Diffeomorphisms on S1, Projective Structures and Integrable Systems

摘要

It is well known that the KdV equation is the avatoar of a scalar Lax equationdefined by a Lax pair of scalar nth order differential (AGD) operator. In this paper the authors derive (formally) the Korteweg-de Vries (KdV) as an evolution equation of the AGD operator (at least for n < or = 4) under the action of Vect(S(sup1)). The solutions of the AGD operator defines an immersion R -> Rp(sup n-1) in homogeneous coordinates. In this paper the authors derive the Schwarzian KdV equation as an evolution of the solution curve associated to Delta(sup (n)), for n < or = 4. The authors 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 Knorrer and Moser. The authors show that certain quadratic function of a projective field satisfy the C. Neumann system.