Translation surfaces in 3-D Minkowski space

There are three kinds of vectors in the 3-D Minkowski space, i.e., space-like, time-like and light-like vectors among which choosing any two vectors as the directions of translation will divide the translation surfaces into six types. A new metric form is chosen to study the Weingarten translation surfaces which are translating in the two null directions in a pseudo-orthogonal frame. Then, the first and second fundamental forms, Gaussian curvature and mean curvature of the surfaces are directly calculated according to the principles of differential geometry. It follows that some theorems of classification of those translation surfaces are given mainly by virtue of the linear and square relationships between the Gaussian curvature and the mean curvature.