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.
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 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.
\(s{\text {}}curves\) 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 \(H=1\)