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Hasimoto surfaces in Galilean space \(G_{3}\)

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 s-curves and t-curves 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 3-space 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 Cayley–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 non-isotropic vector if the first component \(v_{1}\) is not equal to zero. All vectors \(v=(1,v_{2},v_{3})\) are unit non-isotropic 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. 1.

    Lines, which do not cross the absolute line f is called proper non-isotropic lines.

  2. 2.

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

  3. 3.

    All lines of the ideal plane \(\omega\) except f are called proper non-isotropic lines.

  4. 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

$$\begin{aligned} \langle \overrightarrow{u},\overrightarrow{v}\rangle _{G3}= \left\{ \begin{array}{ll} {u_{1}v_{1}} &{} if\, u_{1}\ne 0\, or\, v_{1}\ne 0\\ {u_{2}v_{2}+u_{3}v_{3}} &{}if\, u_{1}=0\, and\, v_{1}=0 \end{array}\right. \end{aligned}$$

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

$$\begin{aligned} \overrightarrow{u}\times \overrightarrow{v}=\left\{ \begin{array}{c} \begin{vmatrix} 0 &{}\quad e_{2} &{}\quad e_{3} \\ x_{1} &{}\quad x_{2} &{}\quad x_{3} \\ y_{1} &{}\quad y_{2} &{}\quad y_{3} \end{vmatrix} {\text { if }}x_{1}\ne 0\;{\text {or}}\;y_{1}\ne 0. \\ \begin{vmatrix} e_{1} &{}\quad e_{2} &{} \quad e_{3} \\ x_{1} &{}\quad x_{2} &{}\quad x_{3} \\ y_{1} &{}\quad y_{2} &{}\quad y_{3} \end{vmatrix} {\text { if }}x_{1}=0\;{\text {and}}\;y_{1}=0. \end{array} \right. \end{aligned}$$

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.

$$\begin{aligned} {{\mathbf {T}}}= & {} \left( 1,y^{\prime },z^{\prime }\right) \\ {\mathbf {N}}= & {} \frac{1}{k}\left( 0,y^{\prime \prime },z^{\prime \prime }\right) \\ {\mathbf {B}}= & {} \frac{1}{k}\left( 0,-z^{\prime \prime },y^{\prime \prime }\right) \end{aligned}$$

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

$$\begin{aligned} k(s)=\sqrt{y^{\prime \prime 2}+z^{\prime \prime 2}} \end{aligned}$$

and \(\tau (s)\) is the torsion function of the admissible curve r(s) and is given by the following equation

$$\begin{aligned} \tau (s)=\frac{1}{k^{2}}\det \left( r^{\prime }(s),r^{\prime \prime }(s),r^{\prime \prime \prime }(s)\right) . \end{aligned}$$

The Frenet equations in Galilean space \(G_{3}\) for the a admissible curve r(s) can be written as

$$\begin{aligned} \left[ \begin{array}{c} {\mathbf {T}}^{^{\prime }} \\ {\mathbf {N}}^{^{\prime }} \\ {\mathbf {B}}^{^{\prime }} \end{array} \right] =\left[ \begin{array}{ccc} 0 &{}\quad k &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad \tau \\ 0 &{}\quad -\tau &{} \quad 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {T}} \\ {\mathbf {N}} \\ {\mathbf {B}} \end{array} \right] \end{aligned}$$
(1)

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.

$$\begin{aligned} I=\left( g_{1}ds+g_{2}dt\right) ^{2}+\epsilon \left( h_{11}ds^{2}+2h_{12}dsdt+h_{22}dt^{2}\right) \end{aligned}$$

where the symbols \(g_{i}=x_{i}\) is the derivatives of the first coordinates function x(st) 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,

$$\begin{aligned} \epsilon =\left\{ \begin{array}{l} 0,\quad if\,ds:dt\;is\;non{\text {-}}isotropic \\ 1,\quad if\,ds:dt\;is\;isotropic \end{array} \right. \end{aligned}$$

Gauss curvature K is denoted by

$$\begin{aligned} K=\dfrac{L_{11}L_{22}-L_{12}^{2}}{W^{2}} \end{aligned}$$
(2)

Mean curvature H is given by

$$\begin{aligned} H=\dfrac{g_{2}^{2}L_{11}-2g_{1}g_{2}L_{12}+g_{1}^{2}L_{22}}{2W^{2}} \end{aligned}$$
(3)

where

$$\begin{aligned} W=\sqrt{\left( x_{t}z_{s}-x_{s}z_{t}\right) ^{2}+\left( x_{s}y_{t}-x_{t}y_{s}\right) ^{2}} \end{aligned}$$

and

$$\begin{aligned} L_{ij}=\frac{x_{s}r_{ij}-x_{ij}r_{s}}{x_{s}}.N, x_{s}=g_{1}\ne 0, \end{aligned}$$
(4)

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 \(\chi (s,t)\).

$$\begin{aligned} S=\frac{1}{W}\left( 0,x_{t}y_{s}-x_{s}y_{t},x_{t}z_{s}-x_{s}z_{t}\right) \end{aligned}$$

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

$$\begin{aligned} \left[ \begin{array}{c} {\mathbf {T}}^{^{\prime }} \\ {\mathbf {N}}^{^{\prime }} \\ {\mathbf {B}}^{^{\prime }} \end{array} \right] =\left[ \begin{array}{ccc} 0 &{}\quad k &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \tau \\ 0 &{}\quad -\tau &{}\quad 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {T}} \\ {\mathbf {N}} \\ {\mathbf {B}} \end{array} \right] \end{aligned}$$
(5)

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

$$\begin{aligned} \left[ \begin{array}{c} {\mathbf {T}}^{\cdot } \\ {\mathbf {N}}^{^{\cdot }} \\ {\mathbf {B}}^{^{\cdot }} \end{array} \right] =\left[ \begin{array}{ccc} 0 &{} \quad -\tau k &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -\tau ^{2} \\ 0 &{}\quad \tau ^{2} &{}\quad 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {T}} \\ {\mathbf {N}} \\ {\mathbf {B}} \end{array} \right] \end{aligned}$$
(6)

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

$$\begin{aligned} \left[ \begin{array}{c} {\mathbf {T}}^{^{\cdot }} \\ {\mathbf {N}}^{\cdot } \\ {\mathbf {B}}^{\cdot } \end{array} \right] =\left[ \begin{array}{ccc} 0 &{} \quad \alpha &{}\quad \gamma \\ \beta &{}\quad 0 &{}\quad \eta \\ -\gamma &{}\quad -\eta &{}\quad 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {T}} \\ {\mathbf {N}} \\ {\mathbf {B}} \end{array} \right] \end{aligned}$$
(7)

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

$$\begin{aligned} \left( \alpha _{s}-\gamma \tau \right) {\mathbf {N}}+\left( \alpha \tau +\gamma _{s}\right) {\mathbf {B}}=\left( k\beta \right) {\mathbf {T}}+\left( k_{t}\right) \mathbf {N+}k\eta {\mathbf {B}} \end{aligned}$$
(8)

i.e.

$$\begin{aligned} \beta =0,\alpha _{s}=\gamma \tau +k_{t},\gamma _{s}=k\eta -\alpha \tau \end{aligned}$$
(9)

From the condition \(\chi _{st}=\chi _{ts}\) we get the the following equations

$$\begin{aligned} \gamma =0,\eta =-\tau ^{2} \end{aligned}$$
(10)

substituting from Eqs. (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

$$\begin{aligned} K=0 \end{aligned}$$
(11)

and the Mean curvature H of \(\chi (s,t)\) will be obtained from the relation

$$\begin{aligned} H=\dfrac{-\left( x_{t}^{2}+2\tau x_{s}x_{t}+\tau ^{2}x_{s}^{2}\right) }{2k} \end{aligned}$$
(12)

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

$$\begin{aligned} L_{ij}=\left( \begin{array}{cc} -k &{}\quad k\tau \\ k\tau &{}\quad \mathbf {-}k\tau ^{2} \end{array} \right) \end{aligned}$$

hence, Gauss curvature K of Hasimoto surfaces \(\chi (s,t)\) identically zero.

Mean curvature H of Hasimoto surface is given by

$$\begin{aligned} H= & {} \dfrac{g_{2}^{2}L_{11}-2g_{1}g_{2}L_{12}+g_{1}^{2}L_{22}}{2W^{2}}= \dfrac{-kx_{t}^{2}-2k\tau x_{s}x_{t}-k\tau ^{2}x_{s}^{2}}{2k^{2}} \\= & {} \dfrac{-\left( x_{t}^{2}+2\tau x_{s}x_{t}+\tau ^{2}x_{s}^{2}\right) }{2k} \end{aligned}$$

\(\square\)

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. 1.

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

  2. 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. 1.

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

  2. 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 Theorem 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. 1.

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

  2. 2.

    \(t{\text {-}}curves\) of Hasimoto surface \(\chi (s,t)\) are principal direction.

Proof

1. For s-parameter 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 t-parameter 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 \(H=-1\)

Fig. 1
figure1

Hasimoto surface \(\chi (s,t)=(s+10,-\cos (s+t),\sin (s+t))\)

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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

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Elzawy, M. Hasimoto surfaces in Galilean space \(G_{3}\). J Egypt Math Soc 29, 5 (2021). https://doi.org/10.1186/s42787-021-00113-y

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Keywords

  • Galilean geometry
  • Hasimoto surface
  • Smoke ring equation

Mathematics Subject Classification

  • 53A35
  • 53B25
  • 53C42