Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

Bauer, Martin; Bruveris, Martins; Harms, Philipp; Michor, Peter W.

doi:10.1007/s10455-011-9294-9

Cited by 46 publications

(79 citation statements)

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“…In the context of shape analysis of submanifolds a similar result has been found by Michor and Mumford for the L 2 -metric on the space of unparametrized submanifolds and on the diffeomorphism group, see [53]. These results have been later extended to spaces of parametrized submanifolds and to fractional-order metrics on diffeomorphism groups, see [8,12]. For Sobolev metrics of order s ≥ 1 on spaces of submanifolds the following theorem shows that this ill-behavior cannot appear, which renders this class of metrics relevant for applications in shape analysis.…”

Section: 1supporting

confidence: 63%

Metric registration of curves and surfaces using optimal control

Bauer

Charon

Younes

2019

Handbook of Numerical Analysis

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This chapter presents an overview of recent developments in the analysis of shapes such as curves and surfaces through Riemannian metrics. We show that several constructions of metrics on spaces of submanifolds can be unified through the prism of Riemannian submersions, with shape space metrics being induced from metrics defined on the top spaces. Computing the resulting Riemannian distances involves solving geodesic matching problems with boundary conditions. To deal efficiently with such variational problems, one can rely on an auxiliary family of "chordal" distances to simplify the treatment of boundary conditions, which we use to come up with a relaxed inexact formulation of the matching problem. This also allows to turn shape matching into optimal control problems and give a common framework to address them in practice. We then specify our analysis to the cases of intrinsic shape metrics defined using invariant Sobolev metrics on parametrized immersions, outer shape metrics induced from metrics on diffeomorphism groups of the ambient space and finally a recent hybrid model that combines those two approaches.2000 Mathematics Subject Classification. 68Q25, 68R10, 68U05.

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Section: 1supporting

confidence: 63%

Metric registration of curves and surfaces using optimal control

Bauer

Charon

Younes

2019

Handbook of Numerical Analysis

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“…For Imm(S 1 , R 2 ) vanishing of the geodesic distance is proven in [5]; the proof makes use of the vanishing of the distance on B i,f (S 1 , R 2 ) and on Diff(S 1 ). Remark 4.3 In fact, this result holds more generally for the space Imm(M, R d ).…”

Section: The Space Of Riemannian Metricsmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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“…The variational formulations allow one to study analytical properties of the PDEs in relation to geometric properties of the underlying infinite-dimensional Riemannian manifold [51,48,4,5,13,34]. Most importantly, local well-posedness of the PDE, including smooth dependence on initial conditions, is closely related to smoothness of the geodesic spray on Sobolev completions of the configuration space [23].…”

Section: Introductionmentioning

confidence: 99%

Fractional Sobolev metrics on spaces of immersions

2020

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We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler-Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

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Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

Abstract: Abstract. The Virasoro-Bott group endowed with the right-invariant L 2 -metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

Cited by 46 publications

References 10 publications

Metric registration of curves and surfaces using optimal control

Metric registration of curves and surfaces using optimal control

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Fractional Sobolev metrics on spaces of immersions

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