Geodesic flow on SO(4), Kac-Moody Lie algebra and singularities in the complex t-plane.

Lesfari, Ahmed

Publicacions Matemàtiques, Tome 43 (1999), p. 261-279 / Harvested from Biblioteca Digital de Matemáticas

Résumé

The article studies geometrically the Euler-Arnold equations associatedto geodesic flow on SO(4) for a left invariant diagonal metric. Such metric were first introduced by Manakov [17] and extensively studied by Mishchenko-Fomenko [18] and Dikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] and Haine [8]. In this problem there are four invariants of the motion defining in C4 = Lie(SO(4) ⊗ C) an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coadjoint action. This method allows us to linearize the problem on a two-dimensional Prym variety Prymσ(C) of a genus 3 Riemann surface C. In section 2, the method consists of requiring that the general solutions have the Painlevé property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski [10] and has developed and used more systematically [3], [4], [8], [13]. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler- Arnold equations occurs on a Prym variety Prymσ(Γ) of an another genus 3 Riemann surface Γ. In the last section the Riemann surfaces are compared explicitly.

Détail

Publié le : 1999-01-01

MR MR1697525 | Zbl 0968.35010

DMLE-ID : 3887