Projective connections, AGD manifold and integrable systems

If u(i)s are periodic function on the line, the operator d(n)/dx(n) + u(n-1 ) d(n-1)/dx(n-1) + u(n-2) d(n-2)/dx(n-2) + ... + u(1) d/dx + u(0), acting o n periodic functions, is called a Adler-Gelfand-Dikii (or AGD) operator. In this paper we consider a projective connection as defined by this nth orde r operator on the circle. In particular, projective connection as defined b y a second order operator can be identified with the dual of Virasoro algeb ra, and it is well known that the KdV equation as a Euler-Arnold equation i n the coadjoint orbit of the Bott-Virasoro group. In this paper rye study ( formally) the evolution equation of the Adler-Gelfand-Dikii operator, Delta ((n)), (at least for n less than or equal to 4), under the action of Vect( S-1). This yields a single generating equation for periodic function u. We also establish a connection between the projective vector field, a vector f ield leaves fixed a given (extended) projective connection, and the C. Neum ann system using the idea of Knorrer and Moser. We show that certain quadra tic function of a projective field satisfies C. Neumann system.