A proof of Lens Rigidity in the category of Analytic Metrics

Vargo, James

doi:10.4310/mrl.2009.v16.n6.a13

Cited by 17 publications

(18 citation statements)

References 16 publications

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“…We replace nextZ( (z − ), z − ) byZ ˜ (z − ),z − =z + modulo errors controlled by (49), (50), so that we can get z + − z − on the left; and use the fact that the latter satisfies (49). This would allow us to conclude that the integral is "small" in U in .…”

Section: ∂Mmentioning

confidence: 99%

Boundary rigidity with partial data

Stefanov

Uhlmann

Vasy

2015

J. Amer. Math. Soc.

146

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We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary.The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces. 1 arXiv:1306.2995v5 [math.DG] 8 Oct 2015 2. Reducing the non-linear problem to a pseudo-linear one

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Section: ∂Mmentioning

confidence: 99%

Boundary rigidity with partial data

Stefanov

Uhlmann

Vasy

2015

J. Amer. Math. Soc.

146

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“…The second follows from the fact that the two systems share the same length data. See [24] for details.…”

Section: Proposition 2φ Is a Well-defined Function On Sm 1 With Valuementioning

confidence: 98%

“…This has been done in the absence of magnetic fields in [24]. The proof carries over in the presence of magnetic fields with only changes in notation.…”

Section: Theorem 4: Construction Of the Magnetic Equivalencementioning

confidence: 99%

“…Croke showed that if a manifold is lens rigid, then so is a finite quotient of that manifold [8]. In [24] lens rigidity was shown if the metrics are a priori assumed to be real analytic. Stefanov and Uhlmann proved lens rigidity for metrics that are a priori known to be close to a given base metric taken from a set of generic metrics including the real analytic ones [22,23].…”

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Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Herreros

Vargo

2010

J Geom Anal

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Consider a compact manifold M with boundary ∂M endowed with a Riemannian metric g and a magnetic field . Given a point and direction of entry at the boundary, the scattering relation 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, ) 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 . In the case that the magnetic field 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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“…Stefanov and Uhlmann [38] proved uniqueness up to diffeomorphisms fixing the boundary for metrics sufficiently close to a generic regular metrics. Vargo [43] showed that a class of analytic metrics are lens rigid. On the other hand, some interesting special cases of the lens rigidity problem with trapped geodesics are studied in [11,13].…”

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confidence: 99%

Stability for the Lens Rigidity Problem

Bao

Zhang

2017

Arch Rational Mech Anal

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Let g be a Riemannian metric for R d (d ≥ 3) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain M . The lens rigidity problem is concerned with recovering the metric g inside M from the corresponding lens relation on the boundary ∂M . In this paper, the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non-trapping metric satisfying "strong fold-regular" condition. A metric g is called strong fold-regular if for each point x ∈ M , there exists a set of geodesics passing through x whose conormal bundle covers T * x M . Moreover, these geodesics contain either no conjugate points or only fold conjugate points with a non-degeneracy condition. Examples of strong fold-regular metrics are constructed. Our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points. The approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow, which is a weighted geodesic X-ray transform problem for symmetric 2-tensor fields. A key ingredient is to show that the kernel of the X-ray transform on symmetric solenoidal 2-tensor fields is of finite dimension. It remains open whether the kernel space is trivial or not.

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A proof of Lens Rigidity in the category of Analytic Metrics

Cited by 17 publications

References 16 publications

Boundary rigidity with partial data

Boundary rigidity with partial data

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Stability for the Lens Rigidity Problem

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