%0 Journal Article
%T Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
%A Peter W. Michor
%A David Mumford
%J Mathematics
%D 2004
%I arXiv
%X 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.
%U http://arxiv.org/abs/math/0409303v2