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Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

Bauer, Martin ; Bruveris, Martins ; Michor, Peter

In: Journal of Nonlinear Science, 2014, vol. 24, no. 5, p. 769-808

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    Title
    • Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
    Author
    • Bauer, Martin. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria
    • Bruveris, Martins. Institut de mathématiques, EPFL, 1015, Lausanne, Switzerland
    • Michor, Peter. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Journal of Nonlinear Science, 2014, vol. 24, no. 5, p. 769-808. Springer US; http://www.springer-ny.com
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:326223
    Summary
    In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space $$\mathrm{Diff }_{1}(\mathbb R)$$ Diff 1 ( R ) equipped with the homogeneous 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 $$L^2$$ L 2 -metric. Here $$\mathrm{Diff }_{1}(\mathbb R)$$ Diff 1 ( R ) denotes the extension of the group of all compactly supported, rapidly decreasing, or $$W^{\infty ,1}$$ W ∞ , 1 -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa-Holm equation.