Semiglobal boundary rigidity for Riemannian metrics

Lassas, Matti; Sharafutdinov, V. A.; Uhlmann, Günther

doi:10.1007/s00208-002-0407-4

Cited by 68 publications

(78 citation statements)

References 15 publications

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“…Before starting the proof of Theorem 2.9 we recall that Michel [130] has proven that for two dimensional Riemannian manifolds with strictly convex boundary one can determine from the boundary distance function, up to the natural obstruction, all the derivatives of the metric at the boundary. This result was generalized to any dimensions in [123]. The proof of Theorem 2.9 consists in showing that from the scattering relation we can determine the traces at the boundary of conjugate harmonic functions, which is equivalent information to knowing the DN map associated to the Laplace-Beltrami operator.…”

Section: The Scattering Relation and The Dirichlet-to-neumann Mapmentioning

confidence: 96%

“…This is known for simple subspaces of Euclidean space (see [67]), simple subspaces of an open hemisphere in two dimensions (see [130]), simple subspaces of symmetric spaces of constant negative curvature [26], simple two dimensional spaces of negative curvature (see [46,149]). If one metric is close to the Euclidean metric boundary rigidity was proven in [123] that was improved in [36]. We remark that simplicity of a compact manifold with boundary can be determined from the boundary distance function.…”

Section: Introductionmentioning

confidence: 91%

“…In [178], it was proven a local result for metrics in a small neighborhood of the Euclidean one. This result was used in [123] to prove a semiglobal solvability result assuming that one metric is close to the Euclidean and the other has bounded curvature. As it was mentioned earlier it is known [167], that a linearization of the boundary rigidity problem near a simple metric g is given by the following integral geometry problem: recover a symmetric tensor of order 2, which in any coordinates is given by f = ( f i j ), by the geodesic X-ray transform…”

Section: Remarksmentioning

confidence: 99%

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Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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Section: The Scattering Relation and The Dirichlet-to-neumann Mapmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 91%

Section: Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Inverse problems: seeing the unseen

2014

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show abstract

“…Before starting the proof of Theorem 2.9 we recall that Michel [131] has proven that for two dimensional Riemannian manifolds with strictly convex boundary one can determine from the boundary distance function, up to the natural obstruction, all the derivatives of the metric at the boundary. This result was generalized to any dimensions in [122]. The proof of Theorem 2.9 consists in showing that from the scattering relation we can determine the traces at the boundary of conjugate harmonic functions, which is equivalent information to knowing the DN map associated to the Laplace-Beltrami operator.…”

Section: The Scattering Relation and The Dirichlet-to-neumann Mapmentioning

confidence: 96%

“…This is known for simple subspaces of Euclidean space (see [68]), simple subspaces of an open hemisphere in two dimensions (see [131] ), simple subspaces of symmetric spaces of constant negative curvature [27], simple two dimensional spaces of negative curvature (see [47] or [150]). If one metric is close to the Euclidean metric boundary rigidity was proven in [122] that was improved in [37]. We remark that simplicity of a compact manifold with boundary can be determined from the boundary distance function.…”

Section: Introductionmentioning

confidence: 91%