Here shape space is either the manifold of simple closed smooth
unparameterized curves in R2 or is the orbifold of immersions from
S1 to R2 modulo the group of diffeomorphisms of S1. We
investige several Riemannian metrics on shape space: L2-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 R2is 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.Comment: 46 pages, some misprints correcte