Invariant distributions and the geodesic ray transform

TL;DR

This paper explores the relationship between the injectivity of the geodesic ray transform on symmetric tensors and the existence of invariant distributions, with implications for understanding geometric structures on manifolds.

Contribution

It establishes an equivalence principle linking solenoidal injectivity of the transform to invariant distributions on compact simple manifolds.

Findings

Equivalence between solenoidal injectivity and invariant distributions.

Results extend to non-trapping manifolds with convex boundary.

Provides new tools for tensor tomography and geometric analysis.

Abstract

We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric $m$-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal $m$-tensors. We work with compact simple manifolds, but several of our results apply to non-trapping manifolds with strictly convex boundary.