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The Hunter-Saxton equation describes the geodesic flow on a sphere
University of California, United States .ORCID iD: 0000-0001-6191-7769
2007 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, Vol. 57, no 10, 2049-2064 p.Article in journal (Refereed) Published
Abstract [en]

The Hunter-Saxton equation is the Euler equation for the geodesic flow on the quotient space Rot (S) {set minus} D (S) of the infinite-dimensional group D (S) of orientation-preserving diffeomorphisms of the unit circle S modulo the subgroup of rotations Rot (S) equipped with the over(H, ̇)1 right-invariant metric. We establish several properties of the Riemannian manifold Rot (S) {set minus} D (S): it has constant curvature equal to 1, the Riemannian exponential map provides global normal coordinates, and there exists a unique length-minimizing geodesic joining any two points of the space. Moreover, we give explicit formulas for the Jacobi fields, we prove that the diameter of the manifold is exactly frac(π, 2), and we give exact estimates for how fast the geodesics spread apart. At the end, these results are given a geometric and intuitive understanding when an isometry from Rot (S) {set minus} D (S) to an open subset of an L2-sphere is constructed.

Place, publisher, year, edition, pages
2007. Vol. 57, no 10, 2049-2064 p.
Keyword [en]
Curvature, Geodesic flow, Infinite-dimensional sphere, Nonlinear wave equations
National Category
Mathematics Physical Sciences
URN: urn:nbn:se:kth:diva-163833DOI: 10.1016/j.geomphys.2007.05.003ISI: 000249227400007ScopusID: 2-s2.0-34547608701OAI: diva2:802362

QC 20150416

Available from: 2015-04-12 Created: 2015-04-12 Last updated: 2015-04-16Bibliographically approved

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Lenells, Jonatan
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