pith. sign in

arxiv: 1209.2836 · v3 · pith:FFVUX3SRnew · submitted 2012-09-13 · 🧮 math.AP · math.DG

The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line

classification 🧮 math.AP math.DG
keywords equationmetricsobolevorderspacediffdiffeomorphismequipped
0
0 comments X
read the original abstract

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.

This paper has not been read by Pith yet.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.