Seminars/Colloquia/Invited Talks

Seminars

Peter Michor

From Riemannian geometries on spaces of plane curves to vanishing geodesic distance on spaces of submanifolds and of diffeomorphisms

PLACE:	Clark 314
EVENT:	CIS Seminar Series
DATE:	May 25, 2005
TIME:	4:00 - 5:00

Abstract

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of immersions from the unit circle to the plane modulo the group of diffeomorphisms of the unit circle, acting as reparameterizations. We investigate metrics G(h,k) defined as integrals over the unit circle of an expression of the form Phi(l(c), kappa(c))(theta) (h(theta).k(theta))|c'(theta)| d theta where Phi is a function of the length l(c) and of the curvature kappa(c) of the curve c and h,k are normal vector fields to c. We know most for the metric with Phi=1+A kappa(c)^2. For A=0, the geodesic distance between any two distinct curves is 0, while for A>0 the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are larger or equal to 0, while for curves with high curvature or perturbations of high frequency, the curvatures are lower or equal to 0. We also investigate the H^n metrics. We compute the momentum mappings for all of the metrics described above. In fact, many results hold in a much more general situation. The L^2-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N,g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L^2-metric. This is in particular true for Burgers' equation. (This is joint work with David Mumford.)