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Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

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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.

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Acknowledgements

We would like to thank the referees for their careful reading of the article as well as the thoughtful comments, that helped us improve the exposition.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Martin Bauer & Peter W. Michor

  2. Institut de Mathématiques, EPFL, 1015, Lausanne, Switzerland

    Martins Bruveris

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  1. Martin Bauer
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  2. Martins Bruveris
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  3. Peter W. Michor
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Correspondence to Martins Bruveris.

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M. Bauer was supported by FWF Project P24625.

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Bauer, M., Bruveris, M. & Michor, P.W. Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. J Math Imaging Vis 50, 60–97 (2014). https://doi.org/10.1007/s10851-013-0490-z

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