The Hunter–Saxton equation describes the geodesic flow on a sphere

Abstract

The Hunter–Saxton equation is the Euler equation for the geodesic flow on the quotient space Rot ( S ) ∖ 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 H ̇ 1 right-invariant metric. We establish several properties of the Riemannian manifold Rot ( S ) ∖ 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 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 ) ∖ D ( S ) to an open subset of an L 2 -sphere is constructed.