Higher-Rank Radon Transforms on Constant Curvature Spaces

TL;DR

This paper investigates higher-rank Radon transforms on constant curvature spaces, establishing conditions for their existence, injectivity, and inversion, across Euclidean, elliptic, and hyperbolic geometries.

Contribution

It provides new sharp conditions for the existence and injectivity of these transforms and explores their behavior across different geometric models.

Findings

Sharp conditions for Radon transform existence and injectivity

Transition formulas between geometric models

Inversion formulas and support theorems

Abstract

We study higher-rank Radon transforms that take functions on $j$-dimensional totally geodesic submanifolds in the $n$-dimensional real constant curvature space to functions on similar submanifolds of dimension $k >j$. The corresponding dual transforms are also considered. The transforms are explored the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.