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.

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海森堡 X 射线变换的注入性

我们开始研究亚黎曼流形上的 X 射线断层扫描，海森堡群展示了最简单的非平凡例子。使用群傅里叶变换的语言，我们证明了傅里叶切片定理的算子值化身，并应用这个新工具来证明海森堡群上的一个足够正则的函数是由它在亚黎曼测地线上的线积分确定的。我们还考虑了近似亚黎曼度量的驯服度量 $g_\epsilon$ 族，并表明相关的 X 射线变换对于所有 $\epsilon>0$ 都是单射的。该结果给出了在具有大量共轭点的几何中进行注入 X 射线变换的具体示例。