Spheres, Kähler geometry, and the two-component Hunter-Saxton equation

Details Spheres, Kähler geometry, and the two-component Hunter-Saxton equation Spheres, Kähler geometry, and the two-component Hunter-Saxton equation

2011-08-12

arxiv.org/abs/1108.2727v1

Mathematics

Differential Geometry

24 pages

Scientific paper

The two-component Hunter-Saxton (2HS) equation is the Euler equation for geodesic flow on a semi-direct product manifold endowed with a right-invariant metric. We show that this manifold is in fact isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulas for the solutions of 2HS. We also show that when restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold which admits a K\"ahler 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.

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