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• W1618708454 abstract "We study the geometry of the space of densities Dens(M), which is the quotient space Di( M)=Di (M) of the dieomorphism group of a com- pact manifold M by the subgroup of volume-preserving diemorphisms, en- dowed with a right-invariant homogeneous Sobolev _ H 1 -metric. We construct an explicit isometry from this space to (a subset of) an innite-dimension al sphere and show that the associated Euler-Arnold equation is a completely integrable system in any space dimension. We also prove that its smooth solutions break down in nite time. Furthermore, we show that the _ H 1 -metric induces the Fisher-Rao (informa- tion) metric on the space of probability distributions, and thus its Riemannian distance is the spherical version of Hellinger distance. We compare it to the Wasserstein distance in optimal transport which is induced by an L 2 -metric on Di( M). The _ H 1 geometry we introduce in this paper can be seen as an innite-dimension al version of the geometric theory of statistical manifolds." @default.
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• W1618708454 date "2011-05-03" @default.
• W1618708454 modified "2023-09-27" @default.
• W1618708454 title "Geometry of diffeomorphism groups, complete integrability and optimal transport" @default.
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