Abstract
We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff μ (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev \({\dot{H}^1}\) -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 \({\dot{H}^1}\) -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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Khesin, B., Lenells, J., Misiołek, G. et al. Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics. Geom. Funct. Anal. 23, 334–366 (2013). https://doi.org/10.1007/s00039-013-0210-2
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DOI: https://doi.org/10.1007/s00039-013-0210-2
Keywords and phrases
- Diffeomorphism groups
- Riemannian metrics
- geodesics
- curvature
- Euler–Arnold equations
- Fisher–Rao metric
- Hellinger distance
- integrable systems