Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 11 (2016), 107 -- 134

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This work is licensed under a Creative Commons Attribution 4.0 International License.

ÉTUDE GÉOMÉTRIQUE ET TOPOLOGIQUE DU FLOT GÉODÉSIQUE SUR LE GROUPE DES ROTATIONS
(GEOMETRIC AND TOPOLOGICAL STUDY OF THE GEODESIC FLOW ON THE ROTATION GROUP)

Ahmed Lesfari

Abstract. The aim of this survey paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps Λ : so(4)→ so(4)* ≡ so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from ℝ to ℂ, and then the regular invariant manifolds are complex instead of real tori; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem.

2010 Mathematics Subject Classification: 37J35, 14H40, 14H70.
Keywords: integrable systems, Jacobians, Prym varieties.

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Ahmed Lesfari
Département de Mathématiques,
Faculté des Sciences, Université Chouaïb Doukkali,
B.P. 20, El-Jadida, Maroc.
e-mail: lesfariahmed@yahoo.fr

http://www.utgjiu.ro/math/sma