On the Funk transform on compact symmetric spaces

TL;DR

This paper proves that on certain compact symmetric spaces, functions are uniquely determined by integrals over shortest closed geodesics, extending the Funk transform's injectivity beyond spheres.

Contribution

It establishes the injectivity of the Funk transform on irreducible compact symmetric spaces not isometric to spheres and proves a support theorem for rank one spaces.

Findings

Functions on these spaces are uniquely determined by geodesic integrals.

Injectivity of the Funk transform is extended beyond spheres.

Support theorem for rank one symmetric spaces.

Abstract

We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.