Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field

TL;DR

This paper proves that for real-analytic compact Riemannian manifolds with magnetic fields, the boundary scattering data uniquely determines the topology, metric, and magnetic field, extending previous boundary rigidity results.

Contribution

It establishes boundary rigidity for magnetic geodesics on real-analytic manifolds, allowing conjugate points and generalizing prior results significantly.

Findings

Unique determination of topology, metric, and magnetic field from scattering data

Generalization of boundary rigidity results to magnetic manifolds with conjugate points

Extension of rigidity results to real-analytic settings with minor restrictions

Abstract

Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of the magnetic geodesic flow, measured at the boundary. Conjugate points are allowed with minor restrictions. In exchange for the real-analytic assumption, prior results in boundary rigidity are greatly generalized.