Rigidity and the distance between boundary points

Croke, Christopher B.

doi:10.4310/jdg/1214446326

Cited by 123 publications

(144 citation statements)

References 11 publications

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“…The theorem still holds in this case. We refer the reader to the appendix of Croke's paper [5] for the proof. In the proof of Proposition 5.2 if the past boundary N * of (M * , g * ) is totally geodesic, then the almost C 2 version of the theorem is not needed.…”

Section: Boundary Rigidity Of Product Manifoldsmentioning

confidence: 99%

“…In [5] Croke extended the Green and Gulliver result to Riemannian metrics on R n without conjugate points and which agree with the standard metric outside a compact set. In doing so he showed that this result follows from the boundary rigidity for domains in Euclidean space, a problem which had already been considered by Gromov [7] and Michel [12].…”

Section: Introductionmentioning

confidence: 99%

“…The resulting metric g * ε is continuous and smooth off the hypersurface N × {0} = N. (In the terminology of Croke [5] the metric is almost C 2 .) Using the lens equivalence of (M, g) and (M * , g * ) it is not hard to see that (M ε , g ε ) and (M * ε , g * ε ) are lens equivalent and it is clear that the past boundary {−ε} × N is totally geodesic in (M * ε , g * ε ).…”

mentioning

confidence: 99%

“…The proof given above adapts without changes to almost C 2 metrics (cf. the appendix to Croke's paper [5]). Proof.…”

mentioning

confidence: 99%

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Boundary and Lens Rigidity of Lorentzian Surfaces

Andersson

Dahl

Howard

1996

Trans. Amer. Math. Soc.

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Abstract. Let g be a Lorentzian metric on the plane R 2 that agrees with the standard metric g 0 = −dx 2 + dy 2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (R 2 , g). Then (R 2 , g) and (R 2 , g 0 ) are isometric. Further, if (M, g) and (M * , g * ) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesics of (M, g) maximize the distances between their points and (M, g) and (M * , g * ) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M * , g * ) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

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Section: Boundary Rigidity Of Product Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…The proof given above adapts without changes to almost C 2 metrics (cf. the appendix to Croke's paper [5]). Proof.…”

mentioning

confidence: 99%

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Boundary and Lens Rigidity of Lorentzian Surfaces

Andersson

Dahl

Howard

1996

Trans. Amer. Math. Soc.

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“…Assume that each integral curve of φ · ∇ x through a point in also intersects G and that the corresponding projection map → G is proper. Then we get a solution of (45) …”

Section: Construction Of Cgo Solutions With a Non-linear Phasementioning

confidence: 99%

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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Inverse scattering by obstacles

2016

Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems 2e

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Rigidity and the distance between boundary points

Cited by 123 publications

References 11 publications

Boundary and Lens Rigidity of Lorentzian Surfaces

Boundary and Lens Rigidity of Lorentzian Surfaces

Inverse problems: seeing the unseen

Inverse scattering by obstacles

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