Abstract
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 eϕ 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 eϕ, 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 , which is very useful to probabilists [2].
On a manifold with density eϕ, the weighted mean curvature of a hypersurface with unit normal N is defined by
where H is the Riemannian mean curvature of the hypersurface [9]. The weighted mean curvature Hϕ of a surface in E3 with density eϕ 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 Hϕ = 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 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 xo = 0, the absolute line f by xo = x1 = 0 and the elliptic involution by
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 Qi = (ri, si, ti), (i = 1, 2) is given by
Moreover, the distance in the Euclidean space between Q1 and Q2 is given by
The Galilean scalar product between two vectors P = (p1, p2, p3) and Q = (q1, q2, q3) is defined by
For this, the Galilean norm of a vector P is . Moreover, the cross product in the Galilean space is defined by
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
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
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,
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 and no isotropic tangents (x.(t) ≠ 0).
Let C be an open subset of R2 and M a surface in G3 parameterized by
In order to specify the partial derivatives we will denote:
Then r is satisfied admissibility criteria if no where it has Euclidean tangent planes. The first fundamental form is given by
where , , and , , h12 = yuyv + zuzv, also
Now, consider the function
hence the isotropic unit normal vector field N of the surface r = r(u, v) is given by
The second fundamental form is obtained by
such that
where i, j = u, v.
Note that the dot “·” denotes the Euclidean scalar product. Therefore, the Gaussian and mean curvature are given by.
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
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
where f(u1) is a differentiable function defined on I.
Theorem 5. Let Γ2be helicoidal surface in G3 defined by
where f(u1) is a differentiable function defined on I. Then the unit normal vector field N of the surface Γ2is given by
The first and the second fundamental forms of the surface Γ2inG3are given respectively by
and
Proof. Let Γ2 be helicoidal surface in G3 defined by
Then the unit normal vector field N of the surface Γ2 is an isotropic vector field defined by
where the positive function ω is given by
Here the partial derivatives of the functions x, y, andz with respect to ui (i = 1, 2) are denoted by xui, yui, and zui, respectively. On the other hand, let us define gi = xui, hij = yuiyuj + zuizuj, i, j = 1, 2. So, the first fundamental form of the surface Γ2 in G3 is given by
where
and
Then
In the sequel, the second fundamental form II of Γ2 is given by
where , and . ■
Corollary 6. The Gaussian curvature K of the surface Γ2is obtained by
Moreover, the mean curvature of the surface Γ2is given by
Proof. Since the Gaussian curvature K is given by , then
The mean curvature of the surface is obtain from
By substituting, we get
4. Weighted Helicoidal Surfaces in G3
Let Γ2 be a helicoidal surface in G3 defined by
where f(u1) is a differentiable function defined on I. Suppose that Γ2 is the surface in G3 with a linear density eϕ, where ϕ = αx + βy + γz, α, β, γ not all zero.
In this case, the weighted mean curvature Hϕ of Γ2 can be expressed as
where ∇ϕ is the gradient of ϕ. If Γ2 is the weighted minimal surface, then
Theorem 7. Let Γ2be weighted minimal helicoidal surface inG3defined by
with a linear densityeϕ, then f(u1) will be one of the following
f(u1) is the solution of the differential equation
f(u1) is the solution of the differential equation
Proof. Let Γ2 be a helicoidal surface in G3 defined by
where f(u1) is a differentiable function defined on I. By substituting in equation (27) we obtain
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
To solve this equation, we make reduction of the order as: Let which gives , substitutes into equation (31) we obtain the differential equation
Integrating factor and hence the solution is given by
i.e.,
which gives
Therefore, Γ2 is determined by
where
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
Case 2.1. If β = 0, therefore
which gives the following differential equation
where , B = γ , , .
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
such that
and
with an arbitrary constants c1, c2. Therefore, in this case, the surface Γ2 is given by
Case 2.2. If γ = 0, then
a simple computations gives the next differential equation
where , , , D = 3 βb cos(u2), E = βsin(u2), , G = −2b cot(u2) and .
Case 2.3. If γ β ≠0, therefore
which gives the following differential equation
where , , , D = 3 βbcos(u2), E = −γ, F = βsin(u2), , , , and . ■
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 eϕ, 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 . As known, 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
Therefore, there exist four types of isotropic vectors r = (0, r2, r3) in : spacelike vectors (if ), timelike vectors (if ) and two types of lightlike vectors (if r2 = ±r3) [15]. Thus, one can define different types of Helicoidal surfaces in .
Statements
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.
Acknowledgments
The authors wish to express their sincere thanks to referee for making several useful comments that improved the first version of the paper.
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.
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Summary
Keywords
Galilean space, helicoidal surfaces, density, mean curvature, Gauss curvature, weight helicoidal surface
Citation
Mosa S and Elzawy M (2020) Helicoidal Surfaces in Galilean Space With Density. Front. Phys. 8:81. doi: 10.3389/fphy.2020.00081
Received
27 November 2019
Accepted
09 March 2020
Published
31 March 2020
Volume
8 - 2020
Edited by
Yang-Hui He, City University of London, United Kingdom
Reviewed by
Yilun Shang, Northumbria University, United Kingdom; Praveen Agarwal, Anand International College of Engineering, India
Updates
Copyright
© 2020 Mosa and Elzawy.
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*Correspondence: Mervat Elzawy mervatelzawy@science.tanta.edu.eg; mrzawy@taibahu.edu.sa
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
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