International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 414-426

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-414

WEIGHTED MINIMAL AND WEIGHTED FLAT SURFACES OF REVOLUTION IN

GALILEAN 3-SPACE WITH DENSITY

AHMET KAZAN1,∗ AND H. BAYRAM KARADAĞ2

1Department of Computer Technologies, Doğanşehir Vahap Küçük Vocational School of Higher Education,

Inonu University, Malatya, Turkey

2Department of Mathematics, Faculty of Arts and Sciences, Inonu University, Malatya, Turkey

∗Corresponding author: ahmet.kazan@inonu.edu.tr

Abstract. In this paper, we obtain the weighted mean and weighted Gaussian curvatures of surfaces of

revolution in Galilean 3-space with density ea1x
2+a2y

2+a3z
2
, a1,a2,a3 ∈ R not all zero. Also, we investigate

some cases of weighted minimal surfaces of revolution according to ai, i = 1, 2, 3 and weighted flat surfaces

of revolution.

1. Introduction

The geometry of surfaces of revolution is an important studying area for geometers and it has been

studied widely in Euclidean 3-space E3, Lorentz-Minkowski space E31 , Galilean 3-space G3, pseudo-Galilean

3-space G13 and also in higher dimensions of these spaces. In these studies, the authors have investigated

lots of characterizations about surfaces of revolution, but one of the most important characterization is

minimal and flat surfaces of revolution. The catenoid which is obtained by rotating a catenary is the

most famous minimal surface of revolution. For another characterizations of surfaces of revolution, we refer

to [1], [3], [5], [6], [7], [8], [11], [12], [14] and etc.

2010 Mathematics Subject Classification. 53A10, 53A20, 53A35.

Key words and phrases. surface of revolution; isotropic rotation; weighted mean curvature; weighted gaussian curvature;

weighted minimal and flat surface of revolution.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

414

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-414


Int. J. Anal. Appl. 16 (3) (2018) 415

On the other hand, the geomerty of weighted manifold with density is a new studying area for geometers.

In this sense, the weighted mean curvature Hφ, which is also called φ-mean curvature, of a surface in Euclidean

3-space E3 with density eφ has been introduced in [9] and it is given by Hφ = H − 12 〈N,∇φ〉 , where H is

the mean curvature, N is the unit normal vector of surface and ∇φ is the gradient of φ. The weighted mean

curvature is a natural generalization of the mean curvature of a surface and a surface with Hφ = 0 is called

a weighted minimal surface or a φ-minimal surface.

Also in [2], the authors have introduced the notion of weighted Gaussian curvature or φ-Gaussian curvature

of a surface which is a generalization of the Gaussian curvature of a surface in a manifold with density eφ

and they have defined it as Gφ = G − ∆φ. Here, G is the Gaussian curvature of a surface and ∆ is the

Laplacian operator. If a surface’s weighted Gaussian curvature is zero everywhere, then we call it a weighted

flat surface or φ-flat surface.

In [18], the translation surfaces in G3 with a log-linear density have been studied and such a surface with

vanishing weighted mean curvature has been classified. In [13], the authors have considered the Euclidean

3-space E3 with a positive density function eφ, where φ = −x2−y2, (x,y,z) ∈ E3 and they have constructed

all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation

with the weighted Gaussian curvature and the mean curvature functions. Also in [10], the authors have

studied the ruled surfaces and translation surfaces in E3 with density ez and as a generalization of this

density, Lopez has used a linear density eax+by+cz, a,b,c ∈ R and classified weighted minimal translation

and cyclic surfaces in E3 [15].

In the present paper, we obtain the weighted mean curvature and weighted Gaussian curvature of three

types of surfaces of revolution in Galilean 3-space with density ea1x
2+a2y

2+a3z
2

, ai ∈ R not all zero and

investigate some cases of weighted minimal surfaces of revolution according to ai, i = 1, 2, 3 and weighted

flat surfaces of revolution. Also, we draw the obtaining surfaces of revolution with the aid of Mathematica.

2. Preliminaries

The Galilean space G3 is a Cayley-Klein space equipped with the projective metric of signature (0, 0, +, +).

The absolute figure of the Galilean geometry consists of an ordered triple {w,f,I}, where w is the ideal

(absolute) plane, f the line (absolute line) in w and I the fixed elliptic involution of points of p.

A vector x = (x1,x2,x3) in G3 is isotropic if x1 = 0 and non-isotropic otherwise. So, for affine ccordinates

(x,y,z), the y-axis and z-axis are isotropic while the x-axis is non-isotropic. Also, the yz-plane is Euclidean

and the xy-plane and xz-plane are isotropic. If x = (x1,x2,x3) and y = (y1,y2,y3) are two vectors in G3,

then the Galilean scalar product of x and y is defined by

〈x, y〉 =


 x1y1, if x1 6= 0 or y1 6= 0,x2y2 + x3y3, if x1 = 0 and y1 = 0.



Int. J. Anal. Appl. 16 (3) (2018) 416

So, the Galilean norm of a vector x in G3 is ‖x‖ =
√
〈x, x〉 and the vector x is a unit vector if ‖x‖ = 1. The

Galilean cross product of x and y is defined by

x × y =

∣∣∣∣∣∣∣∣∣
0 e2 e3

x1 x2 x3

y1 y2 y3

∣∣∣∣∣∣∣∣∣
.

For a more detailed treatment about Galilean space, we refer to [3], [4], [16], [17], [18] and etc.

Furthermore, let M be a surface in G3 parametrized by

Γ(u1,u2) = (x(u1,u2),y(u1,u2),z(u1,u2)).

Then, the unit normal vector N of the surface is defined by

N =
Γ,1 × Γ,2

w
,

where w = ‖Γ,1 × Γ,2‖ and Γ,i = ∂Γ∂ui (u
1,u2) for i ∈{1, 2}. The coefficients of the second fundamental form

are given by

Lij =

〈
Γ,ijx,1 −x,ijΓ,1

x,1
,N

〉
=

〈
Γ,ijx,2 −x,ijΓ,2

x,2
,N

〉
.

Analogous to Euclidean space, in [16], the mean curvature H and the Gaussian curvature K of the surface

are defined by

H =
1

2

2∑
i,j=1

gijLij and K =
L11L22 −L212

w2
,

where gij = gigj, for i,j ∈{1, 2} and g1 = x,2
w

, g2 = −x,1
w

.

Here we always assume that the surface is admissible, that is, its tangent space is nowhere an Euclidean

plane (for detail, see [3]).

3. Weighted Minimal and Weighted Flat Surfaces of Revolution in G3

with Density ea1x
2+a2y

2+a3z
2

In [3], the authors have constructed the surfaces of revolution in Galilean 3-space analogously to how that

is done in Euclidean 3-space and they have obtained 3 types of surfaces of revolution in G3.

Since there are different kinds of planes in G3, we have to consider two possibilities for the supporting

plane of the profile curve which generates the surface of revolution. In this context, the profile curve lies in

an Euclidean plane or it lies in an isotropic plane.

In order to construct a surface of revolution in G3, we use two types of rotations which are defined as

follows:



Int. J. Anal. Appl. 16 (3) (2018) 417

An Euclidean rotation about the non-isotropic x-axis is given by

x′

y′

z′


 =




1 0 0

0 cos θ1 sin θ1

0 −sin θ1 cos θ1





x

y

z


 ,

where θ1 is the Euclidean angle and an isotropic rotation is given by

x′

y′

z′


 =




1 0 0

θ2 1 0

0 0 1





x

y

z


 +




cθ2

c
2
(θ2)

2

0


 ,

where θ2 is the isotropic angle and c ∈ R.

3.1. Weighted Minimal and Weighted Flat Type I Surfaces of Revolution in G3.

Type I surface of revolution has constructed with the aid of an isotropic rotation as

ΓI(u,v) = (cv,f(u) +
c

2
v2,g(u)), (3.1)

where the profile curve α lies in the Euclidean yz-plane and it is parametrized by α(u) = (0,f(u),g(u)) [3].

Here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is

f′2(u) + g′2(u) = 1, ∀u ∈ I. (3.2)

Using (3.2), we obtain the mean curvature and Gaussian curvature of (3.1) as

H =
f′′(u)g′(u) −f′(u)g′′(u)

2
and K =

f′′(u)

c
, (3.3)

respectively. On the other hand, the unit normal vector N of the surface (3.1) is

N = (0,g′(u),−f′(u)). (3.4)

Assume that, M is the surface in G3 with density e
φ, φ = a1x

2 + a2y
2 + a3z

2, a1,a2,a3 ∈ R not all zero.

Then, for this density, from (3.1)-(3.4) the weighted mean curvature and the weighted Gaussian curvature

can be obtained as

Hφ =
f′′g′ −f′g′′

2
− < (a1cv,a2(f +

c

2
v2),a3g), (0,g

′,−f′) > (3.5)

and

Kφ =
f′′

c
− 2(a1 + a2 + a3), (3.6)

respectively.

Firstly, let us assume that the surface of revolution (3.1) is weighted minimal, i.e., Hφ = 0. Then, from

(3.5) we have

f′′g′ −f′g′′ − 2 < (a1cv,a2(f +
c

2
v2),a3g), (0,g

′,−f′) >= 0. (3.7)

Now, we’ll investigate some cases of weighted minimal surfaces of revolution (3.1) according to ai, i = 1, 2, 3.



Int. J. Anal. Appl. 16 (3) (2018) 418

If a1 6= 0, then from (3.7) we get

f′′g′ −f′g′′ = 0. (3.8)

Here, using (3.2) in (3.8) we obtain

f(u) = c1u + c2 (3.9)

and using (3.9) in (3.2) we get

g(u) =
√

1 − c21u + c3, (3.10)

where c1,c2,c3 ∈ R. Hence, we have

Theorem 3.1. Let M be a weighted minimal type I surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, where a1 6= 0, a2 and a3 are constants. Then, M can be parametrized by

ΓI(u,v) = (cv,c1u + c2 +
c

2
v2,
√

1 − c21u + c3), (3.11)

for some constants c, c1 ,c2 and c3.

In Figure 1, one can see the surface of revolution (3.11) for c = 4, c1 = 1/2, c2 = 2 and c3 = 3.

Figure 1.

If a1 = 0, then from (3.7) we get

f′′g′ −f′g′′ − 2(a2(fg′ +
c

2
v2g′) −a3f′g) = 0. (3.12)

Taking a2 = 0 in (3.12) and using (3.2), we get

g′′ − 2a3g(1 − (g′)2) = 0. (3.13)

The equation (3.13) is a second-order nonlinear ordinary differential equation and g(u) = u + c4, c4 ∈ R, is

a special solution for it. So, for this special solution, from (3.2) we get f(u) = c5, c5 ∈ R. Hence, we have



Int. J. Anal. Appl. 16 (3) (2018) 419

Theorem 3.2. Let M be a weighted minimal type I surface of revolution in Galilean 3-space with density

ea3z
2

, where a3 is a non-zero constant. Then, M can be parametrized by

ΓI(u,v) = (cv,c5 +
c

2
v2,u + c4), (3.14)

for some constants c, c4 and c5.

In Figure 2, one can see the surface of revolution (3.14) for c = 1, c4 = 2 and c5 = 3.

Figure 2.

Now, let us investigate the weighted flat surfaces of revolution (3.1). If the surface of revolution (3.1) is

weighted flat, i.e., Kφ = 0, then from (3.6) we have

f′′ − 2c(a1 + a2 + a3) = 0. (3.15)

From (3.15), we have

f(u) = cru2 + c6u + c7 (3.16)

and using (3.16) in (3.2), we have

g(u) = ∓
1

4cr
((c6 + 2cru)

√
1 − c26 − 4cc6ru− 4c2r2u2 + arcsin(c6 + 2cru)) + c8, (3.17)

where r = a1 + a2 + a3 and c, c6, c7, c8 are constants. Hence, we have

Theorem 3.3. Let M be a weighted flat type I surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, a1,a2,a3 ∈ R not all zero. Then, M can be parametrized by

ΓI(u,v) = (cv,cru
2 + c6u + c7 +

c

2
v2, (3.18)

∓
1

4cr
((c6 + 2cru)

√
1 − c26 − 4cc6ru− 4c2r2u2 + arcsin(c6 + 2cru)) + c8),

for some constants c, c6, c7, c8 and r = a1 + a2 + a3.



Int. J. Anal. Appl. 16 (3) (2018) 420

Figure 3 shows the weighted flat type I surface of revolution (3.18) for r = 6, c = 4, c6 = 1, c7 = 2 and

c8 = 3.

Figure 3.

3.2. Weighted Minimal and Weighted Flat Type II Surfaces of Revolution in G3.

Type II surface of revolution can be obtained with the aid of an isotropic rotation as

ΓII(u,v) = (u + cv,g(u),uv +
c

2
v2), (3.19)

where the profile curve α lies in the isotropic xy-plane and it is parametrized by α(u) = (f(u),g(u), 0) [3].

Here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is

f′2(u) = 1, ∀u ∈ I. (3.20)

We obtain the mean curvature and Gaussian curvature of (3.19) as

H =
−c2(ug′′(u) −g′(u))
2(u2 + c2(g′(u))2)

3
2

and K =
c2g′(u)(ug′′(u) −g′(u))

(u2 + c2(g′(u))2)2
, (3.21)

respectively. On the other hand, the unit normal vector N of the surface (3.19) is

N =
1√

u2 + c2(g′(u))2
(0,−u,−cg′(u)). (3.22)

Assume that, M is the surface in G3 with density e
φ, φ = a1x

2 + a2y
2 + a3z

2. Then, for this density, from

(3.19)-(3.22) the weighted mean curvature and the weighted Gaussian curvature can be obtained as

Hφ =
−c2(ug′′ −g′)

2(u2 + c2(g′)2)
3
2

−
1√

u2 + c2(g′)2
< (a1(u + cv),a2g,a3(uv +

c

2
v2)), (0,−u,−cg′) > (3.23)

and

Kφ =
c2g′(ug′′ −g′)
(u2 + c2(g′)2)2

− 2(a1 + a2 + a3), (3.24)

respectively.



Int. J. Anal. Appl. 16 (3) (2018) 421

Firstly, let us assume that the surface of revolution (3.19) is weighted minimal, i.e., Hφ = 0. Then, from

(3.23) we have

c2(ug′′ −g′) + 2(u2 + c2(g′)2) < (a1(u + cv),a2g,a3(uv +
c

2
v2)), (0,−u,−cg′) >= 0. (3.25)

Now, we’ll investigate some cases of weighted minimal surfaces of revolution (3.19) according to ai, i = 1, 2, 3.

If a1 6= 0, then from (3.25) we get

c2(ug′′ −g′) = 0. (3.26)

So, from (3.26) we get

g(u) =
c1
2
u2 + c2, (3.27)

where c1,c2 ∈ R. Hence, we have

Theorem 3.4. Let M be a weighted minimal type II surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, where a1 6= 0, a2 and a3 are constants. Then, M can be parametrized by

ΓII(u,v) = (u + cv,
c1
2
u2 + c2,uv +

c

2
v2), (3.28)

for some constants c, c1 and c2.

The shape of the surface of revolution (3.28) for c = 3, c1 = 1 and c2 = 2 can be seen in Figure 4 as

follows:

Figure 4.

If a1 = 0, then from (3.25) we get

c2(ug′′ −g′) + 2(u2 + c2(g′)2)(−a2ug −a3cg′(uv +
c

2
v2)) = 0. (3.29)

Taking a3 = 0 in (3.29), we get

c2(ug′′ −g′) − 2a2ug(u2 + c2(g′)2) = 0. (3.30)

Hence, we have



Int. J. Anal. Appl. 16 (3) (2018) 422

Theorem 3.5. Let M be a weighted flat type II surface of revolution in Galilean 3-space with density ea2y
2

,

where a2 is a non-zero constant. Then, the real function g which is in (3.19) must satisfy the second-order

nonlinear ordinary differential equation (3.30).

Now, let us investigate the weighted flat surfaces of revolution (3.19). If the surface of revolution (3.19)

is weighted flat, i.e., Kφ = 0, then from (3.24) we have

c2g′(ug′′ −g′) − 2r(u2 + c2(g′)2)2 = 0, (3.31)

where r = a1 + a2 + a3. From (3.31), we have

c2ug′g′′ − c2g′2 − 2ru4 − 4ru2c2g′2 − 2rc4g′4 = 0. (3.32)

Taking p(u) = g′(u) in (3.32), we get

p(u) = ∓
√
−2ru4 −u2 − 2c2c4u2√

2
√
c2ru2 + c4c4

(3.33)

and integrating (3.33), we have

g(u) = ±

u
√

2ru2 + 2c2c4 + 1


 √2 arcsin h(√2√ru2 + c2c4)√ru2 + c2c4

+2(ru2 + c2c4)
√

2ru2 + 2c2c4 + 1




4
√

2r
√
−u2(2ru2 + 2c2c4 + 1)

√
c2ru2 + c4c4

, (3.34)

where r = a1 + a2 + a3, c and c4 are constants. Hence, we have

Theorem 3.6. Let M be a weighted flat type II surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, a1,a2,a3 ∈ R not all zero. Then, M can be parametrized by

ΓI(u,v) = (u + cv, (3.35)

±

u
√

2ru2 + 2c2c4 + 1


 √2 arcsin h(√2√ru2 + c2c4)√ru2 + c2c4

+2(ru2 + c2c4)
√

2ru2 + 2c2c4 + 1




4
√

2r
√
−u2(2ru2 + 2c2c4 + 1)

√
c2ru2 + c4c4

,

uv +
c

2
v2),

for some constants c, c4 and r = a1 + a2 + a3.

Figure 5 shows the weighted flat type II surface of revolution (3.35) for r = −6, c = 2 and c4 = 1.



Int. J. Anal. Appl. 16 (3) (2018) 423

Figure 5.

3.3. Weighted Minimal and Weighted Flat Type III Surfaces of Revolution in G3.

Type III surface of revolution can be obtained with the aid of an Euclidean rotation about the axis x as

ΓIII(u,v) = (u,g(u) cos v,−g(u) sin v), (3.36)

where the profile curve α lies in the isotropic xy-plane and it is parametrized by α(u) = (f(u),g(u), 0) [3].

Here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is

f′2(u) = 1, ∀u ∈ I. (3.37)

Now, we obtain the mean curvature and Gaussian curvature of the surface of revolution (3.36) as

H = −
1

2g(u)
and K = −

g′′(u)

g(u)
, (3.38)

respectively. On the other hand, the unit normal vector N of the surface (3.36) is

N = (0, cos v,−sin v). (3.39)

Assume that, M is the surface in G3 with density e
φ, φ = a1x

2 + a2y
2 + a3z

2. Then, for this density, from

(3.36)-(3.39) the weighted mean curvature and the weighted Gaussian curvature can be obtained as

Hφ = −
1

2g
− < (a1u,a2g cos v,−a3g sin v), (0, cos v,−sin v) > (3.40)

and

Kφ = −
g′′(u)

g(u)
− 2(a1 + a2 + a3), (3.41)

respectively.

Firstly, let us assume that the surface of revolution (3.36) is weighted minimal, i.e., Hφ = 0. Then, from

(3.40) we have

1

2g
+ < (a1u,a2g cos v,−a3g sin v), (0, cos v,−sin v) >= 0 (3.42)

Next, we’ll investigate some cases of weighted minimal surfaces of revolution (3.36) according to ai, i = 1, 2, 3.



Int. J. Anal. Appl. 16 (3) (2018) 424

If a1 6= 0, then from (3.42) we get
1

2g
= 0 (3.43)

and this is a contradiction. Hence, we have

Theorem 3.7. There is no weighted minimal type III surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, where a1 6= 0, a2 and a3 are constants.

If a1 = 0, then from (3.42) we get

1

2g
+ a2g cos

2 v + a3g sin
2 v = 0 (3.44)

Taking a2 = a3 = a 6= 0 in (3.44), we get
1

2g
+ ag = 0 (3.45)

and so,

g(u) = ∓
√
−

1

2a
,a < 0. (3.46)

Hence, we have

Theorem 3.8. Let M be a weighted minimal type III surface of revolution in Galilean 3-space with density

ea(y
2+z2), where a < 0 is a constant. Then, M can be parametrized by

ΓIII(u,v) = (u,∓
√
−

1

2a
cos v,±

√
−

1

2a
sin v). (3.47)

The shape of the surface of revolution (3.47) for a = −1 is given in Figure 6.

Figure 6.

Now, let us investigate the weighted flat surfaces of revolution (3.36). If the surface of revolution (3.36)

is weighted flat, i.e., Kφ = 0, then from (3.41) we have

g′′

g
+ 2r = 0, (3.48)



Int. J. Anal. Appl. 16 (3) (2018) 425

where r = a1 + a2 + a3. From (3.48), we have

g(u) = c1 cos(
√

2ru) + c2 sin(
√

2ru). (3.49)

Hence, we have

Theorem 3.9. Let M be a weighted flat type III surface of revolution in Galilean 3-space with density

ea1x
2+a2y

2+a3z
2

, a1,a2,a3 ∈ R not all zero. Then, M can be parametrized by

ΓIII(u,v) = (u, (c1 cos(
√

2ru) + c2 sin(
√

2ru)) cos v,−(c1 cos(
√

2ru) + c2 sin(
√

2ru)) sin v) (3.50)

for some constants c1, c2 and r = a1 + a2 + a3.

Figure 7 shows the weighted flat type III surface of revolution (3.50) for r = 2, c1 = 1 and c2 = 3.

Figure 7.

References

[1] M. Bekkar and H. Zoubir, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying ∆xi = λixi, Int.

J. Contemp. Math. Sci. 24 (3) (2008), 1173-1185.

[2] I. Corwin, H. Hoffman, S. Hurder, V. Ssum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und.

Math. J. 7 (2006), 1-15.

[3] M. Dede, C. Ekici and W. Goemans, Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space, J. Math. Phys.

in press.

[4] M. Dede, Tubular Surfaces in Galilean Space, Math. Commun. 18 (2013), 209-217.



Int. J. Anal. Appl. 16 (3) (2018) 426

[5] F. Dillen, J. Pas and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990),

239-246.

[6] B. Divjak and Z.M. Sipus, Some Special Surfaces in the Pseudo-Galilean Space, Acta Math. Hungar. 118 (3) (2008),

209-226.

[7] A. Ferrandez and P. Lucas, On Surfaces in the 3-dimensional Lorentz-Minkowski space, Pac. J. Math. 152 (1) (1992),

93-100.

[8] O.J. Garay, On a certain class of finite type surfaces of revolution, Kodai Math. J. 11 (1) (1988), 25-31.

[9] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Func. Anal. 13 (2003), 178-215.

[10] D.T. Hieu and N.M Hoang, Ruled minimal surfaces in R3 with density ez, Pac. J. Math. 243 (2009), 277-285.

[11] G. Kaimakamis and B. Papantoniou, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying ∆II~r =

A~r, J. Geom. 81 (2004), 81-92.

[12] A. Kazan and H.B. Karadağ, A Classification of Surfaces of Revolution in Lorentz-Minkowski Space, Int. J. Contemp.

Math. Sci. 39 (6) (2011), 1915-1928.

[13] D-S. Kim and D.W. Yoon, Constructions of Helicoidal Surfaces in Euclidean Space with Density, Symmetry 9 (2017), Art.

ID 173.

[14] S. Lee and J.H. Varnado, Spacelike constant mean curvature surfaces of revolution in Minkowski 3-space, Differ. Geom.

Dyn. Syst. 8 (1) (2006), 144-165.

[15] R. López, Minimal surface in Euclidean Space with a Log-Linear Density, arXiv:1410.2517 [math.DG] (accessed on 20 July

2017).

[16] O. Röschel, Die Geometrie des Galileischen Raumes, Bericht der Mathematisch-Statistischen Sektion in der Forschungsge-

sellschaft Joanneum, Bericht Nr. 256, Habilitationsschrift, Leoben, (1984).

[17] Z.M. Sipus, Ruled Weingarten Surfaces in the Galilean Space, Period. Math. Hung. 56 (2) (2008), 213-225.

[18] D.W. Yoon, Weighted Minimal Translation Surfaces in the Galilean Space with Density, Open Math. 15 (2017), 459-466.


	1. Introduction
	2. Preliminaries
	3. Weighted Minimal and Weighted Flat Surfaces of Revolution in G3 with Density ea1x2+a2y2+a3z2
	3.1. Weighted Minimal and Weighted Flat Type I Surfaces of Revolution in G3
	3.2. Weighted Minimal and Weighted Flat Type II Surfaces of Revolution in G3
	3.3. Weighted Minimal and Weighted Flat Type III Surfaces of Revolution in G3

	References