 Original research
 Open access
 Published:
Hasimoto surfaces in Galilean space \(G_{3}\)
Journal of the Egyptian Mathematical Society volumeÂ 29, ArticleÂ number:Â 5 (2021)
Abstract
In this article Hasimoto surfaces in Galilean space \(G_{3}\) will be considered, Gauss curvature (K) and Mean curvature (H) of Hasimoto surfaces \(\chi =\chi (s,t)\) will be investigated, some characterization of scurves and tcurves of Hasimoto surfaces in Galilean space \(G_{3}\) will be introduced. Example of Hasimoto surfaces will be illustrated.
Introduction
The geometry of Galilean is one of the Non Euclidean geometry which is very important in special Relativity. For more about Galilean geometry one can read [1,2,3,4].
The Galilean geometry is the geometry that is transferred from Euclidean geometry to special relativity. A long time ago curves and surfaces in Euclidean space were studied. Recently, mathematicians have begun to introduce curves and surfaces in Galilean spaces \(G_{3}\) and \(G_{4}\) the reader can see the following references [5,6,7,8,9,10,11].
Hasimoto surfaces are obtained when the motions of local speed of the curve is proportional to the local curvature of the curve. Hasimoto surfaces is studied in Minkowski 3space reader can see [12]. Generated surfaces via inextensible flows of curves in \(R^{3}\) are studied by Rawa and Samah [13]. Hasimoto surfaces were constructed by many mathematicians [3, 12, 14].
The position vector of the surface \(\chi =\chi (s,t)\) is called Hasimoto surface if the relation \(\chi _{t}=\chi _{s}\times \chi _{ss}\) hold.
In this article Hasimoto surfaces \(\chi =\chi (s,t)\) in Galilean space \(G_{3}\) will be introduced, Gauss curvature (K) and the Mean curvature (H) of Hasimoto surfaces will be obtained. Some conditions for the \(s{\text {}}parameter\) curves and \(t{\text {}}parameter\) curves of Hasimoto surfaces to be geodesic curves, or asymptotic lines in Galilean space \(G_{3}\) will be given. Finally the necessary and sufficient conditions for the curves to be principal curves on the Hasimoto surfaces in \(G_{3}\) will be introduced. Example of Hasimoto surfaces \(\chi =\chi (s,t)\) in Galilean space \(G_{3}\) will be illustrated.
Preliminaries
Galilean space of dimension three \((G_{3})\), is defined to be the space due to Cayleyâ€“Klein model, equipped with the metric of signature \((0,0,+,+)\) which is called projective metric. The triple \((\omega ,f,I)\) are called The absolute of Galilean geometry where \(\omega\) is defined to be the ideal plane (sometimes called the absolute plane), f is a line in the absolute plane \(\omega\) which is called the absolute line and I is defined to be the elliptic involution point \((0,0,x_{2},x_{3})\rightarrow (0,0,x_{3},x_{2})\).
If the plane contains f, it is called the Euclidean plane, if the plane does not contain f it is called isotropic plane, this means that planes \(x=\) constant are Euclidean planes, i.e. the plane \(\omega\) is Euclidean plane. A vector \(v=(v_{1},v_{2},v_{3})\) is called nonisotropic vector if the first component \(v_{1}\) is not equal to zero. All vectors \(v=(1,v_{2},v_{3})\) are unit nonisotropic vectors. The vectors \(v=(0,v_{2},v_{3})\) are isotropic vectors.
In Galilean space \(G_{3}\) we have four types of lines [1]:

1.
Lines, which do not cross the absolute line f is called proper nonisotropic lines.

2.
The lines, which not belong to the ideal plane \(\omega\) but intersect the absolute line f is called the proper isotropic lines.

3.
All lines of the ideal plane \(\omega\) except f are called proper nonisotropic lines.

4.
The absolute line f.
Suppose that \(\overrightarrow{u}=(u_{1},u_{2},u_{3})\) and \(\overrightarrow{v} =(v_{1},v_{2},v_{3})\) are two vectors in Galilean space \(G_{3}\). Galilean scalar product in \(G_{3}\) is
The norm of the vector \(\overrightarrow{u}=(u_{1},u_{2},u_{3})\) can be written as
\(\left\ \overrightarrow{u}\right\ _{G_{3}}=\sqrt{\langle \overrightarrow{u}, \overrightarrow{u}\rangle _{G_{3}}}\).
The vector product of \(\overrightarrow{u}=(u_{1},u_{2},u_{3})\) and \(\overrightarrow{v}=(v_{1},v_{2},v_{3})\) in Galilean space \(G_{3}\) is defined by
The curve \(r(s)=(s,y(s),z(s))\) is called the admissible curve. The associated invariant trihedron (Frenet invariant) \({\mathbf {T,N}}\), and \({{\mathbf {B}}}\) for r(s) is given by the following equations.
where \({\mathbf {T}}\) is the Tangent vector to r(s), \({\mathbf {N}}\) is the Normal vector to r(s), and \({\mathbf {B}}\) is the Binormal of r(s).
Also k(s) is called the curvature function of the admissible curve r(s), and is denoted by the relation
and \(\tau (s)\) is the torsion function of the admissible curve r(s) and is given by the following equation
The Frenet equations in Galilean space \(G_{3}\) for the a admissible curve r(s) can be written as
A \(C^{n}{\text {}}surface\) M, \(n\ge 1\), immersed in Galilean space \(r:U\rightarrow M,U\) belongs to \(R^{2}\), is denoted by \(\chi (s,t)=(x(s,t),y(s,t),z(s,t))\).
First fundamental form for the surface \(\chi (s,t)\) is denoted by I and is given by the following equation.
where the symbols \(g_{i}=x_{i}\) is the derivatives of the first coordinates function x(s,Â t) with respect to s and t, and \(h_{ij}=\tilde{r}_{i}. \tilde{r}_{j}\) the Euclidean inner product of the projection \(\tilde{r}_{k}\) onto \(yz{\text {}}plane\). Furthermore,
Gauss curvature K is denoted by
Mean curvature H is given by
where
and
The vector \(N = \frac{1}{W}(0,x_{t} z_{s}  x_{s} z_{t} ,x_{s} y_{t}  x_{t} y_{s} )\) is the normal vector to the surface \(\chi (s,t)\).
is called a side tangential vector which is tangent plane to surface M.
Main results
In this section we will introduce Frenet equations of curves in both directions s, and t parameters. For Hasimoto surface \(\chi (s,t)\), we will obtain Gauss Curvature (K), Mean Curvature (H), and we will prove that Hasimoto surfaces are Weingarten surfaces. Also we obtain the necessary and sufficient conditions for the \(t{\text {}}curves\) of Hasimoto surface \(\chi (s,t)\) to be geodesic curves, or to be asymtotic curves. Also, we give conditions of the parameter curves to be lines of curvature. Finally, we give characterization for the \(s{\text {}}parameter\) curves to be principal direction for Hasimoto surface \(\chi (s,t)\). At the end of this section example of Hasimoto surface in Galilean space \(G_{3}\) is introduced.
Theorem 1
Let \(\chi =\chi (s,t)\) be Hasimoto surface in Galilean space \(G_{3}\) where \(\chi =\chi (s,t)\) is admissible curve with unit speed for all t. The Frenet equations \({\mathbf {T}}^{^{\prime }}\mathbf {,N}^{^{\prime }}\) and \({\mathbf {B}}^{^{\prime }}\) with respect to the parameter s is given by the following equations
The Frenet Equations \({\mathbf {T}}^{\cdot },\mathbf {N}^{\cdot }\) and \(\ {\mathbf {B}}^{\cdot }\) with respect to the parameter t, is obtained by the following equations
where \(k\ne 0\) is the curvature and \(\tau\) is the torsion for the curve \(\chi =\chi (s,t)\) \(\forall t\).
Proof
Frenet equations \({\mathbf {T}}^{^{\prime }}\mathbf {,N}^{^{\prime }}\) and \({\mathbf {B}}^{^{\prime }}\) with respect to s is given directly from Frenet equation in Galilean space \(G_{3}\) (1). Suppose that we have the differentiable functions \(\alpha ,\beta ,\gamma\) and \(\eta\) where
Our aim is to find \(\alpha ,\beta ,\gamma\) and \(\eta\) functions interms of the curvature and torsion functions for the \(s{\text{}}curve\) \(\chi =\chi (s,t)\) for all t.
Using the conditions \({\mathbf {T}}_{ts}={\mathbf {T}}_{st}\) and \({\mathbf {N}}_{ts}= {\mathbf {N}}_{st}\) we obtain
i.e.
From the condition \(\chi _{st}=\chi _{ts}\) we get the the following equations
substituting from Eqs.Â (8, 9) we give the system in (6). \(\square\)
In the following theorem we will prove that Gaussian curvature K for Hasimoto surface equal to zero and the mean curvature H is equal to \(\dfrac{(x_{t}^{2}+2\tau x_{s}x_{t}+\tau ^{2}x_{s}^{2})}{2k}\).
Theorem 2
Let \(\chi =\chi (s,t)=(x(s,t),y(s,t),z(s,t))\) be a Hasimoto surface in Galilean space \(G_{3}\) where \(s{\text{}}curves\) of the Hasimoto surfaces \(\chi (s,t)\) is curves with unit norm of the speed for all t, then the Gauss curvature K of \(\chi (s,t)\) will be given form the relation
and the Mean curvature H of \(\chi (s,t)\) will be obtained from the relation
k is the curvature function of \(s{\text{}}curves\) of \(\chi (s,t)\) for all t and \(\tau (s)\) is the torsion function of \(s{\text{}}curves\) of \(\chi (s,t)\) for all t.
Proof
Suppose that \(\chi (s,t)=(x(s,t),y(s,t),z(s,t))\) is a parametrization of the surface \(\chi (s,t)\) where the parameters \(s,t\in R\), and \(x(s,t),y(s,t),z(s,t)\in C^{3}\). The normal of the surface is given by \(N= {\mathbf {N}}\)
since \(\chi _{s}={\mathbf {T}}\) we obtain \(\chi _{st}=k\tau {\mathbf {N}}\) from the property of Hasimoto surfaces \(r_{t}=k\mathbf {B,}\) we have \(r_{ts}=k_{s} {\mathbf {B}}k\tau {\mathbf {N}}\) therefore \(k_{s}=0\). By using the statement (4) of the second fundamental form we give
hence, Gauss curvature K of Hasimoto surfaces \(\chi (s,t)\) identically zero.
Mean curvature H of Hasimoto surface is given by
\(\square\)
Since Gauss Curvature of Hasimoto surfaces in Galilean space \(G_{3}\) equal zero the following corollary is true.
Corollary 1
Hasimoto surface \(\chi (s,t)\) is a Weingarten surface in \(G_{3}\).
Proof
The identically Jacobi equation
\(\Phi (H,K)=K_{t}H_{s}H_{t}K_{s}=0\)
Therefore, Hasimoto surface \(\chi (s,t)\) is Weingarten surface. \(\square\)
The curve r(s) is a geodesic curve if and only if it has geodesic curvature equal to zero \((k_{g}=0)\), the curve is called asymptotic is its normal curvature \(k_{n}=0\)
In the following theorems we give some properties for the \(s{\text{}}curves\) and \(t{\text{}}curves\) of Hasimoto surface \(\chi (s,t)\) to be geodesic curves and asymptotic curves in \(G_{3}\).
Theorem 3
Let \(\chi (s,t)\) be a Hasimoto surface in \(G_{3}\). Then the following statements are satisfied

1.
The \(s{\text{}}curves\) of \(\chi (s,t)\) are geodesic curves.

2.
The \(t{\text{}}curves\) of \(\chi (s,t)\) are geodesic curves\(, \Longleftrightarrow\) the curvature of the \(t{\text{}}curves\) of \(\chi (s,t)\) equal to zero for all s \((k_{t}=0)\).
Proof
1. For the \(s{\text{}}curves\) of the Hasimoto \(\chi (s,t)\) for all t, the geodesic curvature is obtain from the following relation
\(k_{g}=S\cdot \chi _{ss}=(N\times \mathbf {T)\cdot (}k\mathbf {N)=0}\), which proof the statement 1.
2. The geodesic curvature for the \(t{\text{}}curves\) of the Hasimoto surface\(\chi (s,t)\) for all s is \(k_{g}=S\cdot \chi _{tt}=(n\times \mathbf {T)\cdot (} k_{t}{\mathbf {B}}+k\tau ^{2}\mathbf {N)=}k_{t}\). \(\square\)
Theorem 4
Suppose that \(\chi (s,t)\) is Hasimoto surface in Galilean space \(G_{3}\). Then the following statement are satisfied.

1.
\(s{\text{}}curves\) are asymptotic \(\Longleftrightarrow\) if \(k=0\) (which means that \(s{\text{}}curves\) not asymptotic curves).

2.
\(t{\text{}}curves\) are asymptotic curves of Hasimoto surface \(\chi (s,t) \Longleftrightarrow \tau =0\).
Proof
1. Let \(\chi (s,t)\) be Hasimoto surface in Galilean space \(G_{3}\). Since normal curvature \(k_{n}=N\cdot \chi _{ss}={\mathbf {N}}\cdot k{\mathbf {N}}=k \mathbf {,}\) then \(s{\text{}}curves\) are asymptotic curves \(\Leftrightarrow\) \(k=0\) (imposble).
2. For \(t{\text{}}curves\) we have \(\chi _{tt}=k_{t}\mathbf {B+}k\tau ^{2} \mathbf {N,}\) \(k_{n}=\mathbf {N\cdot (}k_{t}\mathbf {B+}k\tau ^{2}\mathbf {N)}= k\tau ^{2}\) i.e. \(t{\text{}}curves\) are asymptotic curves of Hasimoto surface \(\Longleftrightarrow k\tau ^{2}=0\) but \(k\ne 0\) therefore \(\tau ^{2}=0\) this means that \(\tau\) must equal zero. \(\square\)
Corollary 2
\(s{\text{}}curves\) and \(t{\text{}}curves\) of Hasimoto surface \(\chi =\chi (s,t)\) in \(G_{3}\) are said to be lines of curvature if and only if \(k\tau =0\).
Proof
\(F=M\mathbf {=0}\) \(\Leftrightarrow\) \(k\tau =0\). \(\square\)
Corollary 3
If \(s{\text {}}curves\) and \(t{\text {}}curves\) of Hasimoto surfaces \(\chi (s,t)\) in \(G_{3}\) are asymptotic curves then \(s{\text {}}curves\) and \(t{\text {}}curves\) are lines of curvatures.
Proof
From TheoremÂ 4 above \(t{\text {}}curves\) are asymptotic curves of Hasimoto surfaces \(\Leftrightarrow\) \(\tau =0\). This implies \(k\tau =0\) which means that \(t{\text {}}curves\) are lines of curvatures. \(\square\)
Principal direction are tangent directions of a curve r(s) on a surface if the normal field of the surface satisfy \(\det (\alpha ^{\cdot },N,N^{\cdot })=0\) this condition essential for principal directions in Euclidean space [15].
Theorem 5
Let, \(\chi (s,t)\) be Hasimoto surfaces in \(G_{3}\), then

1.
\(s{\text {}}curves\) of Hasimoto surface \(\chi (s,t)\) are principal direction for all t if and only if \(\tau =0\).

2.
\(t{\text {}}curves\) of Hasimoto surface \(\chi (s,t)\) are principal direction.
Proof
1. For sparameter curves \(\det (\chi _{s},N,N_{s})=\det ({\mathbf {T}}, {\mathbf {N}},{\mathbf {N}}_{s})=\tau \det (\mathbf {T,N,B}).\)
Hence, \(\det (\chi _{s},N,N_{s})=0\) \(\Longleftrightarrow\) \(\tau =0\).
2. For tparameter curves \(\det (\chi _{t},N,N_{t})=\det (k{\mathbf {B}}, {\mathbf {B}},\tau ^{2}{\mathbf {B}})=0\). \(\square\)
Example 1
Consider Hasimoto surface (Fig. 1) \(\chi (s,t)\) where
\(\chi (s,t)=(s+10,\cos (s+t),\sin (s+t))\), \(0.5\le s,t\le 0.5\), then
The tangent vector for the curve is
\({\mathbf {T}}=(1\), \(\sin (s+t)\), \(\cos (s+t))\)
The normal vector for the curve is
\({\mathbf {N}}=(0\), \(\cos (s+t)\), \(\sin (s+t))\)
The binormal vector for the curve is
\({\mathbf {B}}=(0\), \(\sin (s+t)\), \(\cos (s+t))\)
the curvature function \(k=1\), the torsion function \(\tau =1\)
Mean curvature for \(\chi (s,t)\) is \(H=1\)
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Abbreviations
 \(G_{3}\) :

Galilean space of dimension three
 k(s):

Curvature function
 \(\tau(s)\) :

Torsion function
 K :

Gauss curvature
 H :

Mean curvature
 N :

The normal of the surface
 \(k_{g}\) :

The geodesic curvature
References
Dede, M., Ekici, C.: On parallel ruled surfaces in Galilean space. Kragujev. J. Math. 40(1), 47â€“59 (2016)
Elzawy, M., Mosa, S.: Smarandache curves in the Galilean 4Space G_{4}. J. Egypt. Math. Soc. 25, 53â€“56 (2017)
Aydin, M.E., Mihai, A., Ogrenmis, A.O., Ergut, M.: Geometry of the solutions of localized induction equation in the pseudoGalilean space. Adv. Math. Phys. 2015, Article ID 905978
Mosa, S., Elzawy, M.: Helicoidal surfaces in Galilean space with density. Front. Phys. 8, 1â€“6 (2020)
Ogrenmis, A., Ergut, M., Bekatas, M.: On the helices in the Galilean space G_{3}. Iran. J. Sci. Technol. Trans. Print. A Islamic Repub. Iran 31(A2), 177â€“181 (2007)
Yoon, D.W.: Some classification of translation surfaces in Galilean 3space. Int. J. Math. Anal. 6(28), 1355â€“1361 (2012)
Yoon, D.W., Lee, J.W., Lee, C.W.: Osculating curves in the Galilean 4space. Int. J. Pure Appl. Math. 100(4), 497â€“506 (2015)
Yaglom, I.M.: A Simple NonEuclidean Geometry and Its Physical Basis. Springer, New York (1979)
Dede, M.: Tubuler surfaces in Galilean space. Math. Commun. 18, 209â€“217 (2013)
Elzawy, M., Mosa, S.: Razzaboni surfaces in the Galilean space G^{3}. FJMS 108(1), 13â€“26 (2018)
Dede, M., Ekici, C., Coken, A.: On the parallel surfaces in Galilean space. Hacet. J. Math. Stat. 42(6), 605â€“615 (2013)
Erdogdu, M., Ozdemir, M.: Geometry of Hasimoto surfaces in Minkowski 3space. Math. Phys. Anal. Geom. 17, 169â€“181 (2014)
Hussien, R.H., Mohamed, S.G.: Generated surfaces via inextensible flows of curves in R^{3}. J. Appl. Math. 2016, Article ID 6178961
Abdelall, N.H., Hussien, R.A., Youssef, T.: Hasimoto surfaces. Life Sci. J. 9(3), 556â€“560 (2012)
Sipus, Z.M., Divjak, B.: Surfaces of constant curvature in the pseduoGalilean space. Int. J. Math. Math. Sci. 2012, Article ID 375264
Acknowledgements
The author would like to thank the referees for their helpful suggestions.
Funding
The author is the research funded.
Author information
Authors and Affiliations
Contributions
The author collected the data, performed the calculation, and was a major contributor in writing the manuscript. The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declare that she has no competing interests.
Additional information
Publisherâ€™s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Elzawy, M. Hasimoto surfaces in Galilean space \(G_{3}\). J Egypt Math Soc 29, 5 (2021). https://doi.org/10.1186/s4278702100113y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s4278702100113y