International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 69-72, Pages 3901-3916
doi:10.1155/S0161171204406553

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.