The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
| Autor: | Bauer, Martin, Bruveris, Martins, Michor, Peter W. |
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| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Journal of Nonlinear Science 24, 5 (2014), 769-808 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00332-014-9204-y |
| Popis: | In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space $\operatorname{Diff}_{1}(\mathbb R)$ equipped with the homogenous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat $L^2$-metric. Here $\operatorname{Diff}_{1}(\mathbb R)$ denotes the extension of the group of all either compactly supported, rapidly decreasing or $H^\infty$-diffeomorphisms, that allows for a shift towards infinity. In particular this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton equation. In addition we show that one can obtain a similar result for the two-component Hunter-Saxton equation and discuss the case of the non-homogenous Sobolev one metric which is related to the Camassa-Holm equation. Comment: 31 pages, 6 figures. Misprints corrected, made analogous to the published version |
| Databáze: | arXiv |
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