The L2-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 L2-metric.Comment: 26 pages, LATEX, final versio