Michor, Peter W.; Mumford, David

Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Doc. Math., J. DMV 10, 217-245 (2005)


Summary: 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.

Mathematics Subject Classification

58B20, 58D15, 58E12