Projective structure and integrable geodesic flows on the extension of Bott-Virasoro group

Abstract

This is a sequel to our paper (Lett. Math. Phys. (2000)), triggered from a question posed by Marcel, Ovsienko, and Roger in their paper (1997). In this paper, we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation, and modified dispersionless long wave equation are the geodesic flows with respect to an L2 metric on the semidirect product space Diffs(S1)⋉C∞(S1)kˆ, where Diffs(S1) is the group of orientation preserving Sobolev Hs diffeomorphisms of the circle. We also study the projective structure associated with the matrix Sturm-Liouville operators on the circle.