1,647 research outputs found
Curvature weighted metrics on shape space of hypersurfaces in -space
Let be a compact connected oriented dimensional manifold without
boundary. In this work, shape space is the orbifold of unparametrized
immersions from to . The results of \cite{Michor118}, where
mean curvature weighted metrics were studied, suggest incorporating Gau{\ss}
curvature weights in the definition of the metric. This leads us to study
metrics on shape space that are induced by metrics on the space of immersions
of the form G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}). Here
f \in \Imm(M,\R^n) is an immersion of into and are tangent vectors at . is the standard
metric on , is the induced metric on ,
\vol(f^*\bar g) is the induced volume density and is a suitable smooth
function depending on the mean curvature and Gau{\ss} curvature. For these
metrics we compute the geodesic equations both on the space of immersions and
on shape space and the conserved momenta arising from the obvious symmetries.
Numerical experiments illustrate the behavior of these metrics.Comment: 12 pages 3 figure
Basic differential forms for actions of Lie groups
A section of a Riemannian -manifold is a closed submanifold
which meets each orbit orthogonally. It is shown that the algebra of
-invariant differential forms on which are horizontal in the sense that
they kill every vector which is tangent to some orbit, is isomorphic to the
algebra of those differential forms on which are invariant with
respect to the generalized Weyl group of this orbit, under some condition.Comment: 10 pages, ESI Preprint 87, AmSTe
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
The homotopy type of the space of degree 0 immersed plane curves
The space of all immersed closed curves of rotation degree 0 in the plane
modulo reparametrizations has the same homotopy groups as the circle times the
2-sphere.Comment: 7 page
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