Abstract
In this paper, we show that the geodesic spray associated with the Camassa–Holm equation on the real line is smooth. This improves the earlier C1 result of Lee–Preston [18]. The main idea is to identify the second order vector field associated with the Camassa–Holm equation as a composition of smooth operations. We will also show that the covariant derivative associated with the spray is Riemannian, i.e., compatible with the Sobolev H1 right invariant metric.
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