2011/05/31 by Martin Bauer, Martins Bruveris, Philipp Harms +1 · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Geometry and complex manifolds #Nonlinear Waves and Solitons #math.AP #math.DG #msc:35Q31 #msc:58B20 #msc:58D05
paper · pdf · doi:10.1007/s10455-012-9353-x
published as Ann. Glob. Anal. Geom. 44, 1 (2013), 5-21 · 16 pages. Final version
arxiv created 2012/09/07 · openalex publication_date 2012/09/22 · arxiv updated 2013/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study Sobolev-type metrics of fractional order s≥0 on the group \Diffc(M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M=S1 the geodesic distance on \Diffc(S1) vanishes if and only if s≤\frac12. For other manifolds we obtain a partial characterization: the geodesic distance on \Diffc(M) vanishes for M=\R× N, s<\frac12 and for M=S1× N, s≤\frac12, with N being a compact Riemannian manifold. On the other hand the geodesic distance on \Diffc(M) is positive for dim(M)=1, s>\frac12 and dim(M)≥2, s≥1. For M=\Rn 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 s=\frac 12 and the Camassa-Holm equation for s=1.