TY - JOUR

T1 - Projective connections, AGD manifold and integrable systems

AU - Guha, Partha

N1 - Funding Information:
It is my pleasant duty to acknowledge gratefully for several stimulating conversations with Professors Mitya Alekseevsky, Sasha Kirillov and Richard Montgomery. I would like to thank very much IHES (Bures-sur-Yvette) | where this project was started. This work is partially supported by the S. Chandrasekhar memorial ICSC-World Laboratory fellowship.

PY - 2000/10

Y1 - 2000/10

N2 - If uis are periodic function on the line, the operator dn/dxn + un-1 dn-1/dxn-1 + un-2 dn-2/dxn-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.

AB - If uis are periodic function on the line, the operator dn/dxn + un-1 dn-1/dxn-1 + un-2 dn-2/dxn-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.

UR - http://www.scopus.com/inward/record.url?scp=0034358182&partnerID=8YFLogxK

U2 - 10.1142/S0129055X00000538

DO - 10.1142/S0129055X00000538

M3 - Article

AN - SCOPUS:0034358182

SN - 0129-055X

VL - 12

SP - 1391

EP - 1406

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 10

ER -