Boundary and lens rigidity for non-convex manifolds

@article{Guillarmou2017BoundaryAL,
  title={Boundary and lens rigidity for non-convex manifolds},
  author={Colin Guillarmou and Marco Mazzucchelli and Leo Tzou},
  journal={arXiv: Differential Geometry},
  year={2017}
}
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… 

Figures from this paper

Scattering rigidity for analytic metrics

. For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular

Inverse Boundary Problems for Biharmonic Operators in Transversally Anisotropic Geometries

  • Lili Yan
  • Mathematics
    SIAM Journal on Mathematical Analysis
  • 2021
We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension n ≥ 3. We show that a

4. Integral geometry on manifolds with boundary and applications

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on

Carleman estimates for geodesic X-ray transforms

In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact

Remarks on the anisotropic Calder\'{o}n problem

. We show uniqueness results for the anisotropic Calder´on problem stated on transversally anisotropic manifolds. Moreover, we give a convexity result for the range of Dirichlet-to-Neumann maps on

Resonances and weighted zeta functions for obstacle scattering via smooth models

. We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular

Stitching Data: Recovering a Manifold’s Geometry from Geodesic Intersections

Let ( M ,  g ) be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur along the corresponding geodesics allows for

Inverse problems for nonlinear magnetic Schr\"odinger equations on conformally transversally anisotropic manifolds

We study the inverse boundary problem for a nonlinear magnetic Schrodinger operator on a conformally transversally anisotropic Riemannian manifold of dimension $n\ge 3$. Under suitable assumptions on

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

Dynamical zeta functions for billiards

. Let D ⊂ R d , d (cid:62) 2 , be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let µ j ∈ C , Im µ j > 0 be the resonances of the Laplacian in the exterior

References

SHOWING 1-10 OF 44 REFERENCES

Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for

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

Boundary and lens rigidity of finite quotients

We consider compact Riemannian manifolds (M, ∂M, g) with boundary ∂M and metric g on which a finite group Γ acts freely. We determine the extent to which certain rigidity properties of (M, ∂M, g)

Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the

The inverse problem for the local geodesic ray transform

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a

Rigidity and the distance between boundary points

In this paper we consider some rigidity problems in Riemannian geometry. In particular, we prove Theorem A. Any complete Riemannian metric without conjugate points on R" which is isometric to the

Spectral rigidity and invariant distributions on Anosov surfaces

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on

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