A characterization of parallel surfaces in Minkowski space via lightlike curves and surfaces

We consider a non degenerate surface M in the Minkowski space and a regular curve on it with a well defined causality. Over such curve we construct, locally, a normal surface $$\Sigma $$ as a ruled surface whose rules are non degenerate straight lines orthogonal to M. In the first part we consider immersed timelike surfaces. We prove that if the curve is a lightlike straight line in a timelike surface then it is a line of curvature. This is equivalent to the condition that $$\Sigma $$ is a lightlike surface. As an application we deduce that if at every point of M pass two lightlike straight lines then M is umbilical. Moreover, we prove that through every point of M pass two lightlike geodesics with constant curvatures if and only if M is a parallel surface. Finally, we consider non degenerate immersed surfaces and we give another characterization of parallel surfaces: Through every point of M pass three different curves, of any well defined causality, such that their normal surfaces are either extremal or lightlike if and only if M is a parallel surface.