Spheres, K\"ahler geometry, and the Hunter-Saxton system

TL;DR

This paper explores the geometric structure of the two-component Hunter-Saxton system, revealing its relation to spheres and Kähler manifolds, and providing explicit solutions through geometric interpretations.

Contribution

It demonstrates that the 2HS system describes geodesic flow on a sphere-like manifold and reduces to a Kähler manifold, offering new geometric insights and explicit solution formulas.

Findings

2HS describes geodesic flow on a subset of a sphere

Reduction to a Kähler manifold with complex projective space structure

Provides explicit formulas for solutions of 2HS

Abstract

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, that 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 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.