Projective connections, AGD manifold and integrable systems

If u_is are periodic function on the line, the operator dⁿ/dxⁿ + u_n-1 d^n-1/dx^n-1 + u_n-2 d^n-2/dx^n-2 + · · · u1 d/dx + u0, 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.