  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 an operator-valued
incarnation 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 family of
taming metrics $g_\epsilon$ approximating the sub-Riemannian metric, and show
that the associated X-ray transform is injective for all $\epsilon>0$. This
result gives a concrete example of an injective X-ray transform in a geometry
with an abundance of conjugate points.
