Spheres, Kähler geometry and the Hunter-Saxton system
2013 (English)In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 469, no 2154, 20120726Article in journal (Refereed) Published
Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter- Saxton (2HS) system, which displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold, which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulae for the solutions of 2HS.We also show thatwhen restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold, which admits a Kähler structure. We demonstrate that this manifold is in fact isometric to a subset of complex projective space, and that the above constructions provide an example of an infinite-dimensional Hopf fibration.
Place, publisher, year, edition, pages
2013. Vol. 469, no 2154, 20120726
Curvature, Diffeomorphism groups, Kähler geometry, Nonlinear partial differential equations, Complex projective space, Diffeomorphisms, Geodesic equations, Geometric feature, Infinite-dimensional manifolds, Mathematical physics, Geodesy, Hydraulic structures, Partial differential equations, Spheres, Geometry
IdentifiersURN: urn:nbn:se:kth:diva-163798DOI: 10.1098/rspa.2012.0726ISI: 000318026800009ScopusID: 2-s2.0-84877304468OAI: oai:DiVA.org:kth-163798DiVA: diva2:802394
QC 201504172015-04-132015-04-122015-04-17Bibliographically approved