An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

2006/05/31 by Peter W. Michor, David Mumford, David B. Mumford · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Equivalence of metrics #Geodesic #Geometric Analysis and Curvature Flows #Injective metric space #Mathematical analysis #Mathematics #Metric (unit) #Metric space #Morphological variations and asymmetry #Pure mathematics #Quotient #Riemannian geometry #Sobolev space #Tangent space #math.DG #math.SG #msc:58B20 #msc:58D15 #msc:58E12

paper · pdf · doi:10.1016/j.acha.2006.07.004

published as Applied and Computational Harmonic Analysis 23 (2007), 74-113. · 46 pages, some misprints corrected

arxiv created 2006/07/11 · openalex publication_date 2007/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Here shape space is either the manifold of simple closed smooth unparameterized curves in \mathbb R² or is the orbifold of immersions from S¹ to \mathbb R² modulo the group of diffeomorphisms of S¹. We investige several Riemannian metrics on shape space: L²-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on \mathbb R²is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Citations

• Metrics in the space of curves
• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
• Properties of Sobolev-type metrics in the space of curves
• On the Metrics and Euler-Lagrange Equations of Computational Anatomy

Cited by

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups