Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

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.

38 pages, 10 figures. This version has many more small changes and finally agrees with the published version

References in corpus (5)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• A zoo of diffeomorphism groups on ${\mathbb{R}}^n$
• The Lagrangian Radon Transform and the Weil representation
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• The motion of whips and chains

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