Abstract
In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space \(\mathrm{Diff }_{1}(\mathbb R)\) 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 \(L^2\)-metric. Here \(\mathrm{Diff }_{1}(\mathbb R)\) denotes the extension of the group of all compactly supported, rapidly decreasing, or \(W^{\infty ,1}\)-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.
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Acknowledgments
We thank Yury Neretin for helpful discussions and comments regarding the space \(H^{\infty }(\mathbb R)\). MB was supported by Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 24625.
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Communicated by Tudor Stefan Ratiu.
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Bauer, M., Bruveris, M. & Michor, P.W. Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line. J Nonlinear Sci 24, 769–808 (2014). https://doi.org/10.1007/s00332-014-9204-y
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DOI: https://doi.org/10.1007/s00332-014-9204-y