| Resum: |
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] andDikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] andHaine [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, basedon the Kostant-Kirillov coadjoint action. This methodallo ws us to linearizes the problem on a two-dimensional Prym variety Prymσ(C) of a genus 3 Riemann surface C. In section 2, the methodconsists of requiring that the general solutions have the Painlev'e property, i. e. , have no movable singularities other than poles. It was first adopted by Kowalewski [10] andhas developedandusedmore systematically [3], [4], [8], [13]. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler- Arnoldequations occurs on a Prym variety Prymσ(Γ) of an another genus 3 Riemann surface Γ. In the last section the Riemann surfaces are comparedexplicitly . |
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