PROJECTIVE CONNECTIONS, AGD MANIFOLD AND INTEGRABLE SYSTEMS

If u_is are periodic function on the line, the operator , acting on periodic functions, is called a Adler–Gelfand–Dikii (or AGD) operator. In this paper we consider a projective connection as defined by this nth order operator on the circle. In particular, projective connection as defined by a second order operator can be identified with the dual of Virasoro algebra, and it is well known that the KdV equation as a Euler–Arnold equation in the coadjoint orbit of the Bott–Virasoro group. In this paper we study (formally) the evolution equation of the Adler–Gelfand–Dikii operator, Δ⁽ⁿ⁾, (at least for n ≤ 4), under the action of Vect(S¹). This yields a single generating equation for periodic function u. We also establish a connection between the projective vector field, a vector field leaves fixed a given (extended) projective connection, and the C. Neumann system using the idea of Knörrer and Moser. We show that certain quadratic function of a projective field satisfies C. Neumann system.