The Symplectic Structure of Curves in three dimensional spaces of constant curvature and the equations of mathematical physics

This paper defines a symplectic form on the infinite dimensional Fréchet manifold of framed curves of fixed length over a three dimensional simply connected Riemannian manifold of constant curvature. The framed curves are anchored at the initial point and are further constrained by the condition that the tangent vector of the projected curve coincides with the first leg of the orthonormal frame. Such class of curves are called anchored Darboux curves and in particular include the Serret-Frenet framed curves. The symplectic form ω is defined on the universal covers of the orthonormal frame bundles of the underlying manifolds: SL2(C) for the hyperboloid H , SU2 × SU2 for the sphere S, and the semidirect product E ⊲ SU2 for the Euclidean space E . The form ω is left invariant on each of the above groups, and is induced by the PoissonLie bracket on the appropriate Lie algebra. More precisely, the form ω in each of the above non-Euclidean cases is defined on the Cartan space p corresponding to the decomposition g = p + k