The geometry and curvature of shape spaces

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.