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The boundary distance function and the Dirichlet-to-Neumann map
Abstract
We outline the proof that two dimensional simple Riemannian man-ifolds with boundary are boundary distance rigid. In addition we give, in two dimensions, a reconstruction procedure to recover the index of refraction of a bounded medium in Euclidean space from the travel times of sound waves going through the medium.
... This result is a generalization of Theorem 1.3 of [13] (see also [14]) that deals with the case when the magnetic field is absent. Our proof of Theorem 1.2 is a modification of the arguments in [13], [14] to deal with the additional difficulty of determining also the magnetic field. ...
... This result is a generalization of Theorem 1.3 of [13] (see also [14]) that deals with the case when the magnetic field is absent. Our proof of Theorem 1.2 is a modification of the arguments in [13], [14] to deal with the additional difficulty of determining also the magnetic field. The key ingredient is the interplay between the magnetic ray transform, the fiberwise Hilbert transform, and the Laplace-Beltrami operator of the underlying Riemannian metric. ...
... The needed properties of the magnetic ray transform, I, are established in Section 2. The main result of this section, Theorem 2.1, about surjectivity of the adjoint operator, I * , is a refined version of Theorem 7.3 in [5] and generalizes Theorem 3.3 of [13] (also see [15, Theorem 4.2] and [14]), giving a more precise statement even in that case. It is also valid in any dimension. ...
We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace–Beltrami operator of the underlying Riemannian metric.
... In the Euclidean case, the more precise analysis in Stefanov and Uhlmann (2003) of the singular decomposition allows one to show that the collision kernel k is uniquely determined by ; we shall perform a similar analysis here for a domain with a Riemannian metric. The determination of the metric g follows from the result of Pestov and Uhlmann (to appear) (see also Pestov and Uhlmann, 2004). They show that the scattering relation of MM gg uniquely determines the metric g in dimension two when MM gg is simple, up to a diffeomorphism that fixes the boundary of M. The scattering relation is the set of pairs x v xx vvv ∈ − × + , where xx vv is the " exit " point and direction of the geodesic which " enters " M at x with direction v . ...
... Suppose that we have two albedo operators and coming from measurements made on manifolds MM gg and MM˜ggMM˜ MM˜gg with material parameters a k andã˜kk and˜andãand㘠andã˜kk, respectively. Pestov and Uhlmann, 2004) it is proven that this implies that g = ˜ g, of course up to a choice of coordinates, that is, a diffeomorphism M → M with M = Id M . This completes the proof of part one of Theorem 1. To motivate the analysis that follows, and with Proposition 4 at our disposal, we outline the approach we shall take to prove part two of for xx vv x v ...
Optical tomography means the use of near-infrared light to determine the optical absorption and scattering properties of a medium. In the stationary Euclidean case the dynamics are modeled by the radiative transport equation, which assumes that, in the absence of interaction, particles follow straight lines. Here we shall study the problem in the presence of a Riemannian metric where particles follow the geodesic flow of the metric. In particular, we study the problem in dimension two, where the analysis is more delicate than in the higher dimensional cases.
... For general metrics, not just deformations, a theorem exists in two dimensions [22]: every strong geodesic minimizing manifold with non-positive curvature is boundary rigid. This theorem has been recently generalized to subdomains of simple manifolds in two dimensions [21]. Any compact sub-domain with smooth boundary of any dimension in a constant curvature space (Euclidean space, hyperbolic space or the open hemisphere of a round sphere) is boundary rigid [15,23,24]. ...
... In the case that the manifold is simple the scattering relation is equivalent to the boundary distance function for the two points at the boundary [15]. It has been shown in [21] that in two dimensional simple manifolds the Dirichlet-to-Neumann map is determined by the scattering relation. So in this case the scattering relation, the hodograph and the Dirichlet-to-Neumann map are related. ...
We review boundary rigidity theorems assessing that, under appropriate conditions, Riemannian manifolds with the same spectrum of boundary geodesics are isometric. We show how to apply these theorems to the problem of reconstructing a d+1 dimensional, negative curvature space-time from boundary data associated to two-point functions of high-dimension local operators in a conformal field theory. We also show simple, physically relevant examples of negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity theorems. In those examples, we explicitly show that the spectrum of boundary geodesics is not sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality. Comment: 26 pages, 4 figures
... In the scalar case the following result holds on the solvability of I * 0 [155,156]. ...
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. 65–73, 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?
... In the other direction, there are also beautiful rigidity of inverse problems for metric geometry by Gerver-Nadirashvili [250] and Pestov-Uhlmann [550] on recovering a Riemannian metric when one knows the distance functions between pair of points on the boundary, if the Riemannian manifold is reasonably convex. ...
This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S. S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I [724] worked on fundamental groups of manifolds with non-positive curvature. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much interested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S. Y. Cheng told me about these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [556] published his results right before us by different arguments. Nevertheless our
We consider Riemannian metrics in the two-dimensional disk D (with boundary). We prove that, if a metric g 0 is such that every two interior points of D are connected by a unique geodesic of g 0 , or if g 0 can be extended to a complete metric without conjugate points in R 2 , then the Riemannian area of g 0 is not greater than the area of any other metric g in which the distances between boundary points of D are not less than those in g 0 . Previously this fact was known only in the case when g 0 has constant curvature. We give a generalization of the main result to the Finslerian case and an interpretation of it in terms of simply connected Lipschitz surfaces with a fixed boundary in a Banach space.
We study the inverse problem of determining a Riemannian manifold from the boundary data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to find an unknown conductivity inside a given body from voltage and current measurements made at the boundary of the body. We show that one can reconstruct the conformal class of a smooth, compact Riemannian surface with boundary from the set of Cauchy data, given on a non-empty open subset of the boundary, of all harmonic functions. Also, we show that one can reconstruct in dimension n≥3 compact real-analytic manifolds with boundary from the same information. We make no assumptions on the topology of the manifold other than connectedness.
The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ν⩾,2 with the metric
, we consider the problem of constructing a new metricdu=nds according to the known formula
where ξ,ηεδM and Kξ,η is the geodesic in the metric du, connecting the points ξ, η. One proves uniqueness and one obtains a stability estimate
, where the refraction indices n1, n2 are the solutions of the inverse kinematic problem, constructed relative to the functions τ1, τ2, respectively,
is the differential form on δM×δM
where τ=τ2−τ1,
.
If an electrical potential is applied to the surface of a solid body, the current flux across the surface depends on the conductivity in the interior of the body. We want to consider the inverse problem of determining the conductivity by these boundary measurements
Developments in inverse problems since Calderón's foundational paper, Chapter 19 in Harmonic Analysis and Partial Differential Equations
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Two dimensional simple Riemannian manifolds
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to appear Annals of Math.