Étude géométrique et topologique du flot géodésique sur le groupe des rotations

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.

Keywords

• integrable systems
• Jacobians
• Prym varieties