# Timelike surfaces in Minkowski space with a canonical null direction

Timelike surfaces in Minkowski space with a canonical null direction 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Geometry Springer Journals

# Timelike surfaces in Minkowski space with a canonical null direction

, Volume 109 (2) – May 30, 2018
22 pages

/lp/springer_journal/timelike-surfaces-in-minkowski-space-with-a-canonical-null-direction-NVJmtmwKGO
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Geometry
ISSN
0047-2468
eISSN
1420-8997
D.O.I.
10.1007/s00022-018-0434-2
Publisher site
See Article on Publisher Site

### 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.

### Journal

Journal of GeometrySpringer Journals

Published: May 30, 2018

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