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Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Peter W. Michor·David Mumford
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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.

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Peter W. Michor, David Mumford. 2004-09-17. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. https://arxiv.org/abs/math/0409303

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