203205
20181203023650.0
doi
10.1007/s00332-014-9204-y
0938-8974
ISI
000343140400001
ARTICLE
Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
2014
New York
Springer Verlag
2014
40
Journal Articles
In 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.
Diffeomorphism group
Geodesic equation
Sobolev H1-metric
R-map
Bauer, Martin
Univ Wien, Fak Math, A-1090 Vienna, Austria
246582
Bruveris, Martins
228988
Michor, Peter W.
Univ Wien, Fak Math, A-1090 Vienna, Austria
24
5
769-808
Journal Of Nonlinear Science
252609
CAG2
oai:infoscience.tind.io:203205
article
180122
EPFL-ARTICLE-203205
EPFL
REVIEWED
PUBLISHED
ARTICLE