Journal of Mathematical Imaging and Vision

, Volume 50, Issue 1–2, pp 60–97 | Cite as

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

  • Martin Bauer
  • Martins BruverisEmail author
  • Peter W. Michor


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.


Shape space Diffeomorphism group Manifolds of mappings Landmark space Surface matching Riemannian geometry 



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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Martin Bauer
    • 1
  • Martins Bruveris
    • 2
    Email author
  • Peter W. Michor
    • 1
  1. 1.Fakultät für MathematikUniversität WienViennaAustria
  2. 2.Institut de MathématiquesEPFLLausanneSwitzerland

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