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Two Dimensional Compact Simple Riemannian Manifolds are Boundary Distance Rigid
Abstract
We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.
... . The surjectivity of I * 0 in the above sense was proved in [39] on simple manifolds of any dimension. We will show below how this result is used in the uniqueness proof of tensor tomography in two dimensions. ...
... The following commutator formula for the Hilbert transform and the geodesic vector field, proved in [39], has been a crucial component for many results reviewed in this paper. ...
... . In [39] the following characterization of the space of smooth solutions of the transport equation was given. Here we define ...
We survey recent progress in the problem of recovering a tensor field from its integrals along geodesics. We also propose several open problems.
... A compact Riemannian manifold M is simple if the boundary ∂M is strictly convex and any two points can be joined by a unique distance minimizing geodesic. Michel [18] conjectured that simple manifolds are boundary distance rigid, so far this is known for simple surfaces [20]. More recently, boundary rigidity results are established on manifolds of dimension 3 or larger that satisfy certain global convex foliation condition [25,26]. ...
... Choose p ∈ ∂M 1 and ξ, η ∈ T p ∂M 1 . By (15), (20) and (24) we have (22) is valid. ...
... Since Φ : K 1 → K 2 (see (20)) is a diffeomorphism it holds that DΦ is diffeomorphic on R E ∂M1 (V ). Let p ∈ ∂M 1 . ...
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary.
... If (M, g) is simple, then knowing α is equivalent to knowing the boundary distance function which encodes the distances between any pair of boundary points [12]. On two dimensional simple manifolds, the boundary distance function determines the metric up to an isometry which fixes the boundary [18]; (2) the scattering data of the connection A (see Section 2); (3) the fibrewise Hilbert transform at the boundary (see Section 2). Let S ∞ (∂ + (SM), C n ) denote the set of those w ∈ C ∞ (∂ + (SM), C n ) with the property that the unique solution w ♯ to the transport equation ...
... Recall that α may be extended as a C ∞ diffeomorphism of ∂(SM) [18]. ...
... This can be introduced in various ways (cf. [18,20]), but here we simply indicate that it acts fibrewise and for u k ∈ Ω k , ...
We describe the range of the attenuated ray transform of a unitary connection on a simple surface acting on functions and 1-forms. We use this to determine the range of the ray transform acting on symmetric tensor fields.
... To put our theorems into perspective, let us recall some related results. Under the additional assumption that ∂M is strictly convex, Theorem 2 was proved by Pestov-Uhlmann in [PU05], while Theorem 1 follows from [PU05] together with a result of Mukhometov [Muh81]. Within the class of negatively curved Riemannian metrics on compact 2dimensional disks, the boundary rigidity was proved by Croke [Cro90] and Otal [Ota90]; such metrics are necessarily non-trapping. ...
... To put our theorems into perspective, let us recall some related results. Under the additional assumption that ∂M is strictly convex, Theorem 2 was proved by Pestov-Uhlmann in [PU05], while Theorem 1 follows from [PU05] together with a result of Mukhometov [Muh81]. Within the class of negatively curved Riemannian metrics on compact 2dimensional disks, the boundary rigidity was proved by Croke [Cro90] and Otal [Ota90]; such metrics are necessarily non-trapping. ...
... The main difficulty in our work is the analysis of the glancing and the trapped trajectories where the exit time τ + g has discontinuities. In the non-trapping case, we use the approach of [PU05] to prove the non-linear results (Theorems 1 and 2). The main step is to prove the surjectivity of the adjoint I * 0 . ...
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.
... Moreover, it has a suitable ellipticity property when acting on solenoidal tensors [25]. This has been exploited to great effect to derive surjectivity of I * m knowing injectivity of I m [20,3] for m = 0, 1. Since the range of I * m is contained in the space of solenoidal tensors, by saying I * m is surjective we mean that the range of I * m equals the latter. ...
... Since the range of I * m is contained in the space of solenoidal tensors, by saying I * m is surjective we mean that the range of I * m equals the latter. Surjectivity of I * m for tensors of order 0 and 1 has been the key for the recent success in the solution of several long standing questions in 2D [21,20,15,14,17,5]. However, very little is known about surjectivity for m ≥ 2 and this largely motivates the present paper. ...
... As already mentioned, here we use instead the normal operator I * m I m . The results in [20,3] prove that (1) implies (4) or (5) in Theorem 1.2 for m = 0, 1, so the main contribution in the theorem is to cover the case m ≥ 2 and also to provide additional invariant distributions associated with L 2 solenoidal tensors. The proof of Theorem 1.2 relies on a solenoidal extension of tensor fields. ...
We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric m-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal m-tensors. We work with compact simple manifolds, but several of our results apply to non-trapping manifolds with strictly convex boundary.
... We note that previously known results only give that I * I is injective on L 2 (M ) and surjectivity properties were only obtained after enlarging M (as in [46]). These results are not sufficient to obtain the theorems in Section 2.3, nor do they expose the precise boundary behaviour as we do here. ...
... A property of fundamental importance is that whenever (M, g) has no conjugate points, then, in the interior of M , the operator N is an elliptic pseudo-differential operator (ΨDO) of order −1 with principal symbol c d |ξ| −1 , cf. [27, Section 6.3], [55] or Lemma 3.1 in [46]. [The reference [27] states this property under the so called Bolker condition, which is seen to be equivalent in our case to the absence of conjugate points.] ...
... Let I 1 denote the geodesic ray transform associated to (M 1 , g) and let N 1 = I * 1 I 1 . Following [46] we may cover (S, g) with finitely many simple open sets U k with M ⊂ U 1 , M ∩ U j = ∅ for j ≥ 2, and consider a partition of unity {ϕ k } subordinate to {U k } so that ϕ k ≥ 0, supp ϕ k ⊂ U k and ϕ 2 k = 1. We pick ϕ 1 such that ϕ 1 ≡ 1 on a neighborhood of M compactly supported in U 1 . ...
We consider the statistical inverse problem of recovering a function , where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms of f, corrupted by additive Gaussian noise. For M equal to the unit disk with `flat' geometry and a=0 this reduces to the standard Radon transform, but our general setting allows for anisotropic media M and can further model local `attenuation' effects -- both highly relevant in practical imaging problems such as SPECT tomography. We propose a nonparametric Bayesian inference approach based on standard Gaussian process priors for f. The posterior reconstruction of f corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator . We prove Bernstein-von Mises theorems that entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semi-parametric aspects of f. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semi-parametric Cram\'er-Rao information bound. The proofs rely on an invertibility result for the `Fisher information' operator between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.
... The following commutator formula, which was derived by Pestov and Uhlmann in [27] and generalized in [24], will play an important role. ...
... This formula has been frequently used in recent works on inverse problems, see [22,23,24,26,27,32]. ...
... Surjectivity of I * a . The aim of this section is to prove the following result which is the analogue of the corresponding results in [1,5,22,27,26]. Proof of Theorem 4.2. We embed M into the interior of a compact surface M with boundary and extend the metric g to M and keep the same notation for the extension, choosing ( M , g) to be sufficiently close to (M, g) so that it remains simple. ...
We present two range characterizations for the attenuated geodesic X-ray transform defined on pairs of functions and one-forms on simple surfaces. Such characterizations are based on first isolating the range over sums of functions and one-forms, then separating each sub-range in two ways, first by implicit conditions, second by deriving new inversion formulas for sums of functions and one-forms.
... We say that M is simple (w.r.t. (g, Ω)) if ∂M is strictly magnetic convex and the magnetic exponential map exp µ x : (exp µ x ) −1 (M ) → M is a diffeomorphism for every x ∈ M (cf. the definition of a simple Riemannian manifold [32]). ...
... Here we prove that twodimensional simple magnetic systems are magnetic boundary rigidity. This generalizes the boundary rigidity theorem of [32]. Our proof essentially resembles that in [32], establishing a connection between the scattering relation of a magnetic system and the Dirichlet-to-Neumann map of the Laplace-Beltrami operator of the underlying Riemannian manifold. ...
... This generalizes the boundary rigidity theorem of [32]. Our proof essentially resembles that in [32], establishing a connection between the scattering relation of a magnetic system and the Dirichlet-to-Neumann map of the Laplace-Beltrami operator of the underlying Riemannian manifold. ...
For a compact Riemannian manifold with boundary, endowed with a magnetic potential , we consider the problem of restoring the metric g and the magnetic potential from the values of the Ma\~n\'e action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Ma\~n\'e action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of . For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on . We also show injectivity and stability for g and in a generic class including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple g, . Also, rigidity holds for metrics in a given conformal class, and locally, near any .
... • Ideas from the proof of boundary rigidity for simple surfaces by Pestov and Uhlmann [17], and the recent proof of marked length spectrum rigidity for Anosov surfaces [7]. • A characterisation of biholomorphisms Φ : Z 1 → Z 2 as orientation preserving diffeomorphisms that are fibrewise holomorphic and induce an orbit equivalence φ = Φ| SM 1 : SM 1 → SM 2 between geodesic flows. ...
... Imposing the boundary condition φ| ∂SM = Id implies that α g 1 = α g 2 and vice versa, if the scattering relations agree, then the metrics are related by a boundary fixing conjugacy -as a consequence of the scattering rigidity proved in [17], this enforces that g 1 and g 2 are isometric via a boundary fixing isometry. Our first theorem demonstrates that this rigidity phenomenon persists, if the boundary condition is replaced by the requirement that φ holomorphically extends to transport twistor space. ...
... While our result relies on ideas from [17], the knowledge that no information is lost by focusing on transport twistor spaces suggests that conjugacy rigidity problems (cf. [8, Section 4.6]) might be amenable to using complex geometric methods on twistor space. ...
We prove that biholomorphisms between the transport twistor spaces of simple or Anosov surfaces exhibit rigidity: they must be, up to constant rescaling and the antipodal map, the lift of an orientation preserving isometry.
... A metric g on Ω is said to be simple if the boundary ∂Ω is strictly convex w.r.t. to g and any two points on Ω can be joined by a unique distance minimizing geodesic. Michel conjectured that simple metrics are boundary distance rigid [21], and this has been proved in dimension two [34]. In dimensions ⩾3, this is known for generic simple metrics [36]. ...
... Proof. Combining lemmas 3.5 and 3.6 with theorem 3.2, we get (33) for all sufficiently large k ′ > 0. To get (34), consider the event ...
... Combining this with (33) gives us (34). ...
In this paper, we consider the travel time tomography problem for conformal metrics on a bounded domain, which seeks to determine the conformal factor of the metric from the lengths of geodesics joining boundary points. We establish forward and inverse stability estimates for simple conformal metrics under some a priori conditions. We then apply the stability estimates to show the consistency of a Bayesian statistical inversion technique for travel time tomography with discrete, noisy measurements.
... General results on the inverse kinematic problem have been proven by Stefanov and Uhlmann in [36], Chung et al. [3] and Sharafutdinov [33]; a further uniqueness and stability result can be found in [37]. The 2D problem for anisotropic metrics was solved by Pestov and Uhlmann [25]; the approach contained therein is constructive. The question of a unique solution, the so called boundary rigidity problem, is not entirely solved by now. ...
... For more references concerning analytical results for the inverse kinematic problem and the boundary rigidity problem we refer to the book of Sharafutdinov [32] and the references therein. Based on the Pestov-Uhlmann reconstruction formulas from [25] Monard [19] derived a numerical solver for the linear geodesic ray transform. Another numerical solution scheme which relies on Beylkin's theory [2] is presented in [26]. ...
The article deals with a classical inverse problem: the computation of the refractive index of a medium from ultrasound time-of-flight (TOF) measurements. This problem is very popular in seismics but also for tomographic problems in inhomogeneous media. For example ultrasound vector field tomography needs a priori knowledge of the sound speed. According to Fermat's principle ultrasound signals travel along geodesic curves of a Riemannian metric which is associated with the refractive index. The inverse problem thus consists of determining the index of refraction from integrals along geodesics curves associated with the integrand leading to a nonlinear problem. In this article we describe a numerical solver for this problem scheme based on an iterative minimization method for an appropriate Tikhonov functional. The outcome of the method is a stable approximation of the sought index of refraction as well as a corresponding set of geodesic curves. We prove some analytical convergence results for this method and demonstrate its performance by means of several numerical experiments. Another novelty in this article is the explicit representation of the backprojection operator for the ray transform in Riemannian geometry and its numerical realization relying on a corresponding phase function that is determined by the metric. This gives a natural extension of the conventional backprojection from 2D computerized tomography to inhomogeneous geometries.
... In Riemannian geometry, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Pestov and Uhlmann solved the inverse problem and demonstrated that through knowing the lengths of the geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with a boundary, we can uniquely determine the Riemannian metric [16]. Now, we apply the Ramsey approach to the sets of points belonging to the compact Riemannian manifold. ...
... The problem of the closed geodesics is a celebrated mathematical problem [16][17][18][19][20][21]. Lusternik-Schnirelmann demonstrated that on a surface of the type of the 2-sphere S 2 , there are three simple geodesies [18], and Fet, in turn, demonstrated that on every compact Riemannian manifold M, there is one simple closed geodesic line [19]. ...
We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e. non-geodesic or geodesic lines, necessarily appears in a graph. We also considered the bi-colored complete, Ramsey graphs emerging from intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M_1,g_1) and (M_2,g_2), represented by the Riemann surfaces which intersect along the curve: (M_1,g_1 )∩(M_2,g_2 )=L were addressed. Curve L does not contain geodesic lines in both of manifolds (M_1,g_1) and (M_2,g_2). Consider six points located on the L: {1,…6}⊂L. The points {1,…6}⊂L are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M_1,g_1)/ red links, and, alternatively, with the geodesic lines belonging to the manifold (M_2,g_2)/ green links. Points {1,…6}⊂L form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.
... Given a smooth (C ∞ ) compact Riemannian manifold with smooth boundary, the problem asks to what extent one can recover the metric g by knowing the boundary distance function d g | ∂M ×∂M ( [23]). Assuming that (M, g) is a simple surface (see [25] Section 3.8 for a several equivalent definitions of simple manifolds), the problem was solved in [26], proving that one can recover the metric up to an isometry that fixes the boundary. Recently, the problem was solved in higher dimensions for compact simple manifolds with further geometric conditions (non-positive curvature, non-negative curvature, absence of focal points), and for manifolds with strictly convex boundary satisfying a foliation condition, see [29]. ...
... Finally, in dimension 2 we obtain boundary rigidity for MP-systems up to kgauge equivalence. This generalizes the celebrated result [26] Theorem 1.1. ...
... In 1981, Michel [9] conjectured that every simple compact Riemannian manifold with boundary is boundary rigid. The 2-dimensional case was verified by Pestov and Uhlmann [10]. In higher dimensions, however, the conjecture is wide open and has only been verified for a few classes [3,4]. ...
... Theorem 1 can be viewed as a discrete analogue of the 2-dimensional boundary rigidity result of Pestov and Uhlmann [10]. The discrete case should be more approachable than the continuous one in general. ...
The boundary rigidity problem is a classical question from Riemannian geometry: if is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric induced on the boundary ? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave's result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.
... It was conjectured in [43] that for all compact simple Riemannian manifolds the answer is affirmative. In two dimensions it was solved in [48]. For higher dimensional case the problem is still open, but different variations of it has been considered for instance in [7,9,53,54]. ...
... for all t ∈ (0, 1), where the constant is given by (48) in appendix B. ...
Consider the geometric inverse problem: there is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov–Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense
... Michel conjectured that any simple Riemannian manifold is boundary distance rigid. While this conjecture in its full generality remains open, significant progress has been made in the last few decades; see [14,12,19,29,34] for a few groundbreaking works in this direction. Linearization of the boundary distance function near a fixed simple metric [21] or linearization of the lens data (which consists of the length of geodesics together with the scattering relation) in the general case [31] leads to the integral geometry problem of determining a symmetric 2tensor field f from the knowledge its geodesic ray transform I 2 f . ...
In this article, we study the microlocal properties of the geodesic ray transform of symmetric m-tensor fields on 2-dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier work on the geodesic ray transform of functions in the presence of conjugate points, the normal operator can be decomposed into a sum of a pseudodifferential operator (DO) and a finite number of Fourier integral operators (FIOs) under the assumption of no singular conjugate pairs along geodesics, which always holds in 2-dimensions. In this work, we use the method of stationary phase to explicitly compute the principal symbol of the DO and each of the FIO components of the normal operator acting on symmetric m-tensor fields. Next, we construct a parametrix recovering the solenoidal component of the tensor fields modulo FIOs, and prove a cancellation of singularities result, similar to an earlier result of Monard, Stefanov and Uhlmann for the case of geodesic ray transform of functions in 2-dimensions. We point out that this type of cancellation result is only possible in the 2-dimensional case.
... Then, the λ-scattering relation α : ∂SM → ∂SM is given by α(x, v) := ϕτ (x, v), whereτ is defined as in lemma A.3. Similar to the approach in [PU05], we introduce the operator E : C ∞ (∂ + SM) → C ∞ (∂SM) defined as ...
We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping λ-geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when λ has the vertical Fourier degree at most 2. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when λ has higher Fourier modes.
... In particular, the linearization of the boundary rigidity problem is the geodesic ray transform of symmetric 2-tensors. It has been proved that simple surfaces are boundary rigid [28], in higher dimensions generic simple manifolds are boundary rigid [37] including the analytic ones. See also recent surveys [3,38,44] and the references therein. ...
In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such magnetic ray transforms can be stably inverted, up to natural obstructions, near a strictly convex (with respect to magnetic geodesics) boundary point. Moreover, a global invertibility result follows on a compact Riemannian manifold with strictly convex boundary assuming that some global foliation condition is satisfied.
... This was extended to non-positively curved simple metrics by Croke, Fathi, and Feldman [CFF92]. The conjecture for simple surfaces was finally established by Pestov and Uhlmann [PU05]. As for non simple metrics on compact surfaces, Croke and Herreros [CH16] proved that a negatively-curved cylinder, the flat cylinder, and the flat Möbius strip are lens rigid. ...
We show that, on an oriented compact surface, two sufficiently -close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and same marked boundary distance, extending a result of Croke and Otal.
... In the case of a simple manifold, i.e. a manifold with strictly convex boundary and such that the exponential map is a diffeomorphism at all points (such manifolds are topological balls without trapping and conjugate points), this conjecture was first stated by Michel [Mic82] in 1981, and later proved by Pestov-Uhlmann [PU05] in 2002, in the twodimensional case. It is still an open question in higher dimensions but Stefanov-Uhlmann-Vasy [SUV17] proved the rigidity of a wide range of simple (and also non-simple actually) manifolds satisfying a foliation assumption. ...
Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.
... It is known that on simple manifolds, the boundary rigidity problem and the lens rigidity problem are equivalent [27]. Michel [27] conjectured that simple manifolds are boundary rigid, Pestov and Uhlmann showed that this is true for simple surfaces [39]. In higher dimensions, Stefanov and Uhlmann proved that a generic simple metric is boundary rigid [45], they also gave stability estimates. ...
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 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.
... We also prove a local version of this result. The lens rigidity problem, and the closely related boundary rigidity problem are well studied, see, e.g., [5,6,7,10,13,14,15,17,20] and for some classes of metrics, g can be recovered up to isometry. The last two works deal with non-simple metrics. ...
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
... These inverse boundary problems were solved, at least on the level of uniqueness and sometimes conditional stability, for the Laplace-Beltrami operator and also the other types of scalar operators, in e.g. [1]- [6], [16,18,19], [23]- [25], [28]- [30], [33,34] and monographs [13] or [17] with further references therein. ...
In this paper we consider two inverse problems on a closed connected Riemannian manifold (M,g). The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that M is divided by a hypersurface into two components and we know the eigenvalues of the Laplace operator on (M,g) and also the Cauchy data, on , of the corresponding eigenfunctions , i.e. , where is the normal to . We prove that these data determine (M,g) uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, and only. However, if consists of at least two components, , we are still able to determine (M,g) assuming some conditions on M and . These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting M along , i=1,2, and are of a generic nature. We consider also some other inverse problems on M related to the above with data which is easier to obtain from measurements than the spectral data described.
... Special cases have been proved by Michel [17], Gromov [10], Croke [8], Lassas, Sharafutdinov, and Uhlmann [13], Stefanov and Uhlmann [21], and Burago and Ivanov [5,6]. In two dimensions, the conjecture was settled by Pestov and Uhlmann [19]. Moving away from the simplicity assumption, important recent work of Stefanov, Uhlmann and Vasy solved a local version of the rigidity problem in a neighbourhood of any strictly convex point of the boundary, and obtained a corresponding global rigidity result for manifolds that admit a foliation satisfying a certain convexity condition. ...
We prove that if (M,g) is a topological 3-ball with a -smooth Riemannian metric g, and mean-convex boundary then knowledge of least areas circumscribed by simple closed curves uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) -close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves . We also prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point , we prove that knowledge of the least areas circumscribed by any simple closed curve in a neighbourhood of p uniquely determines the metric near p. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calder\'on inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.
... For further results on the reconstruction of Lorentzian manifolds, we mention Larsson's work [Lar15] using broken causal lens data or sky shadow data (see also the related [KLU10]), and the work by Lassas, Oksanen, and Yang [LOY16] on the reconstruction of the jet of a Lorentzian metric on a timelike hypersurface from time measurements. There is a large amount of literature on inverse problems on Riemannian manifolds with boundary; we refer to [PU05,SUV17] and the references therein. ...
On a time-oriented Lorentzian manifold (M,g) with non-empty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets of sources is uniquely determined by measurements of the intersection of future light cones from points in S with a fixed open subset of the boundary of M; here, light rays are reflected at according to Snell's law. Our proof is constructive, and allows for interior conjugate points as well as multiply reflected and self-intersecting light cones.
... For higher dimensions, it is proven under the additional assumption of real-analyticity. Also it is known that the scattering relation (which is defined through the geodesic in M with end points on ∂M ) determines the simple manifold ( [45], [52]). Here, the simple manifold is a compact Riemannian manifold with strictly convex boundary, whose exponential map exp x : exp −1 ...
We study the inverse scattering for Schr{\"o}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
... The boundary rigidity problem asks if a Riemannian manifold with boundary can be determined by the knowledge of the distances between boundary points. The problem was conjectured [38] to be uniquely solvable for simple manifolds (i.e., with convex boundary and no conjugate points), and proved in two dimensions in [44]. In higher dimensions, boundary rigidity is known for simple conformal metrics [18,39], generic simple manifolds including all analytic ones [47], metrics close to Euclidean or hyperbolic [13,12,19,26,36], and for manifolds foliated by strictly convex hypersurfaces [48,49]. ...
... There is a similar line of research in the continuous setting, known as the boundary rigidity problem (introduced in [11,20]), with can be stated as follows: Given a compact Riemannian manifold (M, g) with boundary ∂M , stablish under which assumptions on ∂M , the geodesic distance d g | ∂M ×∂M , uniquely determines g. For further details on this topic, see [23,29,31]. ...
... There are some important uniqueness and stability results for the TTTP. Since we are focused here on the numerical aspect, we cite only a few of those [17,18,19,21]. We also refer to [22,26] for numerical studies of the TTTP by the methods, which are significantly different from the convexification method of this paper. ...
The travel time tomography problem is a coefficient inverse problem for the eikonal equation. This problem has well known applications in seismic. The eikonal equation is considered here in the circular cylinder, where point sources run along its axis and measurements of travel times are conductes on the whole surface of this cylinder. A new version of the globally convergent convexification numerical method for this problem is developed. Results of numerical studies are presented.
... A lot of progress has been made towards boundary rigidity. Pestov and Uhlmann [21] proved the above conjecture in dimension 2. In higher dimensions, regions in R n (Besikovitch [3]; Gromov [14]), in the open hemisphere S n + (Michel [18]) and in rank-1 symmetric spaces of non-compact type (following the volume entropy rigidity theorem by Besson, Courtois and Gallot [4]) are known to be boundary rigid. Burago and Ivanov proved the boundary rigidity for almost Euclidean ( [7]) and almost real hyperbolic ( [8]) regions. ...
This paper generalizes D. Burago and S. Ivanov’s work (Duke Math J 162:1205–1248, 2013) on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics.
... Michel [15] conjectured that any simple compact Riemannian manifold with boundary is boundary rigid. The case of 2dimensional Riemannian manifolds was confirmed in [14]. In higher dimensions, the conjecture is widely open. ...
In this note, we prove that finite CAT(0) cube complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). This result was conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The reconstruction of a finite cell complex from the boundary distances is the discrete version of the boundary rigidity problem, which is a classical problem from Riemannian geometry. In the proofs, we use the bijection between CAT(0) cube complexes and median graphs and the corner peelings of median graphs.
... In higher dimensions, the geodesic ray transforms are fairly well-understood in negative curvature [25,54,57], and when a manifold has a strictly convex foliation [14,59,67,69]. The geodesic ray transform is closely related to the boundary rigidity problem [60,68] and the spectral rigidity of closed Riemannian manifolds [22,23,56]. Other recent considerations include generalizations of many existing results to some classes of open Riemannian manifolds [18,21,24,44] and to the matrix weighted ray transforms [38,59] as well as their statistical analysis [45,46]. ...
We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.
... In particular, simple manifolds are of Anosov type; in this case, P in (1.1) is simply reduced to ∂ × ∂ (there is a unique geodesic connecting boundary points) and d g is called the boundary distance function. Taking to be a disk, Theorem 1.1 implies as a corollary the boundary rigidity result of Pestov-Uhlmann [36]: two simple disks with same boundary distance function are isometric (via a boundary-preserving isometry). This will be further discussed in Corollary 1.5 below. ...
Let Σ be a smooth compact connected oriented surface with non-empty boundary. A Riemannian metric on Σ is said to be of Anosov type if it has strictly convex boundary, no conjugate points, and a hyperbolic trapped set. We prove that two Riemannian metrics of Anosov type with the same marked boundary distance are isometric (via a boundary-preserving isometry isotopic to the identity). As a corollary, we retrieve the boundary distance rigidity result for simple disks of Pestov and Uhlmann (Ann Math (2) 161(2):1093–1110, 2005). The proof rests on a new transfer principle showing that, in any dimension, the marked length spectrum rigidity conjecture implies the marked boundary distance rigidity conjecture under the existence of a suitable isometric embedding into a closed Anosov Riemannian manifold. Such an isometric embedding result for open Riemannian surfaces of Anosov type was proved by the first author with Chen and Gogolev (Journal de l’École polytechnique-Math. Tome 10:945–987, 2023) while the marked length spectrum rigidity for closed Anosov Riemannian surfaces was established by the second author with Guillarmou and Paternain (Marked length spectrum rigidity for Anosov surfaces. arXiv e-prints, arXiv:2303.12007, 2023).
... Similar to the approach in [PU05], we introduce the operator E : C ∞ (∂ + SM) → C ∞ (∂SM) defined as ...
We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping -geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when has the vertical Fourier degree at most 2. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when has higher Fourier modes.
... In higher dimensions, the geodesic ray transforms are fairly well-understood in negative curvature [GPSU16,PS21,PSU15], and when a manifold has a strictly convex foliation [dHUZ19, PSUZ19, SUV18, UV16]. The geodesic ray transform is closely related to the boundary rigidity problem [PU05,SUV21] and the spectral rigidity of closed Riemannian manifolds [GL19, GLP23,PSU14]. Other recent considerations include generalizations of many existing results to some classes of open Riemannian manifolds [EG22, GGSU19,GLT22,LRS18] and to the matrix weighted ray transforms [IR20,PSUZ19] as well as their statistical analysis [MNP21a,MNP21b]. ...
We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.
... The boundary rigidity problem asks if a Riemannian manifold with boundary can be determined by the knowledge of the distances between boundary points. The problem was conjectured [35] to be uniquely solvable for simple manifolds (i.e., with convex boundary and no conjugate points), and proved in two dimensions in [41]. In higher dimensions, boundary rigidity is known for simple conformal metrics [36,17], generic simple manifolds including all analytic ones [44], metrics close to Euclidean or hyperbolic [25,18,33,12,11], and for manifolds foliated by strictly convex hypersurfaces [45,46]. ...
We consider an inverse problem for a finite graph (X,E) where we are given a subset of vertices and the distances of all vertices . The distance of points is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when (X,E) is a tree and B is the set of leaves of the tree, the graph (X,E) can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph (X,E), or one of those, which has a given number of vertices and the required distances between vertices in B. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only qubits, the same order as the number of elements in the adjacency matrix of (X,E). It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.
... A well-known conjecture in this field is the boundary distance conjecture of Michel [2], which suggests that in a simple Riemannian manifold with boundary, the collection of distances between boundary points determines the Riemannian structure, up to an isometry. To date, Michel's conjecture has been proven only in two dimensions, by Pestov and Uhlmann [3]. ...
The collection of distances between pairs of points in a discrete set can provide information about a Riemannian manifold. For example, in [arXiv:2004.08621] it was shown that if M is a complete and connected Riemannian surface, and the distances between points in some discrete subset of M correspond to those of a net in , then M is isometric to the Euclidean plane. Whether the n-dimensional analog of this theorem holds true is still unknown. In this paper we prove that if M is a complete and connected n-dimensional Riemannian manifold, and is a discrete subset such that the distances between points of X correspond to those of a net in , then X is a net in M. Loosely speaking, this means that there are no mesoscopic portions of the manifold that are oblivious of the embedding.
We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it ) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [Pestov-Uhlmann, IMRN '04] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator . Numerical examples are provided at the end.
We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation known on a lateral boundary. We show that, under a non-conjugacy assumption, every defining function r(x, y) of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric; and each one of and r (up to an elliptic factor) determines the other uniquely. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary/lens rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.
This workshop continues a series of workshops whose current format originated in 1981 under then-organizers Moser and Zehnder, and whose latest iteration took place in July 2023. The general goal of this series of workshops is to discuss the latest developments in the field of dynamical systems, broadly construed, and its connections with neighboring areas of mathematics such as differential geometry, partial differential equations, and more recently contact and symplectic geometry. We continued this tradition, bringing in new participants working in areas of dynamical systems and its connections with other areas of mathematics that are currently highly active and/or showing great promise for future development. Key focus areas for the 2023 workshop include spectral rigidity for planar domains, chaotic and oscillatory motions in celestial mechanics, conformal symplectic dynamics, and relations between dynamics.he workshop by the grant DMS-2230648, “US Junior Oberwolfach Fellows”.
We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C 1 , 1 metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.
Consider a fractional operator , , for connection Laplacian on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension . We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator.
We provide new proofs based on the Myers–Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.
Part 1 Operator equations and inverse problems: definition of quasimonotonicity, the uniqueness theorem inverse problems for hyperbolic equations multidimensional inverse kinematic problems of seismics on the uniqueness of the solution of the Fredholm and Volterra first-kind integral equations on the uniqueness of a solution of integral equations of the first kind with entire kernel existence and uniqueness of a solution to an inverse problem for a parabolic equation formulas in multidimensional inverse problems for evolution equations. Part 2 Inverse problems for kinetic equations: kinetic equations an example of an inverse problems for kinetic equation one-dimensional inverse problems multidimensional inverse problems an uniqueness theorem for the solution of an inverse problem for a kinetic equation the general uniqueness theorem the effect of the "redundant" equation problem of separation differential and integro-differential identities solution-existence problems an inverse problem of mathematical biology. Part 3 Geometry of convex surfaces in the large and inverse problems of scattering theory: geometrical question of scattering theory integral equation of the first kind uniqueness existence stability. Part 4 Integral geometry: inversion formulas the uniqueness and solvability some applications the structure of Riemann spaces and problems of the integral geometry the solvability of a problem in integral geometry by integration along geodesics.
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,
.
Conditions on distinguishability of metrics by hodographs, in Methods and Algorithms of Interpretation of Seismological Informa-tion
- [ Bg
- I N Bernstein
- M L Gerver
[BG] I. N. Bernstein and M. L. Gerver, Conditions on distinguishability of metrics by hodographs, in Methods and Algorithms of Interpretation of Seismological Informa-tion, Computerized Seismology 13 (1980), Nauka, Moscow, 50–73 (in Russian).
The Analysis of liner partial differential operators III, SpringerVerlag, Berlin-Heildelberg Riemannian geometry
- L Hörmander
L. Hörmander, The Analysis of liner partial differential operators III, SpringerVerlag, Berlin-Heildelberg-New York-Tokyo, 1985.
[K]
W. Klingenberg, Riemannian geometry, second edition, de Gruyter Studies in Mathematics, Berlin-New York, 1995.