Vortex filament flows for curves in a 3-dimensional pseudo-Riemannian manifold

Abstract

In this work, we focus on the evolution of the vortex filament flow \(\frac{\partial \gamma }{\partial t} = \frac{\partial \gamma }{\partial s} \wedge \frac{D}{ds}\frac{\partial \gamma }{\partial s}\) for spacelike and timelike curves in a 3-dimensional pseudo-Riemannian manifold. We study the relations between a partial differential equation and the vortex filament flow for spacelike and timelike curves. As a result, we prove that the vortex filament flow of the spacelike curve in a 3-dimensional pseudo-Riemannian manifold with constant sectional curvature is equivalent to the heat equation, and the flow of the timelike curve is equivalent to the non-linear Schrödinger equation. Also, we give some examples to illustrate the vortex filament flow.