Campuses:

Geometries on the Space of Planar Shapes - Geodesics and Curvatures

Tuesday, April 4, 2006 - 9:00am - 10:00am
EE/CS 3-180
Peter Michor (Universität Wien)
The L2 or H0 metric on the space of smooth plane regular
closed curves induces vanishing geodesic distance on the quotient
Imm(S1,R2)/Diff(S1).
This is a general phenomenon and holds on all full
diffeomorphism groups and
spaces Imm(M,N)/Diff(M) for a compact manifold M and a
Riemanninan manifold
N. Thus we have to consider more complicated Riemannian metrics
using lenght or
curvature, and we do this
is a systematic Hamiltonian way, we derive geodesic equation
and split them
into
horizontal and vertical parts, and compute all conserved
quantities via the
momentum mappings of several invariance groups
(Reparameterizations,
motions, and even scalings).
The resulting equations are relatives of well known completely
integrable
systems (Burgers, Camassa Holm, Hunter Saxton).