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The Hunter-Saxton equation describes the geodesic flow on a sphere. (English) Zbl 1125.35085
Author’s summary: The Hunter-Saxton equation is the Euler equation for the geodesic flow on the quotient space Rot$(\Bbb S)\setminus \cal D(\Bbb S)$ of the infinite-dimensional group $\cal D(\Bbb S)$ of orientation-preserving diffeomorphisms of the unit circle $\Bbb S$ modulo the subgroup of rotations Rot$(\Bbb S)$ equipped with the $\dot H^1$ right-invariant metric. We establish several properties of the Riemannian manifold Rot$(\Bbb S)\setminus \cal D(\Bbb S)$: 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 $\frac{\pi}{2}$, 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$(\Bbb S)\setminus \cal D(\Bbb S)$ to an open subset of an $L^{2}$-sphere is constructed.

35Q53KdV-like (Korteweg-de Vries) equations
58B20Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
Full Text: DOI
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