Abstract
We study for which left invariant diagonal metrics λ onSO(N), the Euler-Arnold equations
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 timet∈ℂ, suffices to prove that (under a non-degeneracy assumption on the metric λ) the only such metrics are those which satisfy Manakov's conditions λ ij =(b i −b j ) (a i −a j )−1. The case of degenerate metrics is also analyzed. ForN=4, this provides a new and simpler proof of a result of Adler and van Moerbeke [3].
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Communicated by A. Jaffe
Aspirant Fonds National Belge de la Recherche Scientifique (F.N.R.S.) Supported in part by N.S.F. Grant 8102696 while visiting Brandeis University
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Haine, L. The algebraic complete integrability of geodesic flow onSO(N). Commun.Math. Phys. 94, 271–287 (1984). https://doi.org/10.1007/BF01209305
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DOI: https://doi.org/10.1007/BF01209305