The local magnetic ray transform of tensor fields

TL;DR

This paper investigates the invertibility of the local magnetic ray transform of symmetric tensor fields on Riemannian manifolds, providing stability and global invertibility results under certain geometric conditions.

Contribution

It establishes the stable local inversion and global invertibility of the magnetic ray transform of tensor fields on manifolds with convex boundaries, considering the magnetic flow effects.

Findings

Stable local inversion near convex boundary points

Global invertibility under foliation conditions

Handles tensor combinations of different orders

Abstract

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such magnetic ray transforms can be stably inverted, up to natural obstructions, near a strictly convex (with respect to magnetic geodesics) boundary point. Moreover, a global invertibility result follows on a compact Riemannian manifold with strictly convex boundary assuming that some global foliation condition is satisfied.