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

Partha Guha

doi:10.1155/s0161171204406553

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 anL2metric on the semidirect product spaceDiffs(S1)⋉C∞(S1)kˆ, whereDiffs(S1)is the group of orientation preserving SobolevHsdiffeomorphisms of the circle. We also study the projective structure associated with the matrix Sturm-Liouville operators on the circle.

Highlights

It is known that the periodic Korteweg-de Vries (KdV) and the Camassa-Holm equation [5] can be interpreted as geodesic flow of the right invariant metric on the Bott-Virasoro group, which at the identity is given by the L2 and the Sobolev metric H1-inner product, respectively, [25, 26, 28, 29]
In our earlier papers [10, 11], we have shown that the Ito equation, modified dispersive water wave equation, and modified dispersionless long waves equation arise in a unified geometric setting, all of them are integrable systems which describe geodesic flows
We unify the Ito equation, the dispersive water wave equation, and the long wave equation through a common construction, all are integrable systems which describe geodesic flows with respect to L2 on the extension of the Bott-Virasoro group

Summary

To the memory of Professor Jürgen Moser

(2000)), triggered from a question posed by Marcel, Ovsienko, and Roger in their paper (1997). 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)k, where Diffs (S1) is the group of orientation preserving Sobolev Hs diffeomorphisms of the circle. We study the projective structure associated with the matrix Sturm-Liouville operators on the circle.

Introduction

The result of the paper

Let I be an inertia operator

This means

Conclusion and outlook