Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

TL;DR

This paper reviews different mathematical structures of shape spaces and diffeomorphism groups, focusing on Riemannian metrics, geodesic properties, and their mathematical characteristics.

Contribution

It provides a comprehensive overview of shape space geometries, emphasizing the properties of Riemannian metrics and geodesic equations in these contexts.

Findings

Analysis of geodesic distance and equations

Discussion on metric completeness and curvature

Overview of properties of Riemannian metrics

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.