The Hunter-Saxton Equation: A Geometric Approach

The Hunter–Saxton equation is the Euler equation for the geodesic flow on the quotient space of the infinite-dimensional group of orientation preserving diffeomorphisms of the unit circle modulo the subgroup of rigid rotations equipped with a right-invariant metric. We establish several properties of this quotient space: it has constant sectional 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 manifold. Moreover, we give explicit formulas for the Jacobi fields, we prove that the diameter of the space is π 2 , and give exact estimates for how fast the geodesics spread apart. At the end, these results are given a simple geometric and intuitive understanding when an isometry from the quotient space to an open subset of an L-sphere is constructed.