Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group |
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Authors: | Martin Bauer Martins Bruveris Philipp Harms Peter W Michor |
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Institution: | 1. Fakult?t für Mathematik, Universit?t Wien, Nordbergstrasse 15, 1090, Wien, Austria 2. Insitut de mathématiques, EPFL, 1015, Lausanne, Switzerland 3. Edlabs, Harvard University, 44 Brattle Street, Cambridge, MA, 02138, USA
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Abstract: | We study Sobolev-type metrics of fractional order s ≥ 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 ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , 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 ${s=\frac12}$ , and the Camassa–Holm equation for s = 1. |
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