Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

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.

26 pages, LATEX, final version

References in corpus (1)

• An integrable shallow water equation with peaked solitons

Cited by in corpus (3)

• The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric
• The Completion of the Manifold of Riemannian Metrics
• Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics