Geodesic
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
Documenta mathematica, Tome 10 (2005), pp. 217-245.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 58B20, 58D15, 58E12 
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     title = {Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms},
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Michor, Peter W.; Mumford, David. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta mathematica, Tome 10 (2005), pp. 217-245. http://geodesic.mathdoc.fr/item/DOCMA_2005__10__a14/