DOCUMENTA MATHEMATICA, Vol. 10 (2005), 217-245

Peter W. Michor, David Mumford

Vanishing Geodesic Distance on Spaces of Submanifolds and Diffeomorphisms

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.

2000 Mathematics Subject Classification: Primary 58B20, 58D15, 58E12

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