Tensor tomography and frame flow ergodicity for magnetic flows in higher dimensions

TL;DR

This paper extends geodesic flow results to magnetic flows in higher dimensions, establishing tensor tomography, analyzing magnetic frame flow ergodicity, and developing new Pestov identities and geometric frameworks.

Contribution

It introduces generalized Pestov identities and a magnetic curvature tensor, advancing the understanding of magnetic flows in higher-dimensional manifolds.

Findings

Proved tensor tomography for magnetic flows.

Established ergodicity of magnetic frame flow under pinching conditions.

Developed new Pestov identities for magnetic flows.

Abstract

We extend two results from the theory of geodesic flows to the magnetic setting on manifolds of arbitrary dimension. First, we investigate the magnetic ray transform and establish a tensor tomography result. Second, we define and analyze the ergodicity of the magnetic frame flow under a pinching condition, building on work of Ceki\'{c}-Lefeuvre-Moroianu-Semmelmann. These generalizations rely on new Pestov identities tailored to the magnetic flow, which extend and improve identities derived by Dairbekov-Paternain. In the process, we develop a framework that adapts several concepts of Riemannian geometry to the magnetic context, including covariant differentiation, torsion, curvature, and Jacobi fields. Notably, our curvature tensor generalizes the magnetic sectional curvature recently proposed by Assenza.