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

Abstract

The article studies geometrically the Euler-Arnold equations associated to geodesic flow on $SO(4)$ for a left invariant diagonal metric. Such metric were first introduced by Manakov \cite{17} and extensively studied by Mishchenko-Fomenko \cite{18} and Dikii \cite{6}. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke \cite{4} and Haine \cite{8}. In this problem there are four invariants of the motion defining in $\Bbb{C}^{4}=\operatorname{Lie}(SO(4)\otimes \Bbb{C})$ 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 linearizes the problem on a two-dimensional Prym variety $\operatorname{Prym}_{\sigma }(C)$ of a genus 3 Riemann surface $C$. In section 2, the method consists of requiring that the general solutions have the Painleve property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski \cite{10} and has developed and used more systematically \cite{3}, \cite{4}, \cite{8}, \cite{13}. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler-Arnold equations occurs on a Prym variety $\operatorname{Prym}_{\sigma }(G)$ of an another genus 3 Riemann surface $G$. In the last section the Riemann surfaces are compared explicitly.