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An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
Here shape space is either the manifold of simple closed smooth
unparameterized curves in or is the orbifold of immersions from
to modulo the group of diffeomorphisms of . We
investige several Riemannian metrics on shape space: -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 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 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.Comment: 46 pages, some misprints correcte
Non-natural metrics on the tangent bundle
Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the vertical and horizontal components, with scalar weights. Additionally, we explicitly clarify how to apply our and other induced metrics on the tangent bundle to vector fields where the vertical component is not constant along the fibers. We give application to the Special Orthogonal Group SO(3) as an example.Published versio
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