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

TL;DR

This paper demonstrates that the space of certain diffeomorphisms on the real line with a Sobolev order one metric is flat and isometric to an $L^2$ space, providing explicit solutions for related geodesic equations like Hunter-Saxton.

Contribution

It establishes the flatness and isometry of the Sobolev order one diffeomorphism space and derives explicit solutions for the Hunter-Saxton equations.

Findings

The diffeomorphism space is flat and isometric to an $L^2$ space.

Provides an analytic solution formula for the Hunter-Saxton equation.

Extends results to the two-component Hunter-Saxton and discusses the Camassa-Holm case.

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.