# An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

@article{Michor2006AnOO, title={An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach}, author={Peter W. Michor and David Mumford}, journal={Applied and Computational Harmonic Analysis}, year={2006}, volume={23}, pages={74-113} }

Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1 . We investigate several Riemannian metrics on shape space: L 2 -metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the…

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