Inverse problems for the connection Laplacian

Gabriel P. Paternain; Lauri Oksanen; Yaroslav Kurylev

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Inverse problems for the connection Laplacian

Yaroslav Kurylev, Lauri Oksanen, Gabriel P. Paternain

Analysis of PDEs · Submitted 9 Sep 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian.

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We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian.	contribution 0.58	0.92	0.91	asserted 1.00
s1	The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem.	result 0.78	0.84	0.13	asserted 1.00
s2	As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.	result 0.90	0.93	0.13	asserted 1.00

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The paper

Original abstract

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.

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9 arXiv papers cite this in 13 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Uses its method or tool 1 paper

We refer to [ref] for a detailed exposition of gluing arguments that can be used to construct an isometric copy of $(Ω_1, g)$.

Hyperbolic inverse problem with data on disjoint sets · Reconstruction of the geometry · 0.93

Compares with it 1 paper

We have the following approximate controllability result that is analogous to [ref] and [ref] .

Hyperbolic inverse problem with data on disjoint sets · Approximate controllability · 0.96

Cites it as background 4 papers

We refer to [ref] for state-of-the-art results and to [ref] for reviews.

Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations · Previous literature · 0.98
There exists a large number of variants of the Boundary Control method in the theoretical literature, see e.g. the review [ref] , the monograph [ref] , and the recent theoretical uniqueness [ref] and stability results [ref] .

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction · Introduction · 1.00
Local reconstruction of the geometry from the local source-to-solution map $Λ_χ$ has been studied as a part of iterative schemes, see e.g. [ref] .

Correlation based passive imaging with a white noise source · Previous literature · 0.80

What it cites · 1 sentence without a confident label

Cites as background 1 paper

Let us mention that there is a recent stability result for Gel'fand's inverse interior spectral problem [ref] .

Reconstruction and stability in Gel'fand's inverse interior spectral problem · Introduction · 0.94

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