The X-ray transform on 2-step nilpotent Lie groups of higher rank

TL;DR

This paper establishes the injectivity and support theorem for the X-ray transform on certain 2-step nilpotent Lie groups, using a reduction principle related to geodesic properties.

Contribution

It introduces a reduction principle for manifolds with escaping geodesics and applies it to prove new results for the X-ray transform on specific Lie groups.

Findings

Injectivity of the X-ray transform on these Lie groups

Support theorem established for the transform

Reduction principle for manifolds with escaping geodesics

Abstract

We prove injectivity and a support theorem for the X-ray transform on $2$-step nilpotent Lie groups with many totally geodesic $2$-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.