Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space
In the paper, three types of surfaces of revolution in the Galilean 3- space are defined and studied. The construction of the well-known surface of revolution, defined as the trace of a planar curve rotated about an axis in the supporting plane of the curve, is given for the Galilean 3-space. Then w...
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irk-123456789-1458652019-02-02T01:23:19Z Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space Dede, M. Ekici, C. Goemans, W. In the paper, three types of surfaces of revolution in the Galilean 3- space are defined and studied. The construction of the well-known surface of revolution, defined as the trace of a planar curve rotated about an axis in the supporting plane of the curve, is given for the Galilean 3-space. Then we classify the surfaces of revolution with vanishing Gaussian curvature or vanishing mean curvature in the Galilean 3-space. У статтi визначено та дослiджено три типи поверхонь обертання у тривимiрному просторi Галiлея. Запропоновано конструкцiю поверхнi обертання у тривимiрному просторi Галiлея, визначено обертанням пласко криво навколо осi, що лежить у площинi криво . Класифiковано поверхнi обертання у тривимiрному просторi Галiлея з нульовою гауссовою кривиною та з нульовою середньою кривиною. 2018 Article Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space / M. Dede, C. Ekici, W. Goemans // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 141-152. — Бібліогр.: 13 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.02.141 Mathematical Subject Classification 2010: 53A10, 53A35, 53A40. http://dspace.nbuv.gov.ua/handle/123456789/145865 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the paper, three types of surfaces of revolution in the Galilean 3- space are defined and studied. The construction of the well-known surface of revolution, defined as the trace of a planar curve rotated about an axis in the supporting plane of the curve, is given for the Galilean 3-space. Then we classify the surfaces of revolution with vanishing Gaussian curvature or vanishing mean curvature in the Galilean 3-space. |
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Dede, M. Ekici, C. Goemans, W. |
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Dede, M. Ekici, C. Goemans, W. Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space Журнал математической физики, анализа, геометрии |
| author_facet |
Dede, M. Ekici, C. Goemans, W. |
| author_sort |
Dede, M. |
| title |
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space |
| title_short |
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space |
| title_full |
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space |
| title_fullStr |
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space |
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Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space |
| title_sort |
surfaces of revolution with vanishing curvature in galilean 3-space |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/145865 |
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Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space / M. Dede, C. Ekici, W. Goemans // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 141-152. — Бібліогр.: 13 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT dedem surfacesofrevolutionwithvanishingcurvatureingalilean3space AT ekicic surfacesofrevolutionwithvanishingcurvatureingalilean3space AT goemansw surfacesofrevolutionwithvanishingcurvatureingalilean3space |
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2025-07-10T22:42:42Z |
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1837301622685827072 |
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Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 2, pp. 141–152
doi: https://doi.org/10.15407/mag14.02.141
Surfaces of Revolution with Vanishing
Curvature in Galilean 3-Space
M. Dede, C. Ekici, and W. Goemans
In the paper, three types of surfaces of revolution in the Galilean 3-
space are defined and studied. The construction of the well-known surface
of revolution, defined as the trace of a planar curve rotated about an axis
in the supporting plane of the curve, is given for the Galilean 3-space. Then
we classify the surfaces of revolution with vanishing Gaussian curvature or
vanishing mean curvature in the Galilean 3-space.
Key words: surface of revolution, flat surface, minimal surface, Galilean
3-space.
Mathematical Subject Classification 2010: 53A10, 53A35, 53A40.
1. Introduction
Together with the ruled surfaces, the surfaces of revolution were among the
first studied subjects in differential geometry, besides the curves. Being the trace
of a rotated planar curve, the surfaces of revolution are also to be found widely
in the “real world”. The catenoid, obtained by rotating a catenary, is one of
the elementary minimal surfaces. Although surfaces of revolution or rotational
surfaces have been studied thoroughly for centuries, we believe we can make a
modest contribution to this research area by defining and studying them in a
Galilean 3-space. The geometry of this non-Euclidean space was first studied
intensively by Röschel [11]. In the last decade, this space was used by several
researchers as an ambient space for the well-known Euclidean concepts (see [2,3,
7, 9, 10] for more examples on special surfaces). The study of the surfaces in the
Galilean 3-space can also be found in [1].
In [7], two types of surfaces of revolution in the Galilean 3-space are used
as examples. It is pointed out that they are Weingarten surfaces, that is, there
exists a non-trivial functional dependence between the Gaussian curvature and
the mean curvature. Further, no systematic studies of the curvatures of these
surfaces have been found. The authors of this paper are aimed to study twisted
surfaces, which are surfaces that are traced out by a planar curve on which two
simultaneous rotations are performed, and hence which are a generalization of
surfaces of revolution (see [6] and the references therein). We did it first in [4].
To the best of our knowledge this has not been done before.
c© M. Dede, C. Ekici, and W. Goemans, 2018
https://doi.org/10.15407/mag14.02.141
142 M. Dede, C. Ekici, and W. Goemans
In this paper, we first recall the necessary preliminaries on the Galilean space.
Then we define three different types of surfaces of revolution in the Galilean space
and study the curvature properties of these surfaces. We give the classification
theorems of surfaces of revolution with vanishing Gaussian curvature or vanishing
mean curvature.
2. Preliminaries
For an in-depth study of the Galilean 3-space, see [11]. Here, we recall the
properties that we need from this work.
The Galilean 3-space G3 arises in a Cayley–Klein way by pointing out an
absolute figure {ω, f, I} in the 3-dimensional real projective space. Here ω is the
absolute plane, f is the absolute line and I is the fixed elliptic involution of points
of f . Then the homogeneous coordinates (x0 : x1 : x2 : x3) are introduced such
that ω is given by x0 = 0, f is given by x0 = x1 = 0 and I, by (0 : 0 : x2 :
x3) 7→ (0 : 0 : x3 : −x2). The group of motions of G3 is a six-parameter group.
Regarding this group of motions, except the absolute plane, there exist two classes
of planes in G3: Euclidean planes that contain f and in which the induced metric
is Euclidean and isotropic planes that do not contain f and in which the induced
metric is isotropic. Also, there are four types of lines in G3: isotropic lines that
intersect f , non-isotropic lines that do not intersect f , non-isotropic lines in ω
and the absolute line f .
In affine coordinates defined by (x0 : x1 : x2 : x3) = (1 : x1 : x2 : x3), the
distance between two points Pi = (xi, yi, zi) with i ∈ {1, 2} is defined by
d(P1, P2) =
{
|x2 − x1| if x1 6= x2,√
(y2 − y1)2 + (z2 − z1)2 if x1 = x2.
The vector ~a = (x, y, z) is isotropic if x = 0 and non-isotropic otherwise.
Hence, for standard coordinates (x, y, z), the x-axis is non-isotropic while the
y-axis and the z-axis are isotropic. The yz-plane, x = 0, is Euclidean and the xy-
plane and the xz-plane are isotropic. The Galilean scalar product of two vectors
~a = (x, y, z) and ~b = (x1, y1, z1) is defined by
〈~a,~b〉 =
{
xx1 if x 6= 0 or x1 6= 0,
yy1 + zz1 if x = x1 = 0.
The vector ~a is a unit vector if ‖~a‖ :=
√
〈~a,~a〉 = 1. In [7], the Galilean cross
product of two vectors ~a = (x, y, z) and ~b = (x1, y1, z1) is defined as
~a ∧~b =
∣∣∣∣∣∣
0 e2 e3
x y z
x1 y1 z1
∣∣∣∣∣∣ .
In order to define surfaces of revolution, we need the two types of rotations
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space 143
in G3. A Euclidean rotation about the non-isotropic x-axis is given byx′y′
z′
=
1 0 0
0 cos θ sin θ
0 − sin θ cos θ
xy
z
,
where θ is the Euclidean angle. An isotropic rotation is given byx′y′
z′
=
1 0 0
θ 1 0
0 0 1
xy
z
+
cθ
c
2θ
2
0
,
where θ is the isotropic angle and c ∈ R0. Here the bundle of fixed planes is given
by z = const [8].
Finally, to calculate the curvatures, we have to be able to perform patch
computations for a surface in G3. If a surface in G3 is parameterized by
ϕ(v1, v2) = (x(v1, v2), y(v1, v2), z(v1, v2)),
then we denote the first-order derivatives for i ∈ {1, 2} by ϕ,i = ∂ϕ
∂vi
(v1, v2) [2,11].
Here we will always assume that the surfaces are admissible, that is, the tangent
plane is nowhere Euclidean.
The unit normal vector N of the surface is defined by
N =
ϕ,1 ∧ ϕ,2
w
where w = ‖ϕ,1 ∧ ϕ,2‖ .
The coefficients of the second fundamental form are given by
Lij = 〈ϕ,ijx,1 − x,ijϕ,1
x,1
, N〉 = 〈ϕ,ijx,2 − x,ijϕ,2
x,2
, N〉. (2.1)
The Gaussian curvature K and the mean curvature H of the surface are
defined in [11], analogously to the Euclidean space, by
K =
L11L22 − L2
12
w2
and 2H =
2∑
i,j=1
gijLij , (2.2)
where
g1 =
x,2
w
, g2 = −x,1
w
, and gij = gigj for i, j ∈ {1, 2}.
As proved in [11, Satz 19.5, p. 107], the mean curvature at the point p of a
surface in G3 is the curvature of intersection of the surface with the Euclidean
plane that contains the point p.
144 M. Dede, C. Ekici, and W. Goemans
3. Surfaces of revolution in the Galilean 3-space
We construct a surface of revolution in the Galilean 3-space analogously to
that constructed in the Euclidean 3-space.
Definition 3.1. A surface of revolution in G3 is a surface that is traced
out by a planar curve, the profile curve, rotated in G3. The rotation is either a
Euclidean rotation about an axis in the supporting plane of the profile curve, or
an isotropic rotation for which a bundle of fixed planes is chosen.
Because of the existence of different kinds of planes in G3, we consider two
possibilities for the supporting plane of the profile curve of a surface of revolution
in G3: the profile curve lies either in a Euclidean plane, or in an isotropic plane.
Since a Euclidean plane contains only isotropic vectors, while an isotropic
plane contains both isotropic and non-isotropic vectors, there are three types of
surfaces of revolution to be defined in G3.
3.1. Type I surfaces of revolution in G3. Without losing generality,
we can assume that the profile curve α lies in the Euclidean yz-plane and it is
parameterized by α(t) = (0, f(t), g(t)) where f and g are real functions. On this
profile curve, we perform an isotropic rotation with c ∈ R0, for instance,1 0 0
s 1 0
0 0 1
0
f(t)
g(t)
+
cs
c
2s
2
0
.
Thus, up to a transformation, a type I surface of revolution in G3 is parame-
terized by
ϕ(s, t) =
(
cs, f(t) +
c
2
s2, g(t)
)
.
This type I surface of revolution is one of the two kinds of surfaces of revolution
that are mentioned as examples in [7].
3.2. Type II surfaces of revolution in G3. Now we assume, again
without losing generality, that the profile curve α lies in the isotropic xy-plane
and it is parameterized by α(t) = (f(t), g(t), 0) where f and g are real functions.
Also on this profile curve we perform an isotropic rotation with c ∈ R0,1 0 0
0 1 0
s 0 1
f(t)
g(t)
0
+
cs
0
c
2s
2
.
Then, up to a transformation, a type II surface of revolution in G3 is param-
eterized by
ϕ(s, t) =
(
f(t) + cs, g(t), sf(t) +
c
2
s2
)
.
As far as we know, this type of surfaces of revolution in G3 has not been
defined before.
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space 145
3.3. Type III surfaces of revolution in G3. We start again with a
profile curve α(t) = (f(t), g(t), 0) in the isotropic xy-plane, where f and g are
real functions, but this time we perform a Euclidean rotation about the x-axis
on it, 1 0 0
0 cos s sin s
0 − sin s cos s
f(t)
g(t)
0
.
So, up to a transformation, a type III surface of revolution in G3 is parame-
terized by
ϕ(s, t) = (f(t), g(t) cos s,−g(t) sin s) .
These type III surfaces of revolution are also used as examples in [7].
4. Zero curvature surfaces of revolution in G3
For the three types of surfaces of revolution in G3, we calculate the Gaussian
curvature and the mean curvature, and we examine when they vanish. As in the
Euclidean 3-space, the surfaces with vanishing Gaussian curvature are called flat
and the surfaces with vanishing mean curvature are called minimal. From [11],
we recall the following important theorem which classifies all minimal surfaces in
G3.
Theorem 4.1 ([11]). The minimal surfaces in the Galilean 3-space are cones
whose vertices are on the absolute line f and the ruled surfaces of type C that are
conoidal surfaces having the absolute line f as the directional line at infinity.
A type C ruled surface can be parameterized as follows:
ϕ(s, t) = (s, f(s), 0) + t (0, β2(s), β3(s))
where f , β2 and β3 are three times continuous differentiable real functions such
that β2(s)
2 + β3(s)
2 = 1.
4.1. Zero curvature type I surfaces of revolution in G3. For a type I
surface of revolution in G3, parametrized by
ϕ(s, t) =
(
cs, f(t) +
c
2
s2, g(t)
)
(4.1)
with c ∈ R0, the partial derivatives of type I surfaces of revolution are obtained
as
ϕs = (c, cs, 0), ϕt = (0, f ′(t), g′(t)),
where primes denote derivative respect to t. It follows that
w = ‖ϕs ∧ ϕt‖ = |c|
√
f ′(t)2 + g′(t)2.
By using the partial derivatives of type I surfaces of revolution and (2), one
calculates
g11 = g12 = 0, g22 =
1
f ′(t)2 + g′(t)2
. (4.2)
146 M. Dede, C. Ekici, and W. Goemans
From (2.1), we have the coefficients of the second fundamental form II as
L11 = sgn(c)
−cg′(t)√
f ′(t)2 + g′(t)2
, (4.3)
L12 = 0, (4.4)
L22 = sgn(c)
f ′(t)g′′(t)− f ′′(t)g′(t)√
f ′(t)2 + g′(t)2
. (4.5)
Here, by sgn we mean the sign function. The substituting of (4.3)–(4.5) and
(4.2) into (2.2) gives
K =
g′(f ′′g′ − f ′g′′)
c(f ′2 + g′2)2
and H = sgn(c)
f ′g′′ − f ′′g′
2(f ′2 + g′2)3/2
.
Here and in the remainder of the paper we often drop the parameter of the
functions f and g for reasons of readability. Since the surface is admissible, then
f ′ and g′ can not be both identically zero.
Further, it is immediate that the Gaussian curvature is identically zero if and
only if the mean curvature is identically zero. Hence, a type I surface of revolution
in G3 is flat if and only if it is minimal. Then the following classification theorem
is valid.
Theorem 4.2. A type I surface of revolution in the Galilean 3-space is flat
or, equivalently, minimal, if and only if it is either
1) a parabolic cylinder parameterized by
ϕ(s, t) =
(
cs, a+
c
2
s2, g(t)
)
, (4.6)
2) a part of an isotropic plane, consisting of a family of parabolas, parameterized
by
ϕ(s, t) =
(
cs, f(t) +
c
2
s2, a
)
, (4.7)
3) or a parabolic cylinder parameterized by
ϕ(s, t) =
(
cs, f(t) +
c
2
s2, af(t) + b
)
. (4.8)
Here a, b, c ∈ R with c 6= 0 and a 6= 0 in parameterization (4.8).
Proof. From the expressions for the Gaussian curvature and the mean curva-
ture, it is clear that it is sufficient that f ′g′′ − f ′′g′ = 0.
If f ′ = 0 or g′ = 0, then we obtain parameterization (4.6) or (4.7), respectively.
If f ′ 6= 0 and g′ 6= 0, one has f ′′
f ′ = g′′
g′ . Integrating this equation, we obtain
g(t) = af(t) + b, where a 6= 0 and b are real integration constants. Therefore, the
profile curve is an isotropic straight line and this leads to parameterization (4.8).
Conversely, it is calculated immediately that the surfaces given by the param-
eterizations in the statement are flat and minimal.
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space 147
The following corollary is immediate.
Corollary 4.3. A type I surface of revolution in the Galilean 3-space is flat,
or equivalently, minimal, if and only if its profile curve is an isotropic straight
line.
Remark 4.4. It is easy to see that in parameterization (4.6) the components
satisfy y = a+ x2
2c . Similarly, in parameterization (4.8) a straightforward calcula-
tion shows that y = z−b
a + x2
2c .
Moreover, reparameterize (4.6) such that g(t) = v. Then the resulting pa-
rameterization describes a type C ruled surface
ϕ(s, v) =
(
cs, a+
c
2
s2, 0
)
+ v(0, 0, 1).
One can proceed similarly for parameterization (4.7). Also, reparameterize (4.8)
such that f(t) = v, then it becomes
ϕ(s, v) =
(
cs,
c
2
s2, b
)
+ v(0, 1, a).
Thus, this is a conoidal surface having the absolute line f as the directional
line at infinity. Hence, it is a type C ruled surface. For a drawing of a surface
parameterized by parameterization (4.8), see Figure 4.1a.
(a) A flat and minimal type I sur-
face of revolution, parameter-
ized by (4.8) with a = 1, b =
c = 2 and f any function.
(b) A flat and minimal type II sur-
face of revolution, parameter-
ized by (4.10) with a = c = 1
and b = −2.
Fig. 4.1: Flat and minimal type I and type II surfaces of revolution.
Remark 4.5. As mentioned in [7], since the Gaussian curvature and the
mean curvature of type I surfaces of revolution are the functions of one vari-
able only, these type I surfaces of revolution are Weingarten surfaces. Indeed,
sgn(c)c
(
f ′2 + g′2
)1/2
K = 2g′H.
4.2. Zero curvature type II surfaces of revolution in G3. A type II
surface of revolution in G3 parametrized by
ϕ(s, t) =
(
f(t) + cs, g(t), sf(t) +
c
2
s2
)
148 M. Dede, C. Ekici, and W. Goemans
with c ∈ R0, using (2.2), is calculated to have the Gaussian curvature and the
mean curvature
K =
c2g′
w4
[
f(f ′g′′ − f ′′g′)− f ′2g′
]
and
H =
c2
2w3
[
f(f ′g′′ − f ′′g′)− f ′2g′
]
.
Here w2 = f2f ′2 + c2g′2. Again, since the surface is admissible, then f ′ and
g′ can not be both identically zero.
Also now, the Gaussian curvature is identically zero if and only if the mean
curvature is identically zero. Thus, a type II surface of revolution in G3 is flat
if and only if it is minimal. The following classification theorem can be proved
similarly to that for the type I surfaces of revolution.
Theorem 4.6. A type II surface of revolution in the Galilean 3-space is flat
or, equivalently, minimal, if and only if it is either
1) a part of an isotropic plane, consisting of a family of parabolas, parameterized
by
ϕ(s, t) =
(
f(t) + cs, a, sf(t) +
c
2
s2
)
,
2) a parabolic cylinder parameterized by
ϕ(s, t) =
(
a+ cs, g(t), as+
c
2
s2
)
, (4.9)
3) or a cyclic surface (parabolic sphere) parameterized by
ϕ(s, t) =
(
t+ cs, at2 + b, st+
c
2
s2
)
, (4.10)
where a, b, c ∈ R and c 6= 0.
In order to find parameterization (4.10), in Theorem 4.6 we assume that the
profile curve is parameterized by arclength, then f(t) = t.
Remark 4.7. A straightforward calculation shows that the components of
parameterization (4.9) satisfy
z =
x2
2c
− a2
2c
.
Similarly, for the components of parameterization (4.10) one sees that
x2 − y
a
− 2cz +
b
a
= 0.
Again, reparameterize (4.9) using g(t) = v. Then we get a conoidal surface
that has the absolute line f as a directional line at infinity. Hence, it is a type C
ruled surface
ϕ(s, v) =
(
a+ cs, 0, as+
c
2
s2
)
+ v(0, 1, 0).
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space 149
Similarly, parameterize (4.10), setting u = t+ cs and v = −s
(
t+ c
2s
)
. Then we
have
ϕ(u, v) =
(
u, au2 + b, 0
)
+ v(0, ac,−1).
Remark that in parameterization (4.10) the profile curve is an isotropic circle,
see [11]. In a Galilean space this parameterization represents a cyclic surface
constructed analogously to a Euclidean sphere, see [12]. For a drawing of the
surface, see Figure 4.1b.
Remark 4.8. Also for type II surfaces of revolution, the Gaussian curvature
and the mean curvature are the functions of one variable only, therefore, type II
surfaces of revolution are Weingarten surfaces. Indeed, wK = 2g′H.
4.3. Zero curvature type III surfaces of revolution in G3. For a
type III surface of revolution in G3 given by the parametrization
ϕ(s, t) = (f(t), g(t) cos s,−g(t) sin s) ,
using (2.2), it is calculated that
K =
f ′′g′ − f ′g′′
f ′3g
and H =
sgn(f ′g)
2g
.
Here f ′ should be non-zero in order to have an admissible surface. Also the
function g must be non-zero of course.
For these type III surfaces of revolution in G3, the flat and minimal conditions
are not equivalent. Moreover, it is immediate that there do not exist minimal
type III surfaces of revolution in G3.
This time the proof similar to that of Theorem 4.2 leads to the classification
of flat type III surfaces of revolution in G3.
Theorem 4.9. A type III surface of revolution in the Galilean 3-space is flat
if and only if it is either
1) a cylinder over a Euclidean circle parameterized by
ϕ(s, t) = (f(t), a cos s,−a sin s) , (4.11)
2) or a circular cone with vertex (b, 0, 0) parameterized by
ϕ(s, t) = (ag(t) + b, g(t) cos s,−g(t) sin s) ,
where a, b ∈ R with a 6= 0.
Remark 4.10. We can reparameterize (4.11) by setting f(t) = v, to obtain
ϕ(s, v) = (0, a cos s,−a sin s) + v(1, 0, 0).
In [11], a surface of this kind is called a type B ruled surface.
150 M. Dede, C. Ekici, and W. Goemans
Remark 4.11. As mentioned in [7], since the Gaussian curvature and the mean
curvature of type III surfaces of revolution are the functions of one variable only,
type III surfaces of revolution are Weingarten surfaces. Indeed, sgn(f ′g)f ′2K =
2(f ′′g′ − f ′g′′)H.
Only for type III surfaces of revolution it is possible to determine which
surfaces are of constant mean curvature. In this case, the function g is a non-zero
constant function. Hence the surfaces considered are flat. Moreover, they are
cylinders over Euclidean circles as in parameterization (4.11) of Theorem 4.9.
5. Conclusion and further research
In this work, we defined three types of surfaces of revolution in the Galilean
3-space and studied when they are flat or minimal. For type I and type II
surfaces, the flat and minimal conditions are equivalent. There do not exist
minimal type III surfaces of revolution in G3.
Analogously to how a Minkowski 3-space relates to a Euclidean 3-space, one
has the notion of pseudo-Galilean 3-space G3
1. Without going into detail here,
G3
1 is similar to G3, but the pseudo-Galilean scalar product of two vectors ~a =
(x, y, z) and ~b = (x1, y1, z1) is defined by
〈~a,~b〉 =
{
xx1 if x 6= 0 or x1 6= 0,
yy1 − zz1 if x = x1 = 0.
Therefore, there exist four types of isotropic vectors ~a = (0, y, z) in G3
1: space-
like vectors (if y2 − z2 > 0), timelike vectors (if y2 − z2 < 0) and two types of
lightlike vectors (if y = ±z). In the same way as in a Minkowski 3-space, hy-
perbolic functions have to be used instead of trigonometric functions to describe
rotations. One can also define different types of surfaces of revolution in G3
1. It
was done in [8] but one type seems to be lost there. Thus, this could be the
subject of further research, as well as a more elaborate study of the constancy
of the curvatures of surfaces of revolution in the Galilean and pseudo-Galilean
3-spaces. See also [5] and [13], where the Gauss map of two types of surfaces of
revolution in G3
1 is studied.
The figures of the paper are made by using VisuMath. For more information
see www.visumath.be.
Acknowledgment. The authors wish to thank the referee for valuable com-
ments that improved the first version of the paper. Special thanks are for pointing
out the existence of reference [1].
References
[1] A. Artykbaev and D.D. Sokolov, Geometry in the Large in a Flat Space-Time, FAN,
Tashkent, 1991 (Russian).
[2] M. Dede, Tubular surfaces in Galilean space, Math. Commun. 18 (2013), 209–217.
www.visumath.be
Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space 151
[3] M. Dede, C. Ekici, and A. Ceylan Çöken, On the parallel surfaces in Galilean space,
Hacet. J. Math. Stat. 42 (2013), 605–615.
[4] M. Dede, C. Ekici, W. Goemans, and Y. Ünlütürk, Twisted surfaces with vanish-
ing curvature in Galilean 3-space, Int. J. Geom. Methods Mod. Phys. 15 (2018),
1850001, 13 pp.
[5] B. Divjak and Ž. Milin Šipuš, Some special surfaces in the pseudo-Galilean space,
Acta Math. Hungar. 118 (2008), 209–226.
[6] W. Goemans and I. Van de Woestyne, Twisted surfaces with null rotation axis in
Minkowski 3-space, Results Math. 70 (2016), 81–93.
[7] Ž. Milin Šipuš, Ruled Weingarten surfaces in Galilean space, Period. Math. Hungar.
56 (2008), 213–225.
[8] Ž. Milin Šipuš and B. Divjak, Surfaces of constant curvature in the pseudo-Galilean
space, Int. J. Math. Math. Sci. (2012), Art. ID 375264, 28 pp.
[9] Ž. Milin Šipuš and B. Divjak, Translation surfaces in the Galilean space, Glas. Mat.
Ser. III 46(66) (2011), 455–469.
[10] D. Palman, Drehzykliden des Galileischen Raumes G3, Math. Pannon. 2(1) (1991),
95–104.
[11] O. Röschel, Die Geometrie des Galileischen Raumes, Bericht der Mathematisch-
Statistischen Sektion in der Forschungsgesellschaft Joanneum, Bericht Nr. 256, Ha-
bilitationsschrift, Leoben, 1984.
[12] I.M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis, Springer-
Verlag, New York-Heidelberg, 1979.
[13] D.W. Yoon, Surfaces of revolution in the three dimensional pseudo-Galilean space,
Glas. Mat. Ser. III 48(68) (2013), 415–428.
Received January 9, 2017, revised June 20, 2017.
M. Dede,
Kilis 7 Aralık University, Department of Mathematics, Kilis, 79000, Turkey,
E-mail: mustafadede03@gmail.com
C. Ekici,
Eskişehir Osmangazi University, Department of Mathematics-Computer, Eskişehir,
26480, Turkey,
E-mail: cumali.ekici@gmail.com
W. Goemans,
KU Leuven, Faculty of Economics and Business, Brussels, 1000, Belgium,
E-mail: wendy.goemans@kuleuven.be
Поверхнi обертання з нульовою кривиною у
тривимiрному просторi Галiлея
M. Dede, C. Ekici, and W. Goemans
У статтi визначено та дослiджено три типи поверхонь обертання у
тривимiрному просторi Галiлея. Запропоновано конструкцiю поверхнi
mailto:mustafadede03@gmail.com
mailto:cumali.ekici@gmail.com
mailto:wendy.goemans@kuleuven.be
152 M. Dede, C. Ekici, and W. Goemans
обертання у тривимiрному просторi Галiлея, визначеної обертанням пла-
скої кривої навколо осi, що лежить у площинi кривої. Класифiковано
поверхнi обертання у тривимiрному просторi Галiлея з нульовою гауссо-
вою кривиною та з нульовою середньою кривиною.
Ключовi слова: поверхня обертання, пласка поверхня, мiнiмальна по-
верхня, тривимiрний простiр Галiлея.
Introduction
Preliminaries
Surfaces of revolution in the Galilean 3-space
Type I surfaces of revolution in G3.
Type II surfaces of revolution in G3.
Type III surfaces of revolution in G3.
Zero curvature surfaces of revolution in G3
Zero curvature type I surfaces of revolution in G3.
Zero curvature type II surfaces of revolution in G3.
Zero curvature type III surfaces of revolution in G3.
Conclusion and further research
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