Properties of Sobolev-type metrics in the space of curves

2006/04/30 by Andrea Mennucci, A. C. G. Mennucci, Anthony Yezzi +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Computer science #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Infinity #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Pure mathematics #Sobolev space #Space (punctuation) #Tangent #Tangent space #Type (biology) #math.DG

paper · pdf · doi:10.4171/ifb/196

arxiv created 2006/04/30 · openalex publication_date 2008/12/31 · arxiv updated 2013/06/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We define a manifold M where objects c∈ M are curves, which we parameterize as c:S¹→ \mathbb Rⁿ ( n≥ 2 , S¹ is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics H^j . These metrics have been shown to regularize gradient flows used in computer vision applications, see [13], [14], [16] and references therein. We provide some basic results of H^j metrics; and, for the cases j=1,2 , we characterize the completion of the space of smooth curves. We call these completions H¹ and H² Sobolev-type Riemannian Manifolds of Curves.” This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics. As a byproduct, we prove that the Fréchet distance of curves (see [7]) coincides with the distance induced by the “Finsler L^∞ metric” defined in §2.2 of [18]

Citations

• Metrics in the space of curves
• The inverse function theorem of Nash and Moser
• Fundamentals of Differential Geometry

Cited by

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups