Timelike surfaces in Minkowski space with a canonical null direction

Abstract

Given a constant vector field Z in Minkowski space, a timelike surface is said to have a canonical null direction with respect to Z if the projection of Z on the tangent space of the surface gives a lightlike vector field. For example in the three-dimensional Minkowski space: a surface has a canonical null direction if and only if it is minimal and flat. When the ambient has arbitrary dimension, if a surface has a canonical null direction and has parallel mean curvature vector then it is minimal. We give different ways for building these surfaces in the four-dimensional Minkowski space. On the other hand, we describe several properties in the non ruled general case in four-dimensional Minkowski space. One property is that the tangent part of Z is an asymptotic direction of the surface. We describe these surfaces in the ruled case in arbitrary dimension. Finally use the Gauss map for describe another properties of these surfaces in dimension four.