Tensor tomography in periodic slabs

Abstract

The X-ray transform on the periodic slab $[0,1]\times {\mathbb{T}}^n$, $n\ge 0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\ge 0$. The characterization extends to more general manifolds, twisted slabs, including the Möbius strip as the simplest example.

11 pages