Geodesic
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
Documenta mathematica, Tome 10 (2005), pp. 217-245
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The L2-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 L2-metric.
DOI : 10.4171/dm/187
Classification : 58B20, 58D15, 58E12 
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     author = {Peter W. Michor and David Mumford},
     title = {Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms},
     journal = {Documenta mathematica},
     pages = {217--245},
     year = {2005},
     volume = {10},
     doi = {10.4171/dm/187},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/187/}
}
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Peter W. Michor; David Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta mathematica, Tome 10 (2005), pp. 217-245. doi: 10.4171/dm/187

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