Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Peter W. Michor; David Mumford

Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Differential Geometry 2016-09-07 v2 Analysis of PDEs

Authors: Peter W. Michor , David Mumford

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.

Keywords

geodesic distance submanifolds diffeomorphism groups L^2-metric Fubini-Study metric mean curvature

Cite

@article{arxiv.math/0409303,
  title  = {Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms},
  author = {Peter W. Michor and David Mumford},
  journal= {arXiv preprint arXiv:math/0409303},
  year   = {2016}
}

Comments

26 pages, LATEX, final version

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