Riemannian Geometrical Analysis of the Motion of a Vortex Filament: A System of C8(S1,SO(3)) (original) (raw)

Physical Review Letters, 1996

Abstract

The motion of an isolated vortex filament with small curvature is governed by the local induction equation. The present work is an attempt to study such an integrable system based on the Riemannian geometry of the solution manifold, by reformulating the governing equation as the geodesic equation on an infinite dimensional Lie group C∞S1,SO(3). The sectional curvatures for the vector fields along the geodesic curves and their perturbations are calculated to show their tendency to be positive for some typical filaments such as a straight vortex, a helical vortex, and a vortex soliton.

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