Timelike surfaces in Minkowski space with a canonical null direction

TL;DR

This paper characterizes timelike surfaces in Minkowski space that have a canonical null direction, focusing on ruled and non-ruled cases, and explores their geometric properties and constructions.

Contribution

It provides a detailed description of these surfaces in three and four dimensions, including methods for constructing them and analyzing their properties using the Gauss map.

Findings

Surfaces in 3D Minkowski space with a canonical null direction are minimal and flat.

The paper offers methods to construct such surfaces in 4D Minkowski space.

Properties of these surfaces are analyzed via the Gauss map.

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. In this paper we describe these surfaces in the ruled case. For example when the Minkowski space has three dimensions then a surface with a canonical null direction is minimal and flat. On the other hand, we describe several properties in the non ruled case and we partially describe these surfaces in four-dimensional Minkowski space. We give different ways for building these surfaces in four-dimensional Minkowski space and we finally use the Gauss map for describe another properties of these surfaces.