Authors’ abstract: 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.