Tensor tomography in periodic slabs

TL;DR

This paper characterizes the kernel of the geodesic X-ray transform for tensor fields on periodic slabs and related manifolds, revealing limitations due to symmetry, trapped geodesics, and gauge freedom.

Contribution

It provides a comprehensive characterization of the kernel of the X-ray transform for tensor fields on periodic slabs and extends the results to more general twisted manifolds.

Findings

Kernel characterized for $L^2$-regular tensors of any order

Extension of kernel characterization to twisted slabs and manifolds like the M"obius strip

Identifies non-injectivity issues due to symmetry and trapped geodesics

Abstract

The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, 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 $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the M\"obius strip as the simplest example.