On the s-injectivity of the x-ray transform on manifolds with hyperbolic trapped set**Throughout this paper, we shall work in the smooth category, that is, all the manifolds and coordinate charts are considered to be smooth.

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 Im on symmetric solenoidal tensors and the surjectivity of an operator 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.