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In this article we study Sobolev metrics of order one on diffeomorphismgroups on the real line. We prove that the space$\operatorname{Diff}_{1}(\mathbb R)$ equipped with the homogenous Sobolevmetric of order one is a flat space in the sense of Riemannian geometry, as itis isometric to an open subset of a mapping space equipped with the flat$L^2$-metric. Here $\operatorname{Diff}_{1}(\mathbb R)$ denotes the extensionof the group of all either compactly supported, rapidly decreasing or$H^\infty$-diffeomorphisms, that allows for a shift towards infinity. Inparticular this result provides an analytic solution formula for thecorresponding geodesic equation, the non-periodic Hunter-Saxton equation. Inaddition we show that one can obtain a similar result for the two-componentHunter-Saxton equation and discuss the case of the non-homogenous Sobolev onemetric which is related to the Camassa-Holm equation.

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