Two dimensional compact simple Riemannian manifolds are boundary distance rigid

By L. Pestov, Gunther Uhlmann. 2003

239 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/98140d4465f63292a329f980a136dd8de7ebf336

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Invariant distributions, Beurling transforms and tensor tomography in higher dimensions (2014, background): “In the context of simple manifolds a fundamental property is that I ∗ 0 I0 is an elliptic pseudo-differential operator of order −1 in the interior of M [44] and this combined with injectivity results for I0 gives rise to the desired surjectivity properties for I ∗ 0 .”
Spectral rigidity and invariant distributions on Anosov surfaces (2012, background): “For more details see [38], where it is also proved that the adjoint I∗ 0 is surjective on any simple manifold.”
Invariant distributions and X-ray transform for Anosov flows (2014, unknown): “sition 2.3. We will ﬁrst show that the wave front set of the Schwartz kernel of Π 0 is conormal to the diagonal, and then reduce the computation of the symbol to the case dealt with by Pestov-Uhlmann [PeUh]. We write if Re(λ) >0 R +(λ) = Z ǫ 0 e−λtetXdt+ e−ǫλeǫXR +(λ). where ǫ≥ 0 is small and thi”
Rigidity of broken geodesic flow and inverse problems (2007, background): “The classical boundary rigidity problem is the following (see [13], [14], [17], [19], [30], [35], [36], [37], [40], [41]): Assume that we know the distances dist(x, y) of boundary points x, y ∈ ∂M.”
Lens rigidity for manifolds with hyperbolic trapped set (2014, unknown): “lens equivalent simple surfaces (n= 2), they are isometric. This was proved by Otal [Ot] in negative curvature and by Croke [Cr1] in non-positive curvature. For general simple metrics, Pestov-Uhlmann [PeUh] proved that the scattering data determine the conformal class and, combined with 1), this sho”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Such a space was first introduced in [50] as a ’natural’ space of functions which are mapped into C∞(M) through the traditional adjoint of the X-ray transform, and the second equality is a characterization proved in [50].”
The attenuated ray transform on simple surfaces (2010, methodology): “One of the key tools will be the commutator formula for the geodesic vector field and angular Hilbert transform, established in [27] in the study of the boundary rigidity problem in two dimensions.”
Integral geometry and holography (2015, background): “Pestov and Uhlmann [64] proved that the metric structure of a two-dimensional, compact, simply connected Riemannian manifold with convex boundary and with no conjugate points is uniquely determined (up to diffeomorphism) by the lengths of boundary-toboundary geodesics.”
Boundary rigidity and filling volume minimality of metrics close to a flat one (2010, background): “The authors are grateful to Y. Kurylev and G. Uhlmann for introducing them to and giving them a taste of Inverse Problems and for many useful remarks.”
The gravity dual of a density matrix (2012, background): “A proven result of this form in the mathematics literature [13] is that for two-dimensional simple20 compact Riemannian manifolds with boundary, the bulk geometry is completely fixed by the distance function d(x, y) between points on the boundary (the lengths of the shortest geodesics connecting var”
Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017, background): “Simple surfaces with boundary are boundary rigid [20].”
Determining an Unbounded Potential from Cauchy Data in Admissible Geometries (2011, background): “For the following facts see [3], [20], [22].”