Integral geometry of tensor fields on a manifold of negative curvature

Pestov, Leonid; Sharafutdinov, V. A.

doi:10.1007/bf00969652

Cited by 70 publications

(81 citation statements)

References 1 publication

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“…In the case of Riemannian manifolds, the above operators were denoted in [20,22] by h ∇ and v ∇ respectively. Given a function u ∈ C ∞ (T M \ {0}), note that ∇ · u = 0 if and only if u does not depend on y.…”

Section: 1]mentioning

confidence: 99%

Rigidity properties of Anosov optical hypersurfaces

Dairbekov¹,

Paternain²

2008

Ergod. Th. Dynam. Sys.

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Abstract. We consider an optical hypersurface Σ in the cotangent bundle τ : T * M → M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only τ * θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our work in [7]. Other rigidity issues are also discussed.

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Section: 1]mentioning

confidence: 99%

Rigidity properties of Anosov optical hypersurfaces

Dairbekov¹,

Paternain²

2008

Ergod. Th. Dynam. Sys.

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“…S-injectivity, respectively injectivity for 1-tensors (1-forms) and functions is known, see [167] for references. S-injectivity of I g was proved in [154] for metrics with negative curvature, in [167] for metrics with small curvature and in [170] for Riemannian surfaces with no focal points. A conditional and non-sharp stability estimate for metrics with small curvature is also established in [167].…”

Section: Definition 214 We Say Thatmentioning

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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“…S-injectivity, respectively injectivity for 1-tensors (1-forms) and functions is known, see [168] for references. S-injectivity of I g was proved in [155] for metrics with negative curvature, in [168] for metrics with small curvature and in [171] for Riemannian surfaces with no focal points. A conditional and non-sharp stability estimate for metrics with small curvature is also established in [168].…”

Section: Xi-35mentioning

confidence: 99%

Inverse Problems: Visibility and Invisibility

Uhlmann¹

2013

Journées Équations Aux Dérivées Partielles

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Integral geometry of tensor fields on a manifold of negative curvature

Cited by 70 publications

References 1 publication

Rigidity properties of Anosov optical hypersurfaces

Rigidity properties of Anosov optical hypersurfaces

Inverse problems: seeing the unseen

Inverse Problems: Visibility and Invisibility

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