UC Santa Cruz

We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove a quantized version of the Fourier Slice Theorem and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the families of compatible Riemannian and Lorentzian metrics $g_\epsilon$ and $g_{i\epsilon}$, respectively, taming the sub-Riemannian metric and prove that their associated X-ray transforms are injective for all $\epsilon>0$. In the latter case, we show in particular that a function on $g_{i\epsilon}$-spacetime is determined by integrals over spacelike geodesics. These results provide explicit examples of injective X-ray transforms in a geometry with an abundance of conjugate points and with metrics of mixed curvature.