A characterization of parallel surfaces in Minkowski space via minimal and maximal surfaces

Abstract

We give a characterization of parallel surfaces in the three dimensional Minkowski space. We consider the following construction on a non degenerate surface M. Given a non degenerate curve in the surface we have the ruled surface orthogonal to M along the curve. We prove that if this orthogonal surface is either maximal or minimal then the curve is a geodesic of M. Moreover such geodesic is either a planar line of curvature of M or it has both constant curvature and constant no zero torsion. A first result says that if M is a surface such that through every point pass two non degenerate geodesics, both with constant curvature and torsion, then the surface is parallel. Our main result says that if M is a surface then through every point pass three non degenerate curves whose associated ruled orthogonal surfaces are either maximal or minimal if and only if M is a parallel surface.