Bauer, MartinBruveris, MartinsHarms, PhilippMichor, Peter W.2013-10-012013-10-012013-10-01201310.1007/s10455-012-9353-xhttps://infoscience.epfl.ch/handle/20.500.14299/95296WOS:000319160700002We study Sobolev-type metrics of fractional order s a parts per thousand yen 0 on the group Diff (c) (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S (1), the geodesic distance on Diff (c) (S (1)) vanishes if and only if . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff (c) (M) vanishes for and for , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff (c) (M) is positive for and dim(M) a parts per thousand yen 2, s a parts per thousand yen 1. For , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers' equation for s = 0, the modified Constantin-Lax-Majda equation for , and the Camassa-Holm equation for s = 1.Diffeomorphism groupGeodesic distanceSobolev metrics of non-integral ordertext::journal::journal article::research article