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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 71, Pages 3901-3916

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

1S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Calcutta 700098, India
2Mathematische Physik, Institut für Theoretische Physik, Technische Universität Claustha, Arnold Sommerfeld Straße 6, Clausthal-Zellerfeld 38678, Germany

Received 22 June 2004

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.