A support theorem for the X-ray transform on manifolds with plane covers

TL;DR

This paper proves support theorems for the X-ray transform on certain non-compact manifolds, including all simply connected 2-step nilpotent Lie groups, using plane covers and extending previous results.

Contribution

It establishes support theorems for the X-ray transform on non-compact manifolds with conjugate points, notably for all simply connected 2-step nilpotent Lie groups, and provides new examples.

Findings

Support theorem for all simply connected 2-step nilpotent Lie groups

Examples of non-homogeneous 3D manifolds with support theorems

Extension of support theorems to manifolds with conjugate points

Abstract

This article is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan.We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.