Tensor tomography in periodic slabs

The X-ray transform on the periodic slab [0,1]×Tn, n≥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 L2-regular m-tensors for any m≥0. The characterization extends to more general manifolds, twisted slabs, including the Möbius strip as the simplest example.