Projective structure and integrable geodesic flows on the extension of Bott-Virasoro group (Q1777909)

Summary: This is a sequel to our paper [Lett. Math. Phys. 52, No. 4, 311--328 (2000; Zbl 0997.37050)], triggered from a question posed by \textit{P. Marcel}, \textit{V. Ovsienko} and \textit{C. Roger} [Lett. Math. Phys. 40, 31--39 (1997; Zbl 0881.17019)]. 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 \(L^2\) metric on the semidirect product space \(\widehat{\text{Diff}^s(S^1)\ltimes C^{\infty}(S^1)^k}\), where \(\text{Diff}^s(S^1)\) is the group of orientation preserving Sobolev \(H^s\) diffeomorphisms of the circle. We also study the projective structure associated with the matrix Sturm-Liouville operators on the circle.