A Riemannian Geometrical Analysis of the Filament Equation

The motion of an isolated thin vortex filament with very small curvature is governed by the filament equation. We describe this equation as a geodesic equation on a group manifold $C^{\infty}(L, so(3)) $ in view of the fact that $T_{t}’’ $ plays the role of the angular velocity in the equation of motion for the tangent vector $T_{t} $. We investigate the instability of the filament equation by the calculation of the sectional curvature. We show that the spiral vortex filament is not unstable for the short wave length perturbation. We also investigate the instability of the Hasimoto soliton to find it is not unstable for particular perturbation which may have some relation to the integrability. 1