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Abstract. The Ito equation is shown to be a geodesic flow of $L^2$ metric on the semidirect product space ${\widehat {Diff^s(S^1) \bigodot C^{\infty}(S^1)}}$, where $Diff^s(S^1)$ is the group of orientation preserving Sobolev $H^s$ diffeomorphisms of the circle. We also study a geodesic flow of a $H^1$ metric.
AMSclassification. 58D05, Secondary 35Q53
Keywords. Bott-Virasoro Group, Ito equation