000203205 001__ 203205
000203205 005__ 20181203023650.0
000203205 0247_ $$2doi$$a10.1007/s00332-014-9204-y
000203205 022__ $$a0938-8974
000203205 02470 $$2ISI$$a000343140400001
000203205 037__ $$aARTICLE
000203205 245__ $$aHomogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
000203205 269__ $$a2014
000203205 260__ $$bSpringer Verlag$$c2014$$aNew York
000203205 300__ $$a40
000203205 336__ $$aJournal Articles
000203205 520__ $$aIn this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space 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 -metric. Here denotes the extension of the group of all compactly supported, rapidly decreasing, or -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.
000203205 6531_ $$aDiffeomorphism group
000203205 6531_ $$aGeodesic equation
000203205 6531_ $$aSobolev H1-metric
000203205 6531_ $$aR-map
000203205 700__ $$uUniv Wien, Fak Math, A-1090 Vienna, Austria$$aBauer, Martin
000203205 700__ $$0246582$$g228988$$aBruveris, Martins
000203205 700__ $$aMichor, Peter W.$$uUniv Wien, Fak Math, A-1090 Vienna, Austria
000203205 773__ $$j24$$tJournal Of Nonlinear Science$$k5$$q769-808
000203205 909C0 $$0252609$$pCAG2
000203205 909CO $$particle$$ooai:infoscience.tind.io:203205
000203205 917Z8 $$x180122
000203205 937__ $$aEPFL-ARTICLE-203205
000203205 973__ $$rREVIEWED$$sPUBLISHED$$aEPFL
000203205 980__ $$aARTICLE