Boundary rigidity and filling volume minimality of metrics close to a flat one

Burago, Dmitri; Ivanov, S. V.

doi:10.4007/annals.2010.171.1183

Cited by 79 publications

(105 citation statements)

References 20 publications

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“…Indeed, by Theorem 1 below, the case (i) follows from [52], (ii) follows from [49], and (iii) from [17]. Moreover, there is a conjecture by Gunther Uhlmann [65], that the scattering relation determines any non-trapping compact manifold with boundary.…”

Section: Introductionmentioning

confidence: 91%

Inverse Problem for the Wave Equation with a White Noise Source

Helin

Lassas

Oksanen

2014

Commun. Math. Phys.

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Abstract. We consider a smooth Riemannian metric tensor g on R n and study the stochastic wave equation for the LaplaceBeltrami operator ∂ 2 t u − ∆ g u = F . Here, F = F (t, x, ω) is a random source that has white noise distribution supported on the boundary of some smooth compact domain M ⊂ R n . We study the following formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact subset of M int and that a solution u(t, x, ω 0 ) is produced by a single realization of the source F (t, x, ω 0 ). We ask what information regarding g can be recovered by measuring u(t, x, ω 0 ) on R + × ∂M ? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold (M, g) with probability one. That is, for all geodesics passing through M , the travel times together with the entering and exit points and directions are determined. In particular, if (M, g) is a simple Riemannian manifold and g is conformally Euclidian in M , the measurement determines the metric g in M .

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Section: Introductionmentioning

confidence: 91%

Inverse Problem for the Wave Equation with a White Noise Source

Helin

Lassas

Oksanen

2014

Commun. Math. Phys.

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“…Therefore, we see that ρ 1 (t) = ρ 2 (t) throughout the interval 0 ≤ t ≤ T . In particular, we find that ρ 2 (T ) = 0, which verifies equation (2).…”

Section: Extension Of the Isometry To The Entire Manifoldmentioning

confidence: 59%

“…This has been proved recently in two dimensions [10]. It has also been proved for subdomains of Euclidean space [7], for metrics close to the Euclidean [2], and symmetric spaces of negative curvature [1]. In [13], Stefanov and Uhlmann proved a local boundary rigidity result.…”

Section: An Introduction Including the Results Provedmentioning

confidence: 91%

A proof of Lens Rigidity in the category of Analytic Metrics

Vargo

2009

Mathematical Research Letters

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Abstract. Consider a compact Riemannian manifold with boundary. If all maximally extended geodesics intersect the boundary at both ends, then to each geodesic γ(t) we can form the triple (γ(0),γ(T ), T ), consisting of the initial and final vectors of the segment as well as the length between them. The collection of all such triples comprises the lens data. In this paper, it is shown that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed, but with some restriction.

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“…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: 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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Boundary rigidity and filling volume minimality of metrics close to a flat one

Cited by 79 publications

References 20 publications

Inverse Problem for the Wave Equation with a White Noise Source

Inverse Problem for the Wave Equation with a White Noise Source

A proof of Lens Rigidity in the category of Analytic Metrics

Inverse problems: seeing the unseen

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