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

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.