Helicoidal Surfaces in Galilean Space With Density

In this paper, we construct helicoidal surfaces in the three dimensional Galilean space G³. The First and the Second Fundamental Forms for such surfaces will be obtained. Also, mean and Gaussian curvature given by smooth functions will be derived. We consider the Galilean 3−space with a linear density e^ϕ and construct a weighted helicoidal surfaces in G³ by solving a second order non-linear differential equation. Moreover, we discuss the problem of finding explicit parameterization for the helicoidal surfaces in G³.

M.S.C.2010: 53A35, 51A05

1. Introduction

Due to its applications in probability and statistics, the study of manifolds with density has increased in the last years after Morgan's published his paper “Manifolds with density” [1]. As a new field in geometry, manifolds with density appear in different ways in mathematics, for example as quotients of Riemannian manifolds or as Gauss space [2].

Helicoidal surface is a natural generalization of rotation surface, of which many excellent works have been done, such as Kenmotsu [3].

For helicoidal surface in R³, the cases with prescribed mean curvature or Gauss curvature have been studied by Baikoussis and Koufogiorgos [4]. Also, helicoidal surfaces in three dimensional Minkowski space has been considered by Beneki et al. [5]. A kind of helicoidal surface in 3−dimensional Minkowski space was constructed by Ji and Hou [6].

Construction of helicoidal surfaces in Euclidean space with density by solving second-order non-linear ordinary differential equation with weighted minimal helicoidal surface was introduced in Kim et al. [7]. For weighted type integral inequalities, one can see Agarwal et al. [8].

Mean and Gaussian curvature for surfaces are one of the main objects, which have geometers interest for along time. A manifold with density is a Riemannian manifold Mⁿ with a positive function e^ϕ, known as density, used to weight volume and hypersurface area [2, 9]. A nice example of manifolds with density is Gauss space, the Euclidean space with Gaussian probability density ${(2\pi )}^{\frac{-n}{2}}{e}^{\frac{-r^2}{2}}$, which is very useful to probabilists [2].

On a manifold with density e^ϕ, the weighted mean curvature of a hypersurface with unit normal N is defined by

$\begin{array}{ll}{H}_{\varphi }=H-\frac{1}{n}\frac{d\varphi }{dN}& (1)\end{array}$

where H is the Riemannian mean curvature of the hypersurface [9]. The weighted mean curvature H_ϕ of a surface in E³ with density e^ϕ was introduced by Gromov [10], and it is a natural generalization of the mean curvature H of a surface. The curvature concept is not confined to continuous space, it has been intensively studied in discrete mathematics including networks, for more details one can see Shang [11].

A surface with H_ϕ = 0 is known as a weighted minimal surface or a ϕ−minimal surface in E³ [12]. For more details about manifolds with density and other relative topics, we refer the reader to [1–3, 5–7, 9, 10, 13–16]. In particular, Yoon et al. [17] studied rational surfaces in Pseudo-Galilean space with a log-linear density and investigated ϕ−minimal rotational surfaces. Also, they classified the weighted minimal helicoidal surfaces in the Euclidean space E³ [7].

The purpose of this paper is to construct helicoidal surface in Galilean space G³. Firstly, we choose orthonormal basis as the coordinate frame and define helicoidal surface with density. The first fundamental form ds², the second fundamental form II, the Gaussian and Mean curvature of helicoidal surface will be obtained in section 3. Secondly in section 4, we prescribed the parametrization of a weighted mean curvature $H_{\varphi }=H-\frac{1}{2}<N,\nabla \varphi >$ and solving the non-linear differential equation.

2. Preliminaries

In this part, we give a brief review of curves and surfaces in the Galilean space G³. For more details, one can see [12, 14–16, 18].

The Galilean 3−space G³ can be defined in the three-dimensional real projective space P₃(R) and its absolute figure is an ordered triple {ρ, f, I}, where ρ is the ideal (absolute) plane, f a line in ρ and I is the fixed elliptic involution of the points of f. We introduce homogeneous coordinates in G³ in such a way that the absolute plane ρ is given by x_o = 0, the absolute line f by x_o = x₁ = 0 and the elliptic involution by

$\begin{array}{ll}(0:0:x_2:x_3)\to (0:0:x_3:-x_2)& (2)\end{array}$

A plane is said to be Euclidean if it contains f, otherwise it is called isotropic, i.e., the planes x = const. are Euclidean, in particular the plane ρ. Other planes are isotropic.

The Galilean distance between the points Q_i = (r_i, s_i, t_i), (i = 1, 2) is given by

$\begin{array}{ll}d(Q_1,Q_2)=\{\begin{array}{ll}\mid r_2-r_1\mid ,\hfill & \text{if }{r}_1\ne 0\text{ or }{r}_2\ne 0;\hfill \\ \sqrt{{(s_2-s_1)}^2+{(t_2-t_1)}^2},\hfill & \text{if }{r}_1=0\text{ and }{r}_2=0.\hfill \end{array}& (3)\end{array}$

Moreover, the distance in the Euclidean space between Q₁ and Q₂ is given by

$\begin{array}{l}d(Q_1,Q_2)=\sqrt{{(r_2-r_1)}^2+{(s_2-s_1)}^2+{(t_2-t_1)}^2}\end{array}$

The Galilean scalar product between two vectors P = (p₁, p₂, p₃) and Q = (q₁, q₂, q₃) is defined by

$\begin{array}{ll}<P,Q{>}_G=\{\begin{array}{ll}{p}_1{q}_1,\hfill & \text{if }{p}_1\ne 0\text{ or }{q}_1\ne 0;\hfill \\ p_2{q}_2+p_3{q}_3,\hfill & \text{if }{p}_1=0\text{ and }{q}_1=0.\hfill \end{array}& (4)\end{array}$

For this, the Galilean norm of a vector P is $\parallel P\parallel =\sqrt{<P,P{>}_G}$. Moreover, the cross product in the Galilean space is defined by

$\begin{array}{l}<P{\times }_{G}Q>=\left(0,\left|\begin{array}{cc}{p}_1& p_3\\ q_1& q_3\end{array}\right|,\left|\begin{array}{cc}{p}_1& p_2\\ q_1& q_2\end{array}\right|\right)\end{array}$

A vector P = (p₁, p₂, p₃) is said to be isotropic if p₁ = 0, otherwise it is known as non-isotropic. The following definitions will be helpful [19].

Definition 1. Let a = (1, y₂, y₃) and b = (1, z₂, z₃) be two unit non-isotropic vectors in general position in G³. Then an angle θ between a and b is given by

$\begin{array}{l}\theta =\sqrt{{(z_2-y_2)}^2+{(z_3-y_3)}^2}\end{array}$

Definition 2. An angle ψ between a unit non-isotropic vector a = (1, y₂, y₃) and an isotropic vector c = (0, z₂, z₃) in G³ is given by

$\begin{array}{l}\psi =\frac{y_2{z}_2+y_3{z}_3}{\sqrt{z_2^2+z_3^2}}\end{array}$

Definition 3. An angle η between two isotropic vectors c = (0, y₂, y₃) and d = (0, z₂, z₃) parallel to the Euclidean plane in G³ is equal to the Euclidean angle between them. Namely,

$\begin{array}{l}cos\eta =\frac{y_2{z}_2+y_3{z}_3}{\sqrt{y_2^2+y_3^2}\sqrt{z_2^2+z_3^2}}\end{array}$

Definition 4. The curve α(t) = (x(t), y(t), z(t)) in the Galilean space G³ is said to be admissible if it has no inflection points $({\alpha }^{.}(t){\times }_G{\alpha }^{..}(t)\ne 0)$ and no isotropic tangents (x^.(t) ≠ 0).

Let C be an open subset of R² and M a surface in G³ parameterized by

$\begin{array}{ll}r:C\subseteq R^2\to G^3,r(u,v)=(x(u,v),y(u,v),z(u,v))& (5)\end{array}$

In order to specify the partial derivatives we will denote:

$\begin{array}{ll}{x}_u=\frac{\partial x}{\partial u},x_v=\frac{\partial x}{\partial v}and{x}_{uv}=\frac{{\partial }^{2}x}{\partial u\partial v}& (6)\end{array}$

Then r is satisfied admissibility criteria if no where it has Euclidean tangent planes. The first fundamental form is given by

$\begin{array}{ll}d{s}^2={(g_{1}du+g_{2}dv)}^2+\epsilon (h_{11}d{u}^2+2h_{12}dudv+h_{22}d{v}^2)& (7)\end{array}$

where $g_1=x_u=\frac{\partial x}{\partial u}$, $g_2=x_v=\frac{\partial x}{\partial v}$, and $h_{11}=y_u^2+z_u^2$, $h_{22}=y_v^2+z_v^2$, h₁₂ = y_uy_v + z_uz_v, also

$\begin{array}{ll}\epsilon =\{\begin{array}{l}0,\text{ if the direction }du:dv\text{ is non-isotropic};\hfill \\ 1,\text{ if the direction }du:dv\text{ is isotropic}.\hfill \end{array}& (8)\end{array}$

Now, consider the function

$\begin{array}{ll}\omega =\parallel r_u\times r_v\parallel =\sqrt{{(x_u{z}_v-x_v{z}_u)}^2+{(x_v{y}_u-x_u{y}_v)}^2}& (9)\end{array}$

hence the isotropic unit normal vector field N of the surface r = r(u, v) is given by

$\begin{array}{ll}N=\frac{r_u\times r_v}{\parallel r_u\times r_v\parallel }=\frac{1}{\omega }(0,x_u{z}_v-x_v{z}_u,x_v{y}_u-x_u{y}_v)& (10)\end{array}$

The second fundamental form is obtained by

$\begin{array}{ll}II=L_{11}d{u}^2+2L_{12}dudv+L_{22}d{v}^2& (11)\end{array}$

such that

$\begin{array}{l}{L}_{ij}=\frac{1}{g_1}(g_1(0,y_{ij},z_{ij})-{(g_i)}_j(0,y_u,z_u))·N\\ =\frac{1}{g_2}(g_2(0,y_{ij},z_{ij})-{(g_i)}_j(0,y_v,z_v))·N\end{array}$

where i, j = u, v.

Note that the dot “·” denotes the Euclidean scalar product. Therefore, the Gaussian and mean curvature are given by.

$\begin{array}{ll}K=\frac{L_{11}{L}_{22}-L_{12}^2}{{\omega }^2}andH=\frac{g_2^2{L}_{11}-2g_1{g}_2{L}_{12}+g_1^2{L}_{22}}{2{\omega }^2}& (12)\end{array}$

3. Helicoidal Surfaces in the Galilean Space G³

We will take a regular plane curve α(u₁) = (g(u₁), 0, f(u₁)) with g(u₁) > 0 in the xz− plane which is defined on an open interval I ⊂ R. A surface Γ² in the Galilean space G³ is defined by

$\begin{array}{ll}\chi (u_1,u_2)=(g(u_1)cos(u_2),g(u_1)sin(u_2),f(u_1)+b{u}_2)& (13)\end{array}$

is said to be helicoidal surface with axis oz, a pitch b and the profile curve α.

Without loss of generality, we assume that α(u₁) = (u₁, 0, f(u₁)) is the profile curve in the xz− plane defined on an open interval I of positive real numbers (I ⊂ R⁺). So, the helicoidal surface Γ² in G³ is given by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (14)\end{array}$

where f(u₁) is a differentiable function defined on I.

Theorem 5. Let Γ² be helicoidal surface in G³ defined by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (15)\end{array}$

where f(u₁) is a differentiable function defined on I. Then the unit normal vector field N of the surface Γ² is given by

$\begin{array}{ll}N=\frac{1}{\omega }(0,u_1{f}^{\prime }(u_1)sin(u_2)+bcos(u_2),-u_1)& (16)\end{array}$

The first and the second fundamental forms of the surface Γ² in G³ are given respectively by

$\begin{array}{ll}d{s}^2=co{s}^2(u_2)d{u}_1^2-2u_{1}sin(u_2)cos(u_2)d{u}_{1}d{u}_2+u_1^{2}si{n}^2(u_2)d{u}_2^2& (17)\end{array}$

and

$\begin{array}{ll}II=\frac{1}{\omega }(-u_1{f}^{\prime \prime }(u_1)d{u}_1^2+2bd{u}_{1}d{u}_2-u_1^2{f}^{\prime }(u_1)d{u}_2^2).& (18)\end{array}$

Proof. Let Γ² be helicoidal surface in G³ defined by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (19)\end{array}$

Then the unit normal vector field N of the surface Γ² is an isotropic vector field defined by

$\begin{array}{l}N=\frac{1}{\omega }(0,x_{u_1}{z}_{u_2}-x_{u_2}{z}_{u_1},x_{u_2}{y}_{u_1}-x_{u_1}{y}_{u_2})\\ =\frac{1}{\omega }(0,u_1{f}^{\prime }(u_1)sin(u_2)+bcos(u_2),-u_1)\end{array}$

where the positive function ω is given by

$\begin{array}{ll}\omega =\sqrt{(bcos(u_2)+u_{1}f\prime (u_1)sin{(u_2)}^2+u_1^2}& (20)\end{array}$

Here the partial derivatives of the functions x, y, andz with respect to u_i (i = 1, 2) are denoted by x_ui, y_ui, and z_ui, respectively. On the other hand, let us define g_i = x_ui, h_ij = y_uiy_uj + z_uiz_uj, i, j = 1, 2. So, the first fundamental form of the surface Γ² in G³ is given by

$\begin{array}{ll}d{s}^2=d{s}_1^2+\epsilon d{s}_2^2& (21)\end{array}$

where

$\begin{array}{l}d{s}_1^2={(g_{1}d{u}_1+g_{2}d{u}_2)}^2\\ ={(cos(u_2)d{u}_1-u_{1}sin(u_2)d{u}_2)}^2\end{array}$

and

$\begin{array}{l}d{s}_2^2=h_{11}d{u}_1^2+2h_{12}d{u}_{1}d{u}_2+h_{22}d{u}_2^2\\ =(si{n}^2(u_2)+f{\prime }^2(u_1))d{u}_1^2+2(u_{1}sin(u_2)cos(u_2)\\ +bf\prime (u_1))d{u}_{1}d{u}_2+(u_{1}co{s}^2(u_2)+b^2)d{u}_2^2\end{array}$

Then

$\begin{array}{ll}d{s}^2=co{s}^2(u_2)d{u}_1^2-2u_{1}sin(u_2)cos(u_2)d{u}_{1}d{u}_2+u_1^{2}si{n}^2(u_2)d{u}_2^2& (22)\end{array}$

In the sequel, the second fundamental form II of Γ² is given by

$\begin{array}{ll}II=\frac{1}{\omega }(-u_1{f}^{\prime \prime }(u_1)d{u}_1^2+2bd{u}_{1}d{u}_2-u_1^2{f}^{\prime }(u_1)d{u}_2^2)& (23)\end{array}$

where $L_{11}=\frac{-u_1}{\omega }{f}^{\prime \prime }(u_1)$, $L_{22}=\frac{-u_1^2}{\omega }{f}^{\prime }(u_1)$ and $L_{12}=\frac{b}{\omega }$. ■

Corollary 6. The Gaussian curvature K of the surface Γ² is obtained by

$\begin{array}{ll}K=\frac{1}{{\omega }^4}(u_1^3{f}^{\prime }(u_1)f^{\prime \prime }(u_1)-b^2)& (24)\end{array}$

Moreover, the mean curvature of the surface Γ² is given by

$\begin{array}{l}H=\frac{1}{2{\omega }^3}(-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)\\ -u_1^{2}f\prime (u_1)co{s}^2(u_2))& (25)\end{array}$

Proof. Since the Gaussian curvature K is given by $K=\frac{L_{11}{L}_{22}-L_{12}^2}{{\omega }^2}$, then

$\begin{array}{l}K=\frac{1}{{\omega }^2}(\frac{u_1{f}^{\prime \prime }(u_1)}{\omega }\times \frac{u_1^2{f}^{\prime }(u_1)}{\omega }-\frac{b^2}{{\omega }^2})\\ =\frac{1}{{\omega }^4}(u_1^3{f}^{\prime \prime }(u_1)f^{\prime }(u_1)-b^2)\end{array}$

The mean curvature of the surface is obtain from

$\begin{array}{l}H=\frac{g_2^2{L}_{11}-2g_1{g}_2{L}_{12}+g_1^2{L}_{22}}{2{\omega }^2}\end{array}$

By substituting, we get

$\begin{array}{l}H=\frac{1}{2{\omega }^2}[{(-u_{1}sin(u_2))}^2(\frac{-u_1{f}^{\prime \prime }(u_1)}{\omega })\\ -2cos(u_2)(-u_{1}sin(u_2))(\frac{b}{\omega })+co{s}^2(u_2)(\frac{-u_1^2{f}^{\prime }(u_1)}{\omega })]\\ =\frac{1}{2{\omega }^3}[-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)\\ -u_1^2{f}^{\prime }(u_1)co{s}^2(u_2)]■\end{array}$

4. Weighted Helicoidal Surfaces in G³

Let Γ² be a helicoidal surface in G³ defined by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (26)\end{array}$

where f(u₁) is a differentiable function defined on I. Suppose that Γ² is the surface in G³ with a linear density e^ϕ, where ϕ = αx + βy + γz, α, β, γ not all zero.

In this case, the weighted mean curvature H_ϕ of Γ² can be expressed as

$\begin{array}{ll}{H}_{\varphi }=H-\frac{1}{2}{<N,\nabla \varphi >}_{G^3}& (27)\end{array}$

where ∇ϕ is the gradient of ϕ. If Γ² is the weighted minimal surface, then

$\begin{array}{ll}H=\frac{1}{2}{<N,\nabla \varphi >}_{G^3}& (28)\end{array}$

Theorem 7. Let Γ² be weighted minimal helicoidal surface in G³ defined by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (29)\end{array}$

with a linear density e^ϕ, then f(u₁) will be one of the following

1. $f(u_1)=\frac{2bcot(u_2)}{co{t}^2(u_2)-1}\frac{1}{u_1}+c_1{u}_1^{-co{t}^2(u_2)}$

2. $f(u_1)=\frac{1}{2B}[Aln(u_1)-2ln(\frac{r(u_1)}{p(u_1)})+ln(BD)+ln(u_1)]$

3. f(u₁) is the solution of the differential equation $f^{\prime \prime }(u_1)+[\frac{A}{u_1}+\frac{B}{u_1^2}+C]f^{\prime }(u_1)+\frac{D}{u_1}{f}^{\prime 2}(u_1)+E{f}^{\prime 3}(u_1)+[\frac{F}{u_1}+\frac{G}{u_1^2}+\frac{H}{u_1^3}]=0$

4. f(u₁) is the solution of the differential equation $f^{\prime \prime }(u_1)+(B+\frac{A}{u_1}+\frac{C}{u_1^2})f^{\prime }(u_1)+(E+\frac{D}{u_1})f^{\prime 2}(u_1)+F{f}^{\prime 3}(u_1)+(J+\frac{H}{u_1}+\frac{G}{u_1^2}+\frac{I}{u_1^3})=0$

Proof. Let Γ² be a helicoidal surface in G³ defined by

$\begin{array}{ll}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),f(u_1)+b{u}_2)& (30)\end{array}$

where f(u₁) is a differentiable function defined on I. By substituting in equation (27) we obtain

$\begin{array}{l}-u_1^{3}f″(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)-u_1^{2}f\prime (u_1)co{s}^2(u_2)\\ =({(b\cos (u_2)+u_{1}f\prime (u_1)sin(u_2))}^2+u_1^2)<(0,u_{1}f\prime (u_1)sin(u_2)\\ +bcos(u_2),-u_1),(\alpha ,\beta ,\gamma )>\end{array}$

Now, we can distinguish two cases according to the value of α.

Case 1. If α ≠ 0

In this case the vector (α, β, γ) is non-isotropic, with some simple calculation we can obtain the following differential equation

$\begin{array}{ll}{f}^{\prime \prime }(u_1)+\frac{1}{u_1}co{t}^2(u_2)f^{\prime }(u_1)=\frac{2b}{u_1^2}cot(u_2)& (31)\end{array}$

To solve this equation, we make reduction of the order as: Let $f^{\prime }(u_1)=y(u_1)$ which gives $f^{\prime \prime }(u_1)=y^{\prime }(u_1)$, substitutes into equation (31) we obtain the differential equation

$\begin{array}{ll}{y}^{\prime }(u_1)+\frac{1}{u_1}co{t}^2(u_2)y(u_1)=\frac{2b}{u_1^2}cot(u_2)& (32)\end{array}$

Integrating factor $IF=u_1^{co{t}^2(u_2)}$ and hence the solution is given by

$\begin{array}{ll}y(u_1)=\frac{2bcot(u_2)}{co{t}^2(u_2)-1}\frac{1}{u_1}+c_1{u}_1^{-co{t}^2(u_2)}& (33)\end{array}$

i.e.,

$\begin{array}{ll}{f}^{\prime }(u_1)=\frac{2bcot(u_2)}{co{t}^2(u_2)-1}\frac{1}{u_1}+c_1{u}_1^{-co{t}^2(u_2)}& (34)\end{array}$

which gives

$\begin{array}{ll}f(u_1)=\frac{2bcot(u_2)}{co{t}^2(u_2)-1}ln(u_1)+\frac{c_1}{1-co{t}^2(u_2)}{u}_1^{1-co{t}^2(u_2)}+c_2& (35)\end{array}$

Therefore, Γ² is determined by

$\begin{array}{r}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),\frac{2bcot(u_2)}{co{t}^2(u_2)-1}ln(u_1)\\ +\frac{c_1}{1-co{t}^2(u_2)}{u}_1^{1-co{t}^2(u_2)}+b{u}_2+c_2)& (36)\end{array}$

where

$\begin{array}{l}z(u_1,u_2)=\frac{2bcot(u_2)}{co{t}^2(u_2)-1}ln(u_1)+\frac{c_1}{1-co{t}^2(u_2)}{u}_1^{1-co{t}^2(u_2)}\\ +b{u}_2+c_2& (37)\end{array}$

c₁, c₂ are constants.

Case 2. If α = 0

In this case the vector (0, β, γ) is an isotropic and as before, we obtain

$\begin{array}{l}\frac{1}{{\omega }^2}(-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)\\ -u_1^2{f}^{\prime }(u_1)co{s}^2(u_2))\\ =\beta u_{1}f\prime (u_1)sin(u_2)+\beta bcos(u_2)-\gamma u_1\end{array}$

Case 2.1. If β = 0, therefore

$\begin{array}{r}\frac{1}{{\omega }^2}(-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)\\ -u_1^{2}f\prime (u_1)co{s}^2(u_2))=-\gamma u_1& (38)\end{array}$

which gives the following differential equation

$\begin{array}{ll}{f}^{\prime \prime }(u_1)+A\frac{f^{\prime }(u_1)}{u_1}-B{f}^{\prime 2}(u_1)=\frac{C}{u_1^2}+D& (39)\end{array}$

where $A=co{t}^2(u_2)-2b\gamma cot(u_2)$, B = γ , $C=\gamma b^{2}co{t}^2(u_2)+2bcot(u_2)$ , $D=\frac{\gamma }{si{n}^2(u_2)}$.

By using Bessel's functions of first and second order, a simple computations gives that the solution of Equation (39) can be written in the form

$\begin{array}{ll}f(u_1)=\frac{1}{2B}[Aln(u_1)-2ln(\frac{r(u_1)}{p(u_1)})+ln(BD)+ln(u_1)]& (40)\end{array}$

such that

$\begin{array}{ll}r(u_1)=B\left[c_2{Y}_n(\sqrt{BD}{u}_1)-c_1{J}_n(\sqrt{BD}{u}_1)\right]& (41)\end{array}$

$\begin{array}{r}p(u_1)=[J_{n+1}(\sqrt{BD}{u}_1)Y_n(\sqrt{BD}{u}_1)\\ -J_n(\sqrt{BD}{u}_1)Y_{n+1}(\sqrt{BD}{u}_1)]& (42)\end{array}$

and

$\begin{array}{ll}n=\frac{1}{2}\sqrt{A^2-4BC-2A+1}& (43)\end{array}$

with an arbitrary constants c₁, c₂. Therefore, in this case, the surface Γ² is given by

$\begin{array}{l}\chi (u_1,u_2)=(u_{1}cos(u_2),u_{1}sin(u_2),\frac{1}{2B}[Aln(u_1)\\ -2ln(\frac{r(u_1)}{p(u_1)})+ln(BD)+ln(u_1)]+b{u}_2)& (44)\end{array}$

Case 2.2. If γ = 0, then

$\begin{array}{l}-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)-u_1^2{f}^{\prime }(u_1)co{s}^2(u_2)\\ ={\omega }^2(\beta u_1{f}^{\prime }(u_1)sin(u_2)+\beta bcos(u_2))\end{array}$

a simple computations gives the next differential equation

$\begin{array}{l}{f}^{\prime \prime }(u_1)+[\frac{A}{u_1}+\frac{B}{u_1^2}+C]f^{\prime }(u_1)+\frac{D}{u_1}{f}^{\prime 2}(u_1)\\ +E{f}^{\prime 3}(u_1)+[\frac{F}{u_1}+\frac{G}{u_1^2}+\frac{H}{u_1^3}]=0& (45)\end{array}$

where $A=co{t}^2(u_2)$, $B=3\beta b^{2}cos(u_2)cot(u_2)$, $C=\frac{\beta }{sin(u_2)}$, D = 3 βb cos(u₂), E = βsin(u₂), $F=\frac{\beta bcot(u_2)}{sin(u_2)}$, G = −2b cot(u₂) and $H=\beta b^{3}cos(u_2)co{t}^2(u_2)$.

Case 2.3. If γ β ≠0, therefore

$\begin{array}{l}-u_1^3{f}^{\prime \prime }(u_1)si{n}^2(u_2)+2b{u}_{1}sin(u_2)cos(u_2)-u_1^2{f}^{\prime }(u_1)co{s}^2(u_2)\\ ={\omega }^2(\beta u_1{f}^{\prime }(u_1)sin(u_2)+\beta bcos(u_2)-\gamma u_1)\end{array}$

which gives the following differential equation

$\begin{array}{l}{f}^{\prime \prime }(u_1)+(B+\frac{A}{u_1}+\frac{C}{u_1^2})f^{\prime }(u_1)+(E+\frac{D}{u_1})f^{\prime 2}(u_1)+F{f}^{\prime 3}(u_1)\\ +(J+\frac{H}{u_1}+\frac{G}{u_1^2}+\frac{I}{u_1^3})=0& (46)\end{array}$

where $A=co{t}^2(u_2)-2\gamma bcot(u_2)$, $B=\frac{\beta }{sin(u_2)}$, $C=\frac{3\beta b^{2}co{s}^2(u_2)}{sin(u_2)}$, D = 3 βbcos(u₂), E = −γ, F = βsin(u₂), $G=-(2bcot(u_2)+\gamma b^{2}co{t}^2(u_2))$, $H=\frac{\beta bcos(u_2)}{si{n}^2(u_2)}$, $I=\frac{\beta b^{3}co{s}^3(u_2)}{si{n}^2(u_2)}$, and $J=-\frac{\gamma }{si{n}^2(u_2)}$. ■

5. Conclusion and Further Research

In this work, we constructed helicoidal surfaces in the Galilean 3−space and studied the First and the Second Fundamental Forms. Moreover, we calculated mean and Gaussian curvature for such surfaces. Also, we considered the Galilean 3−space with a linear density e^ϕ, ϕ = αx + βy + γz such that α, β, γ not all zero and constructed a weighted helicoidal surface by solving a second order non-linear differential equation. Moreover, we discussed an explicit parametrization for the helicoidal surfaces in G³.

Analogously to how a Minkowski 3−space relates to a Euclidean 3−space, one has the notion of Pseudo-Galilean 3−space $G_1^3$. As known, $G_1^3$ is similar to G³, but the Pseudo-Galilean scalar product of two vectors r = (r₁, r₂, r₃) and s = (s₁, s₂, s₃) is defined by

$<r,s>=\{\begin{array}{ll}{r}_1{s}_1,\hfill & \text{if }{r}_1\ne 0\text{ or }{s}_1\ne 0;\hfill \\ r_2{s}_2-r_3{s}_3,\hfill & \text{if }{r}_1=s_1=0.\hfill \end{array}$

Therefore, there exist four types of isotropic vectors r = (0, r₂, r₃) in $G_1^3$: spacelike vectors (if $r_2^2-r_3^2>0$), timelike vectors (if $r_2^2-r_3^2<0$) and two types of lightlike vectors (if r₂ = ±r₃) [15]. Thus, one can define different types of Helicoidal surfaces in $G_1^3$.

Data Availability Statement

All datasets generated for this study are included in the article/supplementary material.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors wish to express their sincere thanks to referee for making several useful comments that improved the first version of the paper.

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