r in
TRANSLATION SURFACES OF TYPE 2 IN THE THREE
DIMENSIONAL SIMPLY ISOTROPIC SPACE I13
t
Bull. Korean Math. Soc. 0 (0), No. 0, pp. 1–0
https://doi.org/10.4134/BKMS.b160377
pISSN: 1015-8634 / eISSN: 2234-3016
Bahaddin Bukcu, Murat Kemal Karacan, and Dae Won Yoon
fP
Abstract. In this paper, we classify translation surfaces of Type 2 in
the three dimensional simply Isotropic space I13 satisfying some algebraic
equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form
of the surface. We also give explicit forms of these surfaces.
1. Introduction
m
Ah
ea
do
Let x : M →E be an isometric immersion of a connected n-dimensional
manifold in the m-dimensional Euclidean space Em . Denote by H and ∆
the mean curvature and the Laplacian of M with respect to the Riemannian
metric on M induced from that of Em , respectively. Takahashi proved that
the submanifolds in Em satisfying ∆x = λx, that is, all coordinate functions
are eigenfunctions of the Laplacian with the same eigenvalue λ ∈ R, are either
the minimal submanifolds of Em or the minimal submanifolds of hypersphere
Sm−1 in Em [15].
As an extension of Takahashi theorem, Garay studied hypersurfaces in Em
whose coordinate functions are eigenfunctions of the Laplacian, but not necessarily associated to the same eigenvalu in [8]. He considered hypersurfaces in
Em satisfying the condition
(1.1)
∆x = Ax,
where A ∈ Mat(m, R) is an m × m-diagonal matrix, and proved that such
hypersurfaces are minimal (H = 0) in Em and open pieces of either round
hyperspheres or generalized right spherical cylinders. Related to this, Dillen,
Pas and Verstraelen investigated surfaces in E3 whose immersions satisfy the
Received May 2, 2016.
2010 Mathematics Subject Classification. 53A35, 53B30, 53A40.
Key words and phrases. Laplace operator, simply Isotropic space, translation surfaces.
The authors would like to thank the referees for their valuable comments which helped
to improve the manuscript. The second author was supported by Basic Science Research
Program through the National Research Foundation of Korea(NRF)funded by the Ministry
of Education (2015R1D1A1A01060046).
c
2017
Korean Mathematical Society
1
2
B. BUKCU, M. K. KARACAN, AND D. W. YOON
(1.2)
t
condition
∆x = Ax + B,
∆G = AG,
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(1.3)
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where A ∈ Mat(3, R) is a 3 × 3-real matrix and B ∈ R3 [6]. In other words,
each coordinate function is of 1-type in the sense of Chen [5]. The notion of an
isometric immersion x is naturally extended to smooth functions on submanifolds of Euclidean space or pseudo-Euclidean space. The most natural one of
them is the Gauss map of the submanifold. In particular, if the submanifold is
a hypersurface, the Gauss map can be identified with the unit normal vector
field to it. Dillen, Pas and Verstraelen studied surfaces of revolution in the
three dimensional Euclidean space E3 such that its Gauss map G satisfies the
condition
where A ∈ Mat(3, R) [7]. Yoon studied translation surfaces in the 3-dimensional
Minkowski space whose Gauss map G satisfies the condition (1.3) and also
translation surfaces in a 3-dimensional Galilean space G3 satisfies the condition
(1.4)
∆xi = λi xi ,
Ah
ea
do
where λi ∈ R and provided some examples of new classes of translation surface
[16, 17]. Baba-Hamed, Bekkar and Zoubir classified all translation surfaces
in the 3-dimensional Lorentz-Minkowski space R31 under the condition (1.4)
[2]. Bekkar and Senoussi studied the translation surfaces in the 3-dimensional
Euclidean and Lorentz-Minkowski space under the condition
∆III ri =µi ri ,
where µi ∈ R and ∆III denotes the Laplacian of the surface with respect to the
third fundamental form III [3]. They showed that in both spaces a translation
surface satisfying the preceding relation is a surface of Scherk. Aydin and Sipus
studied constant curvatures of translation surfaces in the three dimensional
simply Isotropic space [1, 14]. Karacan, Yoon and Bukcu classified translation
surfaces of Type 1 satisfying ∆J xi = λi xi , j = 1, 2 and ∆III xi = λi xi , λi ∈ R
[4, 10].
In this paper, we classify the translation surfaces of Type 2 in the three
dimensional simply Isotropic space under the condition ∆J xi = λi xi , J =
I, II, III, where λi ∈ R. ∆J denotes the Laplace operator with respect to the
fundamental forms I, II and III.
2. Preliminaries
A simply isotropic space I13 is a Cayley–Klein space defined from the three
dimensional projective space P(R3 ) with the absolute figure which is an ordered
triple (w, f1 , f2 ), where w is a plane in P(R3 ) and f1 , f2 are two complexconjugate straight lines in w. The homogeneous coordinates in P(R3 ) are
introduced in such a way that the absolute plane w is given by x0 = 0 and the
TRANSLATION SURFACES OF TYPE 2 IN THE I13
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absolute lines f1 , f2 by x0 = x1 + ix2 = 0, x0 = x1 − ix2 = 0. The intersection
point F(0 : 0 : 0 : 1) of these two lines is called the absolute point. The group
of motions of the simply isotropic space is a six-parameter group given in the
affine coordinates x = xx01 , y = xx20 , z = xx30 by
x = a + x cos θ − y sin θ
y = b + x sin θ + y cos θ
(2.1)
z = c + +c1 x + c2 y + z,
fP
where a, b, c, c1 , c2 , θ ∈ R. Such affine transformations are called isotropic congruence transformations or i-motions.
Isotropic geometry has different types of lines and planes with respect to
the absolute figure. A line is called non-isotropic (resp. completely isotropic)
if its point at infinity does not coincide (coincides) with the point F. A plane
is called non-isotropic (resp. isotropic) if its line at infinity does not contain F,
otherwise. Completely isotropic lines and isotropic planes in this affine model
appear as vertical, i.e., parallel to the z-axis. Finally, the metric of the simply
isotropic space I13 is given by
ds2 = dx2 + dy 2 .
Ah
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A surface M immersed in I13 is called admissible if it has no isotropic tangent
planes. For such a surface, the coefficients E, F, G of its first fundamental form
are calculated with respect to the induced metric and the coefficients L, M, N
of the second fundamental form, with respect to the normal vector field of a
surface which is always completely isotropic. The (isotropic) Gaussian and
(isotropic) mean curvature are defined by
(2.2)
K = k1 k2 =
EN − 2F M + GL
LN − M 2
, 2H = k1 + k2 =
,
EG − F 2
EG − F 2
where k1 , k2 are principal curvatures, i.e., extrema of the normal curvature
determined by the normal section (in completely isotropic direction) of a surface. Since EG − F 2 > 0, for the function in the denominator we often put
W 2 = EG − F 2 .The surface M is said to be isotropic flat (resp. isotropic
minimal ) if K (resp.H) vanishes [1, 11, 13, 14].
It is well known in terms of local coordinates {u, v} of M the Laplacian
operators ∆I , ∆II , ∆III of the first, the second and the third fundamental
form on M are defined by
1
∂ Gxu − F xv
∂ F xu − Exv
I
√
√
(2.3)
∆ x = −√
−
,
∂v
EG − F 2 ∂u
EG − F 2
EG − F 2
(2.4)
∆ x = −√
II
1
|LN −M 2 |
∂
∂u
N xu −M xv
√
2
|LN −M |
−
∂
∂v
M xu −Lxv
√
2
|LN −M |
4
B. BUKCU, M. K. KARACAN, AND D. W. YOON
(2.5)

2
EG
−
F

∆III x = −
LN − M 2
Zxu −Y
√ xv
−
(LN −M 2 ) EG−F 2 Y xu −Xx
∂
v
√
∂v (LN −M 2 ) EG−F 2
∂
∂u
where
X = EM 2 − 2F LM + GL2 ,
Y = EM N − F LN + GLM − F M 2 ,
Z = GM 2 − 2F N M + EN 2
[2, 3, 4, 9, 10, 12].
,
fP
3. Translation surfaces in I13

r in
√
t
and
In order to describe the isotropic analogues of translation surfaces of constant
curvatures, we consider translation surfaces obtained by translating two planar
curves. The local surface parametrization is given by
(3.1)
x(u, v) = α(u) + β(v).
Ah
ea
do
Since there are, with respect to the absolute figure, different types of planes
in I13 , there are in total three different possibilities for planes that contain
translated curves: the translated curves can be curves in isotropic planes (which
can be chosen, by means of isotropic motions, as y = 0, resp. x = 0); or
one curve is in a non-isotropic plane (z = 0) and one curve in an isotropic
plane (y = 0); or both curves are curves in non-isotropic perpendicular planes
(y − z = π, resp. y + z = π). Therefore, the obtained translation surfaces allow
the following parametrizations:
Type 1: The surface M is parametrized by
(3.2)
x(u, v) = (u, v, f (u) + g(v)) ,
and the translated curves are α(u) = (u, 0, f (u)), β(v) = (0, v, g(v)) .
Type 2: The surface M is parametrized by
(3.3)
x(u, v) = (u, f (u) + g(v), v) ,
and the translated curves are α(u) = (u, f (u), 0), β(v) = (0, g(v), v) . In order
to obtain admissible surfaces, g 0 (v) 6= 0 is assumed (i.e., g(v) 6=const.).
Type 3: The surface M is parametrized by
1
(3.4)
x(u, v) = (f (u) + g(v), u − v + π, u + v) ,
2
and the translated curves are
1
π
π
π
π
α(u) =
f (u), u + , u −
, β(v) = g(v), − v, + v .
2
2
2
2
2
In order to obtain admissible surfaces, f 0 (u)+g 0 (v) 6= 0 is assumed (i.e., f 0 (u) 6=
−g 0 (v) = a =constant.) [13].
TRANSLATION SURFACES OF TYPE 2 IN THE I13
5
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4. Translation surfaces of Type 2 satisfying ∆I xi = λi xi
t
In this paper, we will investigate the translation surfaces of Type 2.
In this section, we classify translation surface of Type 2 in I13 satisfying the
equation
∆I xi = λi xi ,
(4.1)
where λi ∈R, i=1, 2, 3 and
∆I x = ∆I x1 , ∆I x2 , ∆I x3 ,
where
fP
x1 = u, x2 = f (u) + g(v), x3 = v.
For the translation surface given by (3.3), the coefficients of the first and second
fundamental form are
(4.2)
2
2
E = 1 + f 0 , F = f 0 g0 , G = g0 ,
g 00
f 00
,
M
=
0,
N
=
−
,
g0
g0
respectively. The Gaussian curvature K and the mean curvature H are
02
00
02
g
(v)f
(u)
+
1
+
f
(u)
g 00 (v)
00
00
f (u)g (v)
(4.4)
K=
, H=−
,
g 04 (v)
2g 03 (v)
L=−
Ah
ea
do
(4.3)
respectively.
Suppose that the surface has non zero Gaussian curvature, so
f 00 (u)g 00 (v) 6= 0, ∀ u, v ∈ I.
By a straightforward computation, the Laplacian operator on M with the help
of (4.2) and (2.3) turns out to be


2
2
g 0 f 00 + 1 + f 0 g 00
.
(4.5)
∆I x = 0, 0,
g 03
Suppose that M satisfies (4.1). Then from (4.5), we have

 2
2
g 0 f 00 + 1 + f 0 g 00

 = λv,
(4.6)
g 03
where λ ∈ R. This means that M is at most of 1-type. First of all, we assume that M satisfies the condition ∆I x = 0. We call a surface satisfying that
condition a harmonic surface or isotropic minimal. In this case, we get from
(4.6)
2
2
(4.7)
g 0 f 00 + 1 + f 0 g 00 = 0.
6
B. BUKCU, M. K. KARACAN, AND D. W. YOON
t
Here u and v are independent variables, so each side of (4.7) is equal to a
constant, call it p. Hence, the two equations
r in
g 00
f 00
=
p
=
−
.
1 + f 02
g 02
Thus we get
ln (cos (pu − c2 ))
,
p
ln (pv − c4 )
g(v) = c3 +
,
p
f (u) = c1 −
(4.8)
fP
where p, ci ∈ R. In this case, M is parametrized by
ln (cos (pu − c2 ))
ln (pv − c4 )
(4.9) x(u, v) = u, c1 −
+ c3 +
,v .
p
p
In particular, if p = 0, we have
(4.10)
f (u) = c1 u + c2 ,
g(v) = c3 v + c4 .
where ci ∈ R. In this case, M is parametrized by
(4.11)
x(u, v) = (u, (c1 u + c2 ) + (c3 v + c4 ) , v) .
Ah
ea
do
Using the solution (4.10) gives rise a contradiction with our assumption saying
that the solution must be non-degenerate second fundamental form.
Theorem 4.1. Let M be a translation surface given by (3.3) in I13 . If M is
harmonic or isotropic minimal, then it is congruent to an open part of the
surface (4.9). The obtained surface is the isotropic Schrerk’s surfaces of the
second type.
If λ 6= 0, from (4.6), we have
(4.12)
2
2
3
g 0 f 00 + 1 + f 0 g 00 = λvg 0 .
This differential equation (4.12) admits the solutions
f (u) = c1 , g(v) = c2 ,
arctan
f (u) = c1 , g(v) = ±
(4.13)
√
√
v λ
−λv 2 −2c1
√
λ
+ c2 ,
f (u) = c1 u + c2 ,
p
√ p
−1 − c21 ln vλ + λ λv 2 + 2c4 + 2c21 c4
√
g(v) = c3 ±
,
λ
1
f (u) = c3 u2 vλ + c1 u + c2 , g(v) = c3 v + c4 ,
2
TRANSLATION SURFACES OF TYPE 2 IN THE I13
7
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t
where ci ∈ R. But, using the solutions (4.13) give rise a contradiction with
our assumption saying that the solution must be non-degenerate second fundamental form. Based on the selection of the function f (u), it is possible to
obtain other form of the function g(v), vice versa. For example, if we choose
g(v) = v 2 , we have
√
(u + 2v 2 c2 ) 1 − 4λv 4
,
f (u) = c1 + 2v 2 log cos
2v 2
where c1 , c2 ∈ R. In this case, M is parametrized by
√
(u + 2v 2 c2 ) 1 − 4λv 4
2
2
+
v
,
v
.
(4.14) x(u, v) = u, c1 + 2v log cos
2v 2
fP
Theorem 4.2. Let M be a non harmonic translation surface given by (3.3)
in the three dimensional simply isotropic space I13 . If The surface M satisfies
the condition ∆I xi =λi xi , where λi ∈R, i=1, 2, 3, then it is congruent to an
open part of the surface (4.14). The obtained surface is the isotropic Schrerk’s
surface of the second type.
5. Translation surfaces of Type 2 satisfying ∆II xi = λi xi
In this section, we classify translation surface of Type 2 with non-degenerate
second fundamental form in I13 satisfying the equation
∆II xi = λi xi ,
Ah
ea
do
(5.1)
where λi ∈R, i=1, 2, 3 and
∆II x = ∆II x1 ,∆II x2 ,∆II x3 ,
where
x1 = u, x2 = f (u) + g(v), x3 = v.
By a straightforward computation, the Laplacian operator on M with the help
of (4.3) and (2.4) turns out to be
0 000 0 g
f 0 f 000
g 0 g 000
g 0 g 000
gf
(5.2)
∆II x = − 002 ,
4 − 002 − 002 , − 002 .
2
2f
f
g
2g
The equation (5.1) by means of (5.2) gives rise to the following system of
ordinary differential equations
(5.4)
(5.5)
g 0 f 000
= λ1 u
2f 002
g 0 g 000
− 002
= λ2 (f (u) + g(v)) ,
g
−
(5.3)
g0
2
f 0 f 000
4 − 002
f
−
g 0 g 000
= λ3 v,
2g 002
8
B. BUKCU, M. K. KARACAN, AND D. W. YOON
g 0 (v) (2 + λ1 u + λ3 v) = λ2 (f (u) + g(v)) .
r in
(5.6)
t
where λi ∈ R. This means that M is at most of 3- types. Combining equations
(5.3), (5.4) and (5.5), we have
We discuss six cases according to constants λ1 , λ2 , λ3 .
Case 1: Let λ1 = 0, λ2 6= 0, λ3 6= 0, from (5.6), we obtain
(5.7)
g 0 (2 + λ3 v) = λ2 f + λ2 g.
Here u and v are independent variables, so each side of (5.7) is equal to a
constant, call it p. Hence, the two equations
(5.8)
g 0 (2 + λ3 v) − λ2 g = p = λ2 f.
Their general solutions are
fP
p
,
λ2
(5.9)
λ2
p
g(v) = − + c1 (2 + λ3 v) λ3 ,
λ2
where for some constant c1 6= 0 and λ2 6= 0, λ3 6= 0. In particular, if p = 0,
then we have
f (u) = 0,
(5.10)
λ2
g(v) = c1 (2 + λ3 v) λ3 ,
f (u) =
Ah
ea
do
where c1 ∈ R.
Case 2: Let λ1 = λ2 = 0, λ3 6= 0, from (5.6), we obtain
g 0 (v) (2 + λ3 v) = 0.
Thus we get
(5.11)
g(v) = c1 ,
where c1 ∈ R.
Case 3: Let λ1 6= 0, λ2 = 0, λ3 = 0, from (5.6), we obtain
(5.12)
g 0 (2 + λ1 u) = 0.
Thus we get (5.11).
Case 4: Let λ1 = 0, λ2 6= 0, λ3 = 0, from (5.6), we obtain
(5.13)
2g 0 − λ2 g = λ2 f.
Here u and v are independent variables, so each side of (5.13) is equal to a
constant, call it p. Hence, the two equations
2g 0 − λ2 g = p = λ2 f.
Their general solutions are
p
,
λ2
λ2 v
p
g(v) = − + c1 e 2 ,
λ2
f (u) =
(5.14)
TRANSLATION SURFACES OF TYPE 2 IN THE I13
t
where for some constant c1 6= 0 and λ2 6= 0.
Case 5: Let λ1 = λ2 = λ3 = 0, from (5.6), we obtain
9
r in
2g 0 (v) = 0.
Thus we get (5.11). Here, the function f (u) independent of selection of the
function g(v).
Case 6: Let λ1 6= 0, λ2 6= 0, λ3 = 0, from (5.6), we obtain
g 0 (2 + λ1 u) = λ2 f + λ2 g.
(5.15)
Here u and v are independent variables, so each side of (5.15) is equal to a
constant, call it p. Hence, the two equations
Their general solutions are
fP
2g 0 − λ1 ug 0 − λ2 g = p = λ2 f.
(5.16)
λ2 v
p
p
, g(v) = − + c1 e− −2+λ1 u ,
λ2
λ2
where for some constant c1 6= 0 and λ2 6= 0.
Using the solutions (5.9), (5.10), (5.11), (5.14) and (5.17) give rise a contradiction with our assumption saying that the solution must be non-degenerate
second fundamental form.
(5.17)
f (u) =
Ah
ea
do
Definition. A surface of in the three dimensional simple isotropic space is said
to be II-harmonic if it satisfies the condition ∆II x = 0.
Theorem 5.1. Let M be a translation surface given by (3.3) in the three
dimensional simply isotropic space I13 . Then M is II-harmonic if and only if
it is an open part of the following non isotropic plane
x(u, v) = (u, f (u) + c1 , v) .
The obtained surfaces is a cylindrical surface.
Theorem 5.2. Let M be a non II-harmonic translation surface with nondegenerate second fundamental form given by (3.3) in the three dimensional
simply isotropic space I13 . There are no surfaces satisfy the condition
∆II xi =λi xi ,
where λi ∈R, i=1, 2, 3.
6. Translation surfaces of Type 2 satisfying ∆III xi = λi xi
In this section, we classify translation surface of Type 2 in I13 satisfying the
equation
∆III xi = λi xi ,
(6.1)
where λi ∈R, i=1, 2, 3 and
∆III x = ∆III x1 , ∆III x2 , ∆III x3 ,
10
B. BUKCU, M. K. KARACAN, AND D. W. YOON
t
where
r in
x1 = u, x2 = f (u) + g(v), x3 = v.
Suppose that the surface has non zero Gaussian curvature, so f 00 (u)g 00 (v) 6=
0,∀u ∈ I. By a straightforward computation, the Laplacian operator on M with
the help of (4.2), (4.3) and (2.5) turns out to be
!
2
2
2
2
2
g 0 g 000
−4g 0 f 00 g 00 + f 0 g 0 g 00 f 000 + g 0 f 00 g 000
g 0 f 000
III
, − 002 .
(6.2) ∆ x = − 002 , −
2f
2f 002 g 002
2g
2
2
2
fP
Equation (6.1) by means of (6.2) gives rise to the following system of ordinary
differential equations
0 000 gf
(6.3)
− 002 = λ1 u,
2f
2
2
−4g 0 f 00 g 00 + f 0 g 0 g 00 f 000 + g 0 f 00 g 000
= λ2 (f (u) + g(v)) ,
(6.4)
−
2f 002 g 002
0 000 gg
(6.5)
− 002 = λ3 v,
2g
where λ1 , λ2 and λ3 ∈ R.This means that M is at most of 3- types. Combining
equations (6.3), (6.4) and (6.5), we have
(6.6)
2g 0 + λ1 uf 0 + λ3 vg 0 = λ2 (f (u) + g(v)) .
Ah
ea
do
Here u and v are independent variables, so each side of (6.6) is equal to constant,
call it p. Hence, we have
(6.7)
λ1 uf 0 − λ2 f = p = −λ3 vg 0 − 2g 0 + λ2 g.
Their general solutions are given by
λ2
p
f (u) = − + c1 u λ1 ,
λ2
(6.8)
λ2
p
g(v) =
+ c2 (2 + λ3 v) λ3 ,
λ2
where p, ci ∈ R with λi 6= 0. In this case, M is parametrized by
λ2
λ2
p
1
p
λ3
λ1
u, − + c1 u
(6.9)
x(u, v) =
+
+ c2 (2 + λ3 v)
,v .
2
λ2
λ2
In particular, if p = 0,
λ2
(6.10)
f (u) = c1 u λ1 ,
λ2
g(v) = c2 (2 + λ3 v) λ3 ,
In this case, M is parametrized by
λ2 λ2 (6.11)
x(u, v) = u, c1 u λ1 + c2 (2 + λ3 v) λ3 , v .
We discuss four cases according to constants λ1 , λ2 , λ3 .
TRANSLATION SURFACES OF TYPE 2 IN THE I13
11
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Case 1: Let λ1 = λ2 = λ3 = λ 6= 0, from (6.7), we obtain
and their general solutions are
f (u) = −
(6.12)
p
+ c1 u
λ
r in
λuf 0 − λf = p = −λvg 0 − 2g 0 + λg,
p
+ c2 (2 + λv) ,
λ
where λ, p, ci ∈ R. Using the solution (6.11) give rise a contradiction with our
assumption saying that the solution must be non-degenerate second fundamental form.
Case 2: Let λ1 = λ2 = 0, λ3 6= 0, from (6.7), we obtain
g(v) =
p = −λ3 vg 0 − 2g 0 .
fP
(6.13)
Thus we have
(6.14)
f (u), g(v) = c1 −
p ln (2 + cv)
,
λ3
where c1 , p ∈ R. Here, the function f (u) independent of selection of the function
g(v). In this case, M is parametrized by
p ln (2 + cv)
(6.15)
x(u, v) = u, f (u) + c1 −
,v .
λ3
Ah
ea
do
Case 3: Let λ1 6= 0, λ2 6= 0, λ3 = 0, from (6.7), we obtain
(6.16)
λ1 uf 0 − λ2 f = p = −2g 0 + λ2 g,
and
f (u) = −
(6.17)
λ2
p
+ c1 u λ1 ,
λ2
λ2 v
p
+ ce 2 ,
λ2
where ci , p ∈ R. In this case, M is parametrized by
λ2
λ2 v
p
p
λ1
2
(6.18)
x(u, v) = u, − + c1 u
+
,v .
+ ce
λ2
λ2
g(v) =
Case 4: Let λ1 = λ2 = λ3 = 0, from (4.12), we obtain
p
(6.19)
g(v) = c1 + v,
2
where ci ∈ R. Here, the function f (u) independent of selection of the function
g(v). In this case, M is parametrized by
p (6.20)
x(u, v) = u, v, f (u) + c1 + v .
2
Definition. A surface of in the three dimensional simple isotropic space is said
to be III-harmonic if it satisfies the condition ∆III x = 0.
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B. BUKCU, M. K. KARACAN, AND D. W. YOON
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Theorem 6.1. Let M be a translation surface by (3.3) in the three dimensional
simply isotropic space I13 . If M is III-harmonic, then it is congruent to an open
part of the surface (6.20). The obtained surfaces is a cylindrical surface.
Theorem 6.2 (Classification). Let M be a translation surface with non - degenerate second fundamental form given by (3.3) in the three dimensional simply isotropic space I13 . The surfaces M satisfies the condition ∆III xi =λi xi ,
where λi ∈R, then it is congruent to an open part of the surfaces (6.9), (6.11),
(6.15) and (6.18). The obtained surfaces are the isotropic Schrerk’s surfaces of
the first type.
fP
References
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[1] M. E. Aydin, A generalization of translation surfaces with constant curvature in the
isotropic space, J. Geom. 107 (2016), no. 3, 603–615.
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[12] B. Senoussi and M. Bekkar, Helicoidal surfaces with ∆J r = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math. 60 (2015), no. 13, 437–448.
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Anal. 6 (2012), no. 28, 1355–1361.
TRANSLATION SURFACES OF TYPE 2 IN THE I13
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Dae Won Yoon
Department of Mathematics
Education and RINS
Gyeongsang National University
Jinju 660-701, Korea
E-mail address: [email protected]
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Murat Kemal Karacan
Department of Mathematics
Faculty of Sciences and Arts
Usak University
1 Eylul Campus 64200, Usak, Turkey
E-mail address: [email protected]
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Bahaddin Bukcu
Department of Mathematics
Faculty of Sciences and Arts
Gazi Osman Pasa University
60250, Tokat, Turkey
E-mail address: [email protected]
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