Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2013/05/31 by Martin Bauer, Martins Bruveris, Peter W. Michor · 127 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Completeness (order theory) #Diffeomorphism #Geodesic #Geodesic map #Geometric Analysis and Curvature Flows #Metric (unit) #Metric space #Morphological variations and asymmetry #Riemannian geometry #Space (punctuation) #math.AP #math.DG #msc:35Q31 #msc:58B20 #msc:58D15

paper · pdf · doi:10.1007/s10851-013-0490-z

published in Journal of Mathematical Imaging and Vision 50(1-2), 60-97 (Springer Science+Business Media) · 38 pages, 10 figures. This version has many more small changes and finally agrees with the published version

arxiv created 2013/10/31 · openalex publication_date 2014/01/08 · arxiv updated 2014/10/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• The Cauchy problem for quasi-linear symmetric hyperbolic systems
• The inverse function theorem of Nash and Moser
• Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation
• The energy functional on the Virasoro–Bott group with the L 2-metric has no local minima
• Properties of Sobolev-type metrics in the space of curves
• The Camassa–Holm equation as a geodesic flow on the diffeomorphism group
• Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces
• On the Metrics and Euler-Lagrange Equations of Computational Anatomy
• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
• A zoo of diffeomorphism groups on ℝⁿ R n

Cited by

• Shape Alignment via Allen-Cahn Nonlinear-Convection