Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics
2013 (English)In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 23, no 1, 334-366 p.Article in journal (Refereed) Published
We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diffμ(M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev Ḣ1 -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 Ḣ1-metric induces the Fisher-Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.
Place, publisher, year, edition, pages
2013. Vol. 23, no 1, 334-366 p.
curvature, Diffeomorphism groups, Euler-Arnold equations, Fisher-Rao metric, geodesics, Hellinger distance, integrable systems, Riemannian metrics
IdentifiersURN: urn:nbn:se:kth:diva-163801DOI: 10.1007/s00039-013-0210-2ISI: 000317008300007ScopusID: 2-s2.0-84875665207OAI: oai:DiVA.org:kth-163801DiVA: diva2:802392
QC 201504162015-04-132015-04-122015-04-16Bibliographically approved