On the s-injectivity of the X-ray transform on manifolds with hyperbolic trapped set

TL;DR

This paper proves the injectivity of the X-ray transform on certain manifolds with hyperbolic trapped sets, linking it to the surjectivity of an associated operator, and extends results to all tensor orders on surfaces.

Contribution

It establishes an equivalence principle connecting X-ray transform injectivity and operator surjectivity on manifolds with hyperbolic trapped sets, and proves injectivity for all tensor orders on surfaces.

Findings

Injectivity of the X-ray transform on solenoidal tensors is equivalent to surjectivity of a specific operator.

Injectivity is established for tensors of any order on surfaces under the given conditions.

The results apply to manifolds with strictly convex boundary, no conjugate points, and hyperbolic trapped sets.

Abstract

For smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set, we prove an equivalence principle concerning the injectivity of the X-ray transform $I_m$ on symmetric solenoidal tensors and the surjectivity of an operator ${\pi_m}_*$ on the set of solenoidal tensors. This allows us to establish the injectivity of the X-ray transform on solenoidal tensors of any order in the case of a surface satisfying these assumptions.