  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.
