Bour's Theorem of Spacelike Surfaces in Minkowski 4--Space

TL;DR

This paper extends Bour's theorem to spacelike helicoidal surfaces in Minkowski 4-space, establishing isometries with rotational surfaces and analyzing their geometric properties and parametrizations.

Contribution

It generalizes Bour's theorem from Minkowski 3-space to Minkowski 4-space for spacelike helicoidal surfaces and explores their geometric properties and parametrizations.

Findings

Established isometry between helicoidal and rotational surfaces in Minkowski 4-space.

Derived parametrizations for isometric pairs of surfaces.

Provided examples and visualizations using Mathematica.

Abstract

In this paper, we study on three kinds of spacelike helicoidal surfaces in Minkowski $4$--space. First, we give an isometry between such helicoidal surfaces and rotational surfaces which is a kind of generalization of Bour theorem in Minkowski $3$--space to Minkowski $4$--space. Then, we investigate geometric properties for such isometric surfaces having same Gauss map. By using these results, we give the parametrizations of isometric pair of surfaces. As a particular case, we examine the right helicoidal surfaces in view of Bour's theorem. Also, we present some examples by choosing the components of the profile curves and the parameters of the surfaces via Mathematica.