Geodesic Distance for Right Invariant Sobolev Metrics of Fractional Order on the Diffeomorphism Group

We study Sobolev-type metrics of fractional order on the group of compactly supported diffeomorphisms Diffc(M), where M is a Riemannian manifold of bounded geometry. We prove that the geodesic distance, induced by the Riemannian metric, vanishes if the order s satisfies 0 ≤ s < 1 2 . For M 6= R we show the vanishing of the geodesic distance also for s = 1 2 , and for dim(M) = 1 we show that the distance is positive for 1 2 < s. For M = Rn we derive and discuss the geodesic equations for these metrics. It is a known fact that for specific values of s one recovers well known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin-LaxMajda equation for s = 1 2 or the Camassa-Holm equation for s = 1.