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

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)D(S) of orientation-preserving diffeomorphisms of the unit circle SS modulo the subgroup of rotations Rot(S) equipped with the Ḣ1 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 π2, 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 L2L2-sphere is constructed.