modified solenoidal gauge condition to establish invertibility of the tensorial X-ray transform on asymptotically conic spaces, addressing potential tensor obstructions.

Findings

Invertibility of the geodesic X-ray transform on tensors established.

Use of 1-cusp pseudodifferential operator algebra and semiclassical foliation.

Extension of invertibility results from functions to tensor fields.

Abstract

In this paper we show the invertibility of the geodesic X-ray transform on one forms and 2-tensors on asymptotically conic manifolds, up to the natural obstruction, allowing existence of certain kinds of conjugate points. We use the 1-cusp pseudodifferential operator algebra and its semiclassical foliation version introduced and used by Vasy and Zachos, who showed the same type invertibility on functions. The complication of the invertibility of the tensorial X-ray transform, compared with X-ray transform on functions, is caused by the natural kernel of the transform consisting of `potential tensors'. We overcome this by arranging a modified solenoidal gauge condition, under which we have the invertibility of the X-ray transform.