Journal
GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 23, Issue 1, Pages 334-366Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00039-013-0210-2
Keywords
Diffeomorphism groups; Riemannian metrics; geodesics; curvature; Euler-Arnold equations; Fisher-Rao metric; Hellinger distance; integrable systems
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Funding
- Simonyi Fund
- NSERC Research Grant
- EPSRC, UK
- James D. Wolfensohn Fund
- Friends of the Institute for Advanced Study
- NSF [1105660]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1105660] Funding Source: National Science Foundation
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We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff (mu) (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler-Arnold equation is a completely integrable system in any space dimension whose smooth solutions break down in finite time. We also show that the -metric induces the Fisher-Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.
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