The boundary rigidity problem in the presence of a magnetic field

Abstract

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α. For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α. We also show injectivity and stability for g and α in a generic class G including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple ( g , α ) , rigidity for metrics in a given conformal class, and locally, near any ( g , α ) ∈ G . We also show that simple magnetic systems on two-dimensional manifolds are always rigid.