Injectivity of the Heisenberg X-ray transform

We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which Heisenberg group exhibits simplest nontrivial example. With language Fourier transform, we prove an operator-valued incarnation Slice Theorem, and apply this new tool to show that a sufficiently regular function is determined by its line integrals over geodesics. also consider family taming metrics g? approximating metric, associated transform injective all ?>0. This result gives concrete example in geometry with abundance conjugate points.