JournalsdmVol. 10pp. 217–245

Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

  • Peter W. Michor

    Fakultät für Mathematik Division of Applied Mathematics Universität Wien Brown University Nordbergstrasse 15 Box F, Providence, RI 02912, USA A-1090 Wien, Austria David

  • David Mumford

    and Erwin Schrödinger Institut für Mathematische Physik Boltzmanngasse 9 A-1090 Wien, Austria
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms cover
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Abstract

The -metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type in a Riemannian manifold 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 -metric.

Cite this article

Peter W. Michor, David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10 (2005), pp. 217–245

DOI 10.4171/DM/187