The geometry and curvature of shape spaces

@inproceedings{Mumford2012TheGA,
  title={The geometry and curvature of shape spaces},
  author={David Mumford},
  year={2012}
}
The idea that the set of all smooth submanifolds of a fixed ambient finite dimensional differentiable manifold forms a manifold in its own right, albeit one of infinite dimension, goes back to Riemann. We quote his quite amazing Habilitatsionschrift: There are, however, manifolds in which the fixing of position requires not a finite number but either an infinite series or a continuous manifold of determinations of quantity. Such manifolds are constituted for example by the possible shapes of a… 
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TLDR
This paper presents an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes, and shows how the group of diffeomorphisms of S1 acts as a group of isometries on the space of shapes and can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.
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TLDR
This paper presents an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes, and shows how the group of diffeomorphisms of S1 acts as a group of isometries on the space of shapes and can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.
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TLDR
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