Abstract
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 figure in space, etc.
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Mumford, D. (2012). The geometry and curvature of shape spaces. In: Zannier, U. (eds) Colloquium De Giorgi 2009. Colloquia, vol 3. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-387-1_4
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DOI: https://doi.org/10.1007/978-88-7642-387-1_4
Publisher Name: Edizioni della Normale, Pisa
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