On Radon transforms on compact Lie groups

TL;DR

This paper characterizes when the Radon transform on compact Lie groups is injective, showing it depends on the topology of the group's connected components, with implications for smooth functions and distributions.

Contribution

It provides a complete topological characterization of injectivity of the Radon transform on compact Lie groups, linking it to the absence of certain geodesic structures.

Findings

Radon transform is injective iff components are not homeomorphic to S^1 or S^3

Injectivity holds for both smooth functions and distributions

Key techniques involve geodesic tori and symmetric operator families

Abstract

We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.