Tensor tomography in periodic slabs

Abstract

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.