### A note on bidifferential calculi and bihamiltonian systems

In this note we discuss the geometrical relationship between bi-Hamiltonian systems and bi-differential calculi, introduced by Dimakis and Möller–Hoissen.

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In this note we discuss the geometrical relationship between bi-Hamiltonian systems and bi-differential calculi, introduced by Dimakis and Möller–Hoissen.

The Ito equation is shown to be a geodesic flow of ${L}^{2}$ metric on the semidirect product space $\widehat{{\mathrm{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}}^{s}\left({S}^{1}\right)\u2a00{C}^{\infty}\left({S}^{1}\right)}$, where ${\mathrm{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}}^{s}\left({S}^{1}\right)$ is the group of orientation preserving Sobolev ${H}^{s}$ diffeomorphisms of the circle. We also study a geodesic flow of a ${H}^{1}$ metric.

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