Article
Kyungpook Mathematical Journal 2022; 62(3): 497507
Published online September 30, 2022
Copyright © Kyungpook Mathematical Journal.
On the Gauss Map of Tubular Surfaces in Pseudo Galilean 3Space
Yılmaz Tunçer∗, Murat Kemal Karacan, Dae Won Yoon
Department of Mathematics, Usak University, Usak 64200, Turkey
email : yilmaz.tuncer@usak.edu.tr
Department of Mathematics, Usak University, Usak 64200, Turkey
email : murat.karacan@usak.edu.tr
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
email : dwyoon@gsnu.ac.kr
Received: March 21, 2021; Revised: December 7, 2021; Accepted: December 10, 2021
Abstract
In this study, we define tubular surfaces in Pseudo Galilean 3space as type1 or type2. Using the
Keywords: Tubular surfaces, Gauss map, pointwise 1type Gauss map, Pseudo Galilean 3Space
1. Introduction
Due to their physical importance in curve and surface theory, Galilean and Pseudo Galilean geometries have been widely studied in recent years. The Cayley Klein geometry with projective signature
for any vectors
A vector
The cross product is defined by
All unit nonisotropic vectors are of the form
The curvature
An admissible curve has no inflection points, no isotropic tangents or normals whose projections on the absolute plane would be lightlike vectors. The Frenet trihedron is given by
where
Thus the principal normal vector, or simply normal, is spacelike if
for derivatives of the tangent vector
and
then
where
is called a side tangential vector. Throughout the study we will consider the surfaces with
where
and
[8, 10]. For a vector
In this study, we denote the components of
2. Tubular Surfaces in Pseudo Galilean 3Space
In this section, we will classify the admissible tubular surfaces in
Type1: If
Type2: If
Let
and from (1.4), (1.5) and (1.6), we have
which are the components of F.F.F., so we obtain the
By a direct computation using the equation (1.7), the Laplacian operator Δ on
Suppose that
so it is easy to see that the equality
Theorem 2.1. There is not any harmonic type1 tubular surface given by (2.1) in
For the other cases, we give the following theorem.
Theorem 2.2.Let
Taking the derivative (2.6) with respect to
Combining (2.6) and (2.7) we can obtain the following two equation
On the other hand, from (2.4), (2.7) and (2.8)
and differentiating (2.9) with respect to
By taking the derivative (2.11) with respect to
By considering
Thus, this completes the proof.
From the first equation of (2.5), we can obtain
and by using the second equation of (2.5), we have
Equation (2.15) has complex solutions but in the case of
Corollary 2.3. Let
then
where

Figure 1. (a)

Figure 2. (b)
or
where

Figure 3. (c)

Figure 4. (d)
where
and from (2.3), we find
Thus, it is easy to see that, following theorem holds.
Theorem 2.4. Let
i. There are no type1 tubular surface given by (2.1) in
ii. All type1 tubular surfaces satisfy
iii. All type1 tubular surface satisfy
As a result of Theorem 2.4., we can say
Assume that
and so
Theorem 2.5. Let
where
Let
and we have
which are the components of F.F.F.
and the Laplacian operator Δ on
The Gauss map
and Laplacians of
and
respectively. We can also obtain similar results for type2 surfaces in
Example 2.2. Timelike tube with timelike centered curve satisfying
Spacelike tube with spacelike centered curve satisfying
Timelike tube with timelike centered curve satisfying
Spacelike tube with spacelike centered curve satisfying
Acknowledgements.
The authors are indebted to the referees for helpful suggestions and insights concerning the presentation of this paper.
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