Guha, Partha Projective structure and integrable geodesic flows on the extension of Bott-Virasoro group. (English) Zbl 1126.37319 Int. J. Math. Math. Sci. 2004, No. 69-72, 3901-3916 (2004). 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 P. Marcel, V. Ovsienko and 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. Cited in 1 Document MSC: 37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds Citations:Zbl 0997.37050; Zbl 0881.17019 PDFBibTeX XMLCite \textit{P. Guha}, Int. J. Math. Math. Sci. 2004, No. 69--72, 3901--3916 (2004; Zbl 1126.37319) Full Text: DOI EuDML