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

Abstract

Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige 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 horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on $\mathbb R^2$is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.