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Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

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Abstract

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.

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Authors and Affiliations

  1. Mathematisches Institut, University of Münster, 48149, Münster, Germany

    Pilar Herreros

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    James Vargo

Authors
  1. Pilar Herreros
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  2. James Vargo
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Correspondence to James Vargo.

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Communicated by Michael Taylor.

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Herreros, P., Vargo, J. Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field. J Geom Anal 21, 641–664 (2011). https://doi.org/10.1007/s12220-010-9162-z

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  • DOI: https://doi.org/10.1007/s12220-010-9162-z

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