Equations of Camassa–Holm type and the geometry of loop groups

We recall some of Arnold’s classical work on the Riemannian geometry of diffeomorphism groups, we provide easily computable formulae for the sectional curvature in the case of the diffeomorphism group of the circle, and we observe that this group has a CR-manifold structure. We also present examples of bona fide hamiltonian equations in infinitely many derivatives.