Evolution of space-like curves and special time-like ruled surfaces in the Minkowski space

In this paper, we get the time evolution equations of the curvature and the torsion of a space-like curve in the three-dimensional Minkowski space. Also, we give inextensible evolutions of a time-like ruled surface that is produced by a time-like normal vector and a space-like binormal vector of a space-like curve and derive the necessary conditions for an inelastic surface evolution. As a result, we give applications of the time evolution equations for the curvature and the torsion of the space-like curve in terms of the velocities and get the exact solutions for these equations.