The boundary rigidity problem in the presence of a magnetic field

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dc.creatorDairbekov, N. S.
dc.creatorPaternain, G. P.
dc.creatorStefanov, P.
dc.creatorUhlmann, G.
dc.date2006-11-25
dc.date2007-05-16
dc.date.accessioned2026-07-07T08:01:45Z
dc.date.available2026-07-07T08:01:45Z
dc.descriptionFor 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 $\mathcal{G}$ including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple $g$, $α$. Also, rigidity holds for metrics in a given conformal class, and locally, near any $(g,α)\in \mathcal{G}$.
dc.descriptionThis revised version contains a proof that 2D simple magnetic systems are boundary rigid. Some references have been added
dc.identifierhttps://arxiv.org/abs/math/0611788
dc.identifierhttp://arxiv.org/abs/math/0611788
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/129012
dc.subjectDifferential Geometry
dc.subjectAnalysis of PDEs
dc.titleThe boundary rigidity problem in the presence of a magnetic field
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