Abstract
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 approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all . This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points.
Original language | English |
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Article number | 108886 |
Journal | Journal of Functional Analysis |
Volume | 280 |
Issue number | 5 |
Early online date | 11 Dec 2020 |
DOIs | |
Publication status | Published - 1 Mar 2021 |