Lens Rigidity Research Articles

Local lens rigidity for manifolds of Anosov type

Local lens rigidity for manifolds of Anosov type

Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M \times \partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem. We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points. The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric $g$, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow weights. This is handled by refining and extending our earlier results in the solenoidal gauge.

Boundary and lens rigidity for non-convex manifolds

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.

Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in R^2

It was proved in [L. Noakes and L. Stoyanov, Proc. Amer. Math. Soc., 143 (2015), pp. 3879-3893] that obstacles K in R^d that are finite disjoint unions of strictly convex domains with C^3 boundaries are uniquely determined by the traveling times of billiard trajectories in their exteriors and also by their so-called scattering length spectra. However, the case d = 2 is not covered by that article. In the present paper we give a separate and different proof of this result for the case when d = 2.

Métodos ultrassonográficos utilizados para o planejamento para a cirurgia de catarata em cães

Cataract is an important disease on veterinary ophthalmology. Different techniques facilitate the classification of the cataracts, adding information obtained by the conventional eye exam. This review discusses different modalities of ultrasonography methods available for diagnosis and surgical planning in dogs with cataracts. A-Mode ultrasound is used to evaluate thickness and length of the lens. B-Mode also has the capacity to evaluate features of echogenicity, echotexture, position of the lens and the identification of retinal and vitreous degenerations. The ultrasonic biomicroscopy is useful in assessing the anterior capsule and the positioning of the lens. The Doppler method it is useful to detection of vascular changes in patients with cataracts. The elastography method however allows measuring the lens rigidity, which is extremely important for planning and surgical prognosis.

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we 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 geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.

Lens Rigidity for a Particle in a Yang\u2013Mills Field

We consider the motion of a classical colored spinless particle under the influence of an external Yang–Mills potential A on a compact manifold with boundary of dimension {geq 3}. We show that under suitable convexity assumptions, we can recover the potential A, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points.

Inverting the local geodesic X-ray transform on tensors

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n ≥ 3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.

Lens rigidity with partial data in the presence of a magnetic field

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension \begin{document}$≥ 3$\end{document} with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in \(\mathbb {R}^n\) satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

An inverse problem from condensed matter physics

We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross–Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.

Stability for the Lens Rigidity Problem

Let g be a Riemannian metric for \({\mathbb{R}^d}\) (\({d \geqq 3}\)) 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 \({\partial M}\). 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 the “strong fold-regular” condition. A metric g is called strong fold-regular if for each point \({x\in M}\), 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.

Lens rigidity for manifolds with hyperbolic trapped sets

For a Riemannian manifold ( M , g ) (M,g) with strictly convex boundary ∂ M \partial M , the lens data consist of the set of lengths of geodesics γ \gamma with end points on ∂ M \partial M , together with their end points ( x − , x + ) ∈ ∂ M × ∂ M (x_-,x_+)\in \partial M\times \partial M and tangent exit vectors ( v − , v + ) ∈ T x − M × T x + M (v_-,v_+)\in T_{x_-} M\times T_{x_+} M . We show deformation lens rigidity for such manifolds with a hyperbolic trapped set and no conjugate points. This class contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension 2 2 , we prove that the set of end points and exit vectors of geodesics (i.e., the scattering data) determines the Riemann surface up to conformal diffeomorphism.

Lens rigidity with trapped geodesics in two dimensions

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat M\obius strip are determined by their lens data. We also see by example that the flat M\obius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.

Boundary rigidity with partial data

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.

Simple Riemannian surfaces are scattering rigid

Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this implies that two-dimensional simple manifolds (such as the flat disk) are scattering rigid since they are lens/boundary rigid (Pestov--Uhlmann, 2005).

Evaluation of iridociliary and lenticular elasticity using shear-wave elastography in rabbit eyes.

A previous study has employed shear-wave ultrasound elastographic imaging to assess corneal rigidity in an ex-vivo porcine eye model. This study employs the same modality in vivo in a rabbit eye model in order to assess lens, ciliary body and total ocular rigidity changes following the instillation of atropine and pilocarpine. Ten non-pigmented female rabbits were examined. Measurements of the lens, ciliary body and total ocular rigidity as well as lens thickness and anterior chamber depth were taken with the Aixplorer system (SuperSonic Imagine, Aix-en-Provence, France) with the SuperLinear™ SL 15-4 transducer in both eyes at baseline as well as after pilocarpine and atropine instillation. The IOP was also measured with the TonoPen tonometer. Changes in rigidity in the examined areas following atropine instillation were statistically not significant. Ciliary body rigidity was significantly increased whereas lens and total ocular rigidity were significantly reduced following pilocarpine instillation. The decrease in lens rigidity following pilocarpine was significantly associated with the respective increase in ciliary body rigidity. Shear-wave ultrasound elastography can detect in vivo rigidity changes in the anterior segment of the rabbit eye model and may potentially be applied in human eyes, providing useful clinical information on conditions in which rigidity changes play an important role, such as glaucoma, pseudoexfoliation syndrome or presbyopia.

Recovery of the C ∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the “boundary rigidity problem”. If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the Taylor series of the metric tensor (C ∞ jet) at the boundary (see Lassas et al. (Math Ann 325:767–793, 2003), Uhlmann et al. (Adv Appl Math 31:379–387, 2003)). In this paper we give a new reconstruction procedure for the C ∞ jet at non-concave points on the boundary using the localized boundary distance function. A closely related problem is the “lens rigidity problem”, which asks whether the lens data determine metric structure uniquely. Lens data include information of boundary distance function, the lengths of all geodesics, and the locations and directions where geodesics hit the boundary. We give the first examples that show that lens data cannot uniquely determine the C ∞ jet. The example include two manifolds with the same boundary and the same lens data, but different C ∞ jets. With some additional careful work, we can find examples with different C 1 jets, which means the boundaries in the two lens-equivalent manifolds have different second fundamental forms.

Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds

Let σ be the scattering relation on a compact Riemannian manifold M with nonnecessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let ` be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of σ and ` restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T ∗M ; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that σ and ` restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.

Influence of lens material and intra-ocular pressure on the outcome of non-contact tonometry over soft contact lenses

To quantify the influence of soft lens rigidity and power on the difference between intraocular pressure values (DeltaIOP) obtained by non-contact tonometry through soft lenses in situ and over the cornea (IOP). Both eyes of 25 patients, attending for either routine contact lens check up, were fitted with a (I) low water content silicone hydrogel lenses of relatively high modulus of rigidity (modulus of rigidity 1.2MPa, Focus Night and Day, Cibavision) and (II) high water content daily disposable of relatively low modulus of rigidity (0.91MPa, Focus Dailies, Cibavision). IOP measurements were taken over the cornea and repeated over the lens after insertion. Lens powers ranged from -7.50D to +6.00D. ANOVA revealed DeltaIOP was associated with both lens power and material (p<0.05). Multiple linear regression revealed relationships between DeltaIOP (y) lens power (x(1)) and IOP (x(2)) as follows (I), right eyes, y = 0.899x(1) - 0.172x(2) + 5.659 (F = 15.615, r = 0.766, p < 0.001, n = 25) (I), left eyes, y = 0.993x(1) - 0.101x(2) + 4.694 (F = 23.368, r = 0.825, p < 0.001, n = 25) (II) right eyes, y = 0.399x(1) - 0.370x(2) + 6.595 (F = 11.804, r = 0.719, p < 0.001, n = 25) (II) left eyes, y = 0.561(1) - 0.225x(2) + 4.153 (F = 28.736, r = 0.723, p < 0.001, n = 25) DeltaIOP appears to be related to lens power, material and to a lesser extent IOP. Practitioners should derive their own empirical relationship between DeltaIOP, lens power and IOP for the specific types of soft lenses they commonly use. This would improve efficiency in the screening of IOP in more susceptible soft lens wearers.