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The algebraic complete integrability of geodesic flow on SO(N). (English) Zbl 0584.58023

Author’s abstract: ”We study for which left invariant diagonal metrics \(\lambda\) on SO(N), the Euler-Arnold equations \(\dot X=[X,\lambda (X)]\), \(X=(x_{ij})\in SO(N)\), \(\lambda (X)_{ij}=\lambda_{ij}x_{ij}\), \(\lambda_{ij}=\lambda_{ji}\) can be linearized on an abelian variety, i.e. are solvable by quadratures. We show that, merely by requiring that the solutions of the differential equations be single-valued functions of complex time \(t\in {\mathbb{C}}\), suffices to prove that (under a non- degeneracy assumption on the metric \(\lambda)\) the only such metrics are those which satisfy Manakov’s conditions \(\lambda_{ij}=(b_ i-b_ j)(a_ i-a_ j)^{-1}\). The case of degenerate metrics is also analyzed. For \(N=4\), this provides a new and simpler proof of a result of Adler and van Moerbeke”.
Reviewer: S.Goodman

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H40 Jacobians, Prym varieties
14K99 Abelian varieties and schemes
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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