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

In this work, we focus on the evolution of the vortex filament flow partial derivative gamma|partial derivative iota = partial derivative gamma|partial derivative s boolean AND D|ds partial derivative gamma|partial derivative 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 nonlinear Schr<spacing diaeresis>odinger equation. Also, we give some examples to illustrate the vortex filament