Classifications of translation surfaces in isotropic geometry with constant curvature
UDC 515.12 We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvatures under the condition that at least one of translating curves lies in a plane.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507036583723008 |
|---|---|
| author | Aydin , M. E. Aydin, M. E. |
| author_facet | Aydin , M. E. Aydin, M. E. |
| author_sort | Aydin , M. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-07-02T03:56:29Z |
| description | UDC 515.12
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvatures under the condition that at least one of translating curves lies in a plane. |
| doi_str_mv | 10.37863/umzh.v72i3.505 |
| first_indexed | 2026-03-24T02:02:56Z |
| format | Article |
| fulltext |
UDC 515.12
M. E. Aydin (Firat Univ., Elazig, Turkey)
CLASSIFICATIONS OF TRANSLATION SURFACES
IN ISOTROPIC GEOMETRY WITH CONSTANT CURVATURE
КЛАСИФIКАЦIЇ ТРАНСЛЯЦIЙНИХ ПОВЕРХОНЬ
В IЗОТРОПНIЙ ГЕОМЕТРIЇ ЗI СТАЛОЮ КРИВИНОЮ
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvatures under
the condition that at least one of translating curves lies in a plane.
Запропоновано класифiкацiю трансляцiйних поверхонь в iзотропнiй геометрiї з довiльною сталою iзотропною
гауссовою та середньою кривиною за умови, що принаймнi одна з трансляцiйних кривих лежить у площинi.
1. Introduction. A translation surface in a Euclidean space \BbbR 3 that is expressed as the sum of two
curves can be locally parameterized by [5]
r(x, y) = \alpha (x) + \beta (y), (1.1)
where \alpha and \beta are referred to as translating curves. Recent results and progress on translation
surfaces in \BbbR 3 with constant Gaussian and mean curvatures were well-structured in the papers [14 –
16, 25].
If \alpha and \beta lie in orthogonal planes, then, up to a change of coordinates, the surface is locally
described in explicit form
z(x, y) = f(x) + g(y),
where f, g are smooth real-valued functions of one variable. In this case, beside the planes, only
minimal translation surface (i.e., mean curvature vanishes identically) is the Scherk surface, namely,
the graph of [34]
z(x, y) =
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \mathrm{c}\mathrm{o}\mathrm{s}(cy)\mathrm{c}\mathrm{o}\mathrm{s} (cx)
\bigm| \bigm| \bigm| \bigm| , c \in \BbbR - \{ 0\} .
Many generalizations on this result in (semi-) Euclidean and homogeneous spaces were done so far,
see, for example, [8 – 10, 12, 17, 19, 20, 23, 24, 26, 28, 29, 35, 37, 38].
Recently, Liu and Yu [22] introduced a new class of translation surfaces in \BbbR 3, so-called affine
translation surfaces, as the graphs of
z(x, y) = f(x) + g(y + ax), a \in \BbbR - \{ 0\} . (1.2)
By the change of coordinates x = u, y = v - au in (1.2) one can be locally parametrized as
r(u, v) = (u, v - au, f(u) + g(v)) ,
where the translating curves lie in the planes x = 0 and ax + y = 0. Because a \not = 0 these planes
are not orthogonal to each other and the obtained surface is a natural generalization of the classical
c\bigcirc M. E. AYDIN, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 291
292 M. E. AYDIN
translation surface. In same paper, the authors conjectured that, besides planes, only minimal graph
surface of the form (1.2), usually called affine Scherk surface, is given in explicit form
z(x, y) =
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \bigm| \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
c
\surd
1 + a2x
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s} (c[y + ax])
\bigm| \bigm| \bigm| \bigm| \bigm| , c \in \BbbR - \{ 0\} .
We also refer to [18, 21, 39, 40] for more recent results on this kind of surfaces.
Following Liu and Yu [22] we introduce and classify a new type of translation surfaces in
isotropic geometry with constant isotropic Gaussian curvature (CIGC) and constant isotropic mean
curvature (CIMC). In addition, we obtain the surfaces of CIGC and CIMC whose one translating
curve is planar and other one space curve.
2. Preliminaries. For fundamental notions of curves and surfaces in isotropic geometry that
is one of the Cayley – Klein geometries, we refer the reader to [3, 4, 6, 7, 11, 30 – 33]. Those can
briefed by the arguments from Projective Geometry as in next paragraphs.
Let \BbbP 3 denote the projective space and \Gamma a plane in \BbbP 3. Then an affine space can be obtained
from \BbbP 3 by substracting \Gamma which we call absolute plane. If \Gamma involves a pair of complex-conjugate
straight lines l1 and l2, so-called the absolute lines, then the obtained affine space becomes an
isotropic space \BbbI 3, where the triple (\Gamma , l1, l2) is referred to as the absolute figure of \BbbI 3 .
Let a quadruple
\bigl(
\~t : \~x : \~y : \~z
\bigr)
be the projective coordinates, i.e.,
\bigl(
\~t : \~x : \~y : \~z
\bigr)
\not = (0 : 0 : 0 : 0).
Then \Gamma and l1, l2 are, respectively, parameterized by \~t = 0 and \~t = \~x \pm i\~y = 0. The intersection
point of l1 and l2 is said to be absolute, i.e., (0 : 0 : 0 : 1) .
We are interested in an affine model of \BbbI 3. Thus, by means of the affine coordinates x =
\~x
\~t
,
y =
\~y
\~t
, z =
\~z
\~t
, \~t \not = 0, the group of motions of \BbbI 3 is a six-parameter group given by
(x, y, z) \mapsto - \rightarrow (x\prime , y\prime , z\prime ) :
\left\{
x\prime = a+ x \mathrm{c}\mathrm{o}\mathrm{s} \theta - y \mathrm{s}\mathrm{i}\mathrm{n} \theta ,
y\prime = b+ x \mathrm{s}\mathrm{i}\mathrm{n} \theta + y \mathrm{c}\mathrm{o}\mathrm{s} \theta ,
z\prime = c+ dx+ ey + z,
(2.1)
where a, b, c, d, e, \theta \in \BbbR . The metric invariants of \BbbI 3 under (2.1), such as isotropic distance and
angle, are Euclidean invariants in the Cartesian plane.
A line in \BbbI 3 is said to be isotropic provided its point at infinity agrees with the absolute point. In
the affine model of \BbbI 3, it corresponds to a line parallel to z-axes. Otherwise, it is called non-isotropic
line.
A plane in \BbbI 3 involving an isotropic line is said to be isotropic and then its line at infinity
involves the absolute point. Otherwise, it is called non-isotropic plane. For example, the equation
ax + by + cz = d, a, b, c, d \in \BbbR , determines a non-isotropic (resp., isotropic) plane when c \not = 0
(resp., c = 0).
A unit speed curve has the form
\alpha : I \subseteq \BbbR - \rightarrow \BbbI 3, s \mapsto - \rightarrow (f(s), g(s), h(s)) , (f \prime )2 + (g\prime )2 = 1,
where the derivative with respect to s is denoted by a prime. Therefore, the curvature \kappa and the
torsion \tau are given by
\kappa =
\sqrt{}
(f \prime \prime )2 + (g\prime \prime )2 or \kappa = f \prime g\prime \prime - f \prime \prime g\prime
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 293
and
\tau =
\mathrm{d}\mathrm{e}\mathrm{t} (\alpha \prime , \alpha \prime \prime , \alpha \prime \prime \prime )
\kappa 2
, \kappa \not = 0. (2.2)
A curve that lies in an isotropic (resp., non-isotropic) plane is said to be isotropic (resp., non-isotropic)
planar. Otherwise, we call it space curve and then \tau \not = 0.
Let M2 be an admissible surface in \BbbI 3, that is, a surface in which tangent plane at each point is
non-isotropic. Then the tangent plane Tp
\bigl(
M2
\bigr)
at some point p \in M2 has a Euclidean metric. For
such a surface, the components E,F,G of the first fundamental form is obtained by the metric on
M2 induced from \BbbI 3.
The unit isotropic direction U = (0, 0, 1) is assumed to be the normal vector field of M2
which is indeed orthogonal to all tangent vectors in Tp
\bigl(
M2
\bigr)
. Hence, the components of the second
fundamental form are computed with respect to U, namely,
l =
\mathrm{d}\mathrm{e}\mathrm{t} (rxx, rx, ry)\surd
EG - F 2
, m =
\mathrm{d}\mathrm{e}\mathrm{t} (rxy, rx, ry)\surd
EG - F 2
, n =
\mathrm{d}\mathrm{e}\mathrm{t} (ryy, rx, ry)\surd
EG - F 2
,
where r = r(x, y) refers to a local parameterization on M2 and rx =
\partial r
\partial x
, rxy =
\partial 2r
\partial x\partial y
, etc. Note
that the admissibility of M2 implies EG - F 2 \not = 0.
The isotropic Gaussian (so-called relative) K and the mean curvatures H are defined by
K =
ln - m2
EG - F 2
, H =
En - 2Fm+Gl
2 (EG - F 2)
.
A surface for which H (resp., K ) vanishes identically is said to be isotropic minimal (resp., flat).
Moreover, a surface is said to have CIMC (resp., CIGC) if H (resp., K ) is a constant function on
whole surface.
3. Categorization of translation surfaces. The translation surfaces in \BbbI 3 that are locally given
by (1.1) can be categorized in terms of the translating curves and the absolute figure as below:
Type I \alpha and \beta are planar:
Type I.1: \alpha and \beta are isotropic planar,
Type I.2: \alpha is isotropic planar and \beta non-isotropic planar,
Type I.3: \alpha and \beta are non-isotropic planar.
Type II: \alpha is isotropic planar and \beta space curve.
Type III: \alpha is non-isotropic planar and \beta space curve.
Type IV: \alpha and \beta are space curves.
A surface which belongs to one type is no equivalent to that of another type up to the absolute figure.
Let us assume for a surface of Type 1 that the translating curves lie orthogonal planes. Denoting
f and g smooth functions, after a change of coordinates, such a surface can be locally given in one
of the following explicit forms:
Type I.1*: Both translating curves are isotropic planar z(x, y) = f(x) + g(y).
Type I.2*: One translating curve is non-isotropic planar and other one isotropic planar y(x, z) =
= f(x) + g(z).
Type I.3*: Both translating curves are non-isotropic planar
x(y, z) =
1
2
\biggl[
f
\biggl(
y + z - \pi
2
\biggr)
+ g
\biggl(
- y + z + \pi
2
\biggr) \biggr]
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
294 M. E. AYDIN
These surfaces with CIMC and CIGC were obtained in [27, 36].
Next let assume for a surface of Type 1 that the translating curves lie in arbitrary planes. Let
[aij ] be a 2\times 2 matrix and | aij | = a11a22 - a12a21 \not = 0. More generally the surfaces of Type I.1 are
locally given by
r(u, v) =
\biggl(
a22u
| aij |
- a12v
| aij |
, - a21u
| aij |
+
a11v
| aij |
, f(u) + g(v)
\biggr)
. (3.1)
Up to a change of coordinates (3.1) turns to the graph of the form
z = f (a11x+ a12y) + g (a21x+ a22y) . (3.2)
Such surfaces with CIMC and CIGC, which we call translation graphs of first kind, were presented
in [2]. In this paper, we are interested in the surfaces of Types I.2 – III.
In the case that one curve is isotropic planar and another one without condition the translation
surfaces with CIMC and CIGC were provided in [1].
4. Surfaces of Types I.2 and I.3. Let [aij ] denote a 2 \times 2 matrix and \omega = | aij | \not = 0. We
consider the following translation surface generated by planar curves:
r(u, v) =
\Bigl( a22u
\omega
- a12v
\omega
, f(u) + g(v), - a21u
\omega
+
a11v
\omega
\Bigr)
, (4.1)
where the translating curves and the planes involving them are given by
\alpha (u) =
\Bigl( a22u
\omega
, f(u), - a21u
\omega
\Bigr)
, \Gamma \alpha : a21x+ a22z = 0
and
\beta (v) =
\Bigl(
- a12v
\omega
, g(v),
a11v
\omega
\Bigr)
, \Gamma \beta : a11x+ a12z = 0.
Remark 4.1. Since the roles of f and g are symmetric we only discuss the cases depending on
f throughout the section.
For a surface given by (4.1) we have:
The planes \Gamma \alpha and \Gamma \beta are orthogonal to each other, provided [aij ] is an orthogonal matrix.
If a12 = 0 then, due to \omega \not = 0, \Gamma \alpha becomes a non-isotropic plane and \Gamma \beta an isotropic plane.
Thereby the obtained surface belongs to Type I.2.
If a12 \not = 0, then by symmetry a22 \not = 0 and the planes \Gamma \alpha , \Gamma \beta are non-isotropic. Therefore, the
obtained surface belongs to Type I.3.
After the change of coordinates, so-called affine parameter coordinates,
u = a11x+ a12z, v = a21x+ a22z,
the local surface given by (4.1) turns to the graph of the form
y = f(u) + g(v). (4.2)
We call the surface of the form (4.2) translation graph of second kind. Notice that it is no equivalent
to the graph of the form (3.2) up to the absolute figure. The positive side of this notion is to express
the surfaces of Types I.2 and I.3 into one format.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 295
Now we purpose to present the translation graphs of second kind in \BbbI 3 with CIGC. For this, the
admissibility implies that
a12f
\prime + a22g
\prime \not = 0, f \prime =
df
du
, g\prime =
dg
dv
.
By a calculation, the Gaussian curvature K turns to
K =
\omega 2f \prime \prime g\prime \prime
(a12f \prime + a22g\prime )
4 . (4.3)
It is seen from (4.3) that K vanishes identically provided f \prime \prime = 0, namely, the surface is a generalized
cylinder (see [13, p. 439]) with non-isotropic rulings. So, next result can be stated in order for K to
be a non-vanishing constant.
Remark 4.2. In order to provide convenience in calculations, we denote nonzero constants by c1,
c2, . . . and some constants by d1, d2, . . . throughout the paper unless otherwise stated.
Theorem 4.1. For a translation graph of second kind in \BbbI 3 with nonzero CIGC the following
holds:
f(u) =
c1
2
u2 + d1u+ d2, g(v) =
- c1a
2
11
2K0a222
\biggl(
- 3K0a
2
22
c1a211
v + d3
\biggr) 2
3
+ d4. (4.4)
Proof. Because K = K0 \not = 0, K0 \in \BbbR , in (4.3), we have f \prime \prime g\prime \prime \not = 0. The partial derivative of
(4.3) with respect to u gives
4K0
\omega 2
\bigl(
a12f
\prime + a22g
\prime \bigr) 3 \bigl( a12f \prime \prime \bigr) = f \prime \prime \prime g\prime \prime . (4.5)
We distinguish two cases to proceed (4.5):
Case 1: a12 = 0. Then a11a22 \not = 0 due to \omega \not = 0. By (4.5) we have f \prime \prime = c1 and, thus, it
follows from (4.3) that
K0a
2
22
c1a211
=
g\prime \prime
(g\prime )4
. (4.6)
Solving the equations f \prime \prime = c1 and (4.6) leads to (4.4).
Case 2: a12 \not = 0. By symmetry, we deduce a22 \not = 0. Then (4.5) can be arranged as
(a12f
\prime + a22g
\prime )3
g\prime \prime
=
\omega 2
4K0a12
\biggl(
f \prime \prime \prime
f \prime \prime
\biggr)
. (4.7)
The partial derivative of (4.7) with respect to v yields
3a22(g
\prime \prime )2 -
\bigl(
a12f
\prime + a22g
\prime \bigr) g\prime \prime \prime = 0, (4.8)
where g\prime \prime \prime \not = 0 because a22g
\prime \prime \not = 0. After taking partial derivative of (4.8) with respect to u we
immediately achieve a contradiction.
Theorem 4.1 is proved.
By a calculation, the isotropic mean curvature H of (4.2) is
H = -
\bigl[
a212 + (\omega g\prime )2
\bigr]
f \prime \prime +
\bigl[
a222 + (\omega f \prime )2
\bigr]
g\prime \prime
2 (a12f \prime + a22g\prime )
3 . (4.9)
First, we concern minimality case via the following result.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
296 M. E. AYDIN
Theorem 4.2. For a minimal translation graph of second kind in \BbbI 3 one of the following occurs:
(1) it is a non-isotropic plane;
(2) f(u) =
1
c1a211a
2
22
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \mathrm{c}\mathrm{o}\mathrm{s} \bigl( c1a11a222u+ d1
\bigr) \bigm| \bigm| + d2,
g(v) =
- 1
c1a211a
2
22
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| c1a211a222v + d3
\bigm| \bigm| + d4,
(3) f(u) =
1
c1\omega 2
\mathrm{l}\mathrm{o}\mathrm{g} | \mathrm{c}\mathrm{o}\mathrm{s} (\omega c1a22u+ d1)| + d2,
g(v) =
- 1
c1\omega 2
\mathrm{l}\mathrm{o}\mathrm{g} | \mathrm{c}\mathrm{o}\mathrm{s} (\omega c1a12v + d3)| + d4.
Proof. Since H vanishes identically, (4.9) reduces to\bigl[
a212 + (\omega g\prime )2
\bigr]
f \prime \prime +
\bigl[
a222 + (\omega f \prime )2
\bigr]
g\prime \prime = 0. (4.10)
f \prime \prime = g\prime \prime = 0 is a solution for (4.10), and, in this case, the surface is a non-isotropic plane. Suppose
that f \prime \prime g\prime \prime \not = 0. Hence, (4.10) implies
- f \prime \prime
a222 + (\omega f \prime )2
= c1 = - g\prime \prime
a212 + (\omega g\prime )2
. (4.11)
We have two cases:
Case 1: a12 = 0. Then we have a11a22 \not = 0 because \omega \not = 0. By solving (4.11), we obtain the
item (2) of the theorem.
Case 2: a12 \not = 0. By symmetry, we get a22 \not = 0 and solving (4.11) leads to last item of the
theorem.
Theorem 4.3. For a translation graph of second kind in \BbbI 3 with nonzero CIMC, we have
(a) f(u) = c1u+ d1, g(v) = - H0
a12c1
a221
v2 + d2v + d3,
or
(b) f(u) = c1u+ d1, g(v) =
a222 + (\omega c1)
2
2H0a222
\Biggl(
4H0a22
a222 + (\omega c1)
2 v + d2
\Biggr) 1
2
- a12c1
a22
v + d3.
Proof. Assume H = H0 \not = 0, H0 \in \BbbR , in (4.9). The partial derivatives of (4.9) with respect to
u and v yield
- 6H0\omega
- 2a12a22
\bigl(
a12f
\prime + a22g
\prime \bigr) (f \prime \prime g\prime \prime ) = g\prime g\prime \prime f \prime \prime \prime + f \prime f \prime \prime g\prime \prime \prime . (4.12)
The situation for which both f \prime \prime and g\prime \prime vanish is a solution for (4.12), however we omit this one
because H0 \not = 0. We distinguish the remaining cases:
Case 1: f = c1u + d1 and g\prime \prime \not = 0. This assumption is a solution for (4.12). Then, from (4.9),
we derive
g\prime \prime
(a12c1 + a22g\prime )
3 =
- 2H0
a222 + (\omega c1)2
. (4.13)
We have two cases:
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 297
(1.1) a22 = 0. Here a12a21 \not = 0 due to \omega \not = 0. Solving (4.13) implies the first item of the
theorem.
(1.2) a22 \not = 0. By symmetry we have a12 \not = 0. After solving (4.13) we obtain the second item
of the theorem.
Case 2: f \prime \prime g\prime \prime \not = 0. By dividing (4.12) with f \prime \prime g\prime \prime , one can be rewritten as
- 6H0\omega
- 2a12a22
\bigl(
a12f
\prime + a22g
\prime \bigr) = g\prime
f \prime \prime \prime
f \prime \prime + f \prime g
\prime \prime \prime
g\prime \prime
. (4.14)
We have again cases:
(2.1) a12 = 0. Then \omega \not = 0 implies a11a22 \not = 0. (4.14) turns to
f \prime \prime \prime
f \prime f \prime \prime = d1 = - g\prime \prime \prime
g\prime g\prime \prime
,
which gives that f \prime \prime = c1e
d1f and g\prime \prime = c2e
- d1g . By substituting those in (4.9), we derive
- 2H0a22(g
\prime )3 = c1a
2
11(g
\prime )2ed1f +
\bigl[
c2 + c2a
2
11(f
\prime )2
\bigr]
e - d1g. (4.15)
Put f \prime = p and g\prime = q in (4.15). Then taking partial derivative of (4.15) with respect to f yields
0 = d1c1q
2ed1f + 2c2p \.pe
- d1g, (4.16)
where \.p =
dp
df
=
f \prime \prime
f \prime . If d1 = 0 in (4.16), then we obtain the contradiction \.p = 0. Otherwise, we
have
d1c1e
d1f
2c2p \.p
= c3 = - e - d1g
q2
. (4.17)
By substituting the second equality in (4.17) into (4.15), we conclude
- 2H0a22q (g) = c1a
2
11e
d1f - c2c3
\Bigl[
1 + a211p (f)
2
\Bigr]
. (4.18)
The left-hand side in (4.18) is a function of g however other side is a function of f. This is not
possible.
(2.2) a12 \not = 0 in (4.14). The symmetry implies a22 \not = 0. By dividing (4.14) with f \prime g\prime , we write
D
\biggl(
a12
g\prime
+
a22
f \prime
\biggr)
=
f \prime \prime \prime
f \prime f \prime \prime +
g\prime \prime \prime
g\prime g\prime \prime
, (4.19)
where D = - 6H0\omega
- 2a12a22.
It follows from (4.19) that
f \prime \prime \prime =
\bigl(
- d1f
\prime +Da22
\bigr)
f \prime \prime and g\prime \prime \prime =
\bigl(
d1g
\prime +Da12
\bigr)
g\prime \prime . (4.20)
On the other hand, by taking partial derivative of (4.9) with respect to v and considering the second
equality in (4.20), we obtain
- 6Ha22
\bigl(
a12f
\prime + a22g
\prime \bigr) 2 = 2\omega 2g\prime f \prime \prime +
\bigl[
a222 + (\omega f \prime )2
\bigr] \bigl(
d1g
\prime +Da12
\bigr)
,
which is a polynomail equation on g\prime . The leading coefficient coming from the term (g\prime )2 is - 6Ha322
which cannot vanish. This gives a contradiction.
Theorem 4.3 is proved.
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298 M. E. AYDIN
5. Surfaces of Type II. Let \alpha and \beta denote the isotropic planar and space curves given by,
respectively,
\alpha (x) = (x, ax, f(x)) , \beta (y) = (y, g(y), h(y)) ,
where a \in \BbbR . Because the torsion of \beta is non-vanishing, we deduce from (2.2) that
g\prime \prime h\prime \prime \prime - g\prime \prime \prime h\prime \prime \not = 0, (5.1)
where g\prime =
dg
dy
, h\prime =
dh
dy
and so on. Then the obtained translation surface belongs to Type II and
has the form
r(x, y) = (x+ y, ax+ g(y), f(x) + h(y)) . (5.2)
The assumption (5.1) ensures the admissibility of (5.2), i.e., g\prime - a \not = 0. Hence, by a calculation, the
Gaussian curvature K turns to
K =
f \prime \prime [h\prime \prime (g\prime - a) - g\prime \prime (h\prime - f \prime )]
(g\prime - a)3
, (5.3)
where f \prime =
df
dx
, etc.
Theorem 5.1. A translation surface in \BbbI 3 of the form (5.2) with CIGC (K0) is a generalized
cylinder with non-isotropic rulings, i.e., K0 = 0.
Proof. If K0 \not = 0, then (5.3) can be rewritten as
K0
f \prime \prime =
h\prime \prime
(g\prime - a)2
- g\prime \prime
(g\prime - a)3
(h\prime - f \prime ). (5.4)
Taking partial derivative of (5.4) with respect to x yields
- K0
f \prime \prime \prime
(f \prime \prime )3
=
g\prime \prime
(g\prime - a)3
and solving this one
f(x) =
1
3c21
( - 2c1x+ d1)
3
2 + d2x+ d3 (5.5)
and
g(y) =
1
K0c1
(2K0c1y + d4)
1
2 + ay + d5. (5.6)
Substituting (5.5) and (5.6) into (5.4) gives
0 =
h\prime \prime
h\prime - d2
+
K0c1
2K0c1y + d4
. (5.7)
By solving (5.7), we find
h(y) =
c2
K0c1
(2K0c1y + d4)
1
2 + d2y + d6. (5.8)
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 299
Comparing (5.6) with (5.8) gives a contradiction due to (5.1). Next assume that K0 = 0. If f \prime \prime \not = 0,
we have h\prime \prime (g\prime - a) = g\prime \prime (h\prime - f \prime ). Taking partial derivative of this one with respect to x yields
the contradiction g\prime \prime = 0 due to (5.1). Henceforth only possibility is that f \prime \prime = 0, namely, \alpha is a
non-isotropic line.
Theorem 5.1 is proved.
By a direct calculation, the mean curvature H is
2H =
\bigl[
1 + (g\prime )2
\bigr]
(g\prime - a)f \prime \prime +
\bigl(
1 + a2
\bigr)
[h\prime \prime (g\prime - a) - g\prime \prime (h\prime - f \prime )]
(g\prime - a)3
. (5.9)
Theorem 5.2. A translation surface in \BbbI 3 of the form (5.2) cannot be isotropic minimal.
Proof. We prove by contradiction. If H = 0, then (5.9) reduces to\bigl[
1 + (g\prime )2
\bigr]
(g\prime - a)f \prime \prime +
\bigl(
1 + a2
\bigr) \bigl[
h\prime \prime (g\prime - a) - g\prime \prime (h\prime - f \prime )
\bigr]
= 0. (5.10)
The partial derivative of (5.10) with respect to x yields\bigl[
1 + (g\prime )2
\bigr]
(g\prime - a)f \prime \prime \prime +
\bigl(
1 + a2
\bigr)
g\prime \prime f \prime \prime = 0. (5.11)
We have to distinguish two cases:
Case 1: f \prime \prime = 0, i.e., f(x) = d1x+ d2. By (5.10) we deduce
h\prime \prime
h\prime - d1
=
g\prime \prime
g\prime - a
,
which implies
h = c1g + (d1 - ac1) y - d3.
This is not possible due to (5.1).
Case 2: f \prime \prime \not = 0. (5.11) implies
f \prime \prime \prime
f \prime \prime = -
\bigl(
1 + a2
\bigr)
g\prime \prime
[1 + (g\prime )2] (g\prime - a)
. (5.12)
Hence, it follows from (5.12) that
f \prime \prime = c1f
\prime + d1,
\bigl[
1 + (g\prime )2
\bigr]
(g\prime - a)c1 = -
\bigl(
1 + a2
\bigr)
g\prime \prime . (5.13)
By considering (5.13) into (5.10), we conclude
0 =
g\prime \prime
g\prime - a
- h\prime \prime
h\prime +
d1
c1
. (5.14)
Solving (5.14) gives
g = c2h+
\biggl(
a+
c2d1
c1
\biggr)
y + d2,
which is not possible due to (5.1).
Theorem 5.2 is proved.
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300 M. E. AYDIN
Theorem 5.3. For a translation surface in \BbbI 3 of the form (5.2) with nonzero CIMC (H0) one of
the following occurs:
(1) \alpha (x) = (x, ax, d1x+ d2) and
\beta (y) =
\biggl(
y, g,
H0
1 + a2
(g - ay)2 + d1y + d3(g - ay) + d4
\biggr)
,
(2) \alpha (x) = (x, ax, c1 \mathrm{e}\mathrm{x}\mathrm{p}(c2x) + d1x+ d2) and
\beta (y) =
\biggl(
y, g,
H0
(1 + a2)
(g - ay)2 - d1
c2
y + d3(g - ay) + d4
\biggr)
,
where g = g(y) is a non-linear function and
g - ay \not = - 1
c3
\sqrt{}
- 2c3y + d5 + d6. (5.15)
Proof. We seperate the proof into two cases:
Case 1: f \prime \prime = 0, f(x) = d1x+ d2. By substituting it into (5.9), we derive
2H0
1 + a2
(g\prime - a) =
\biggl(
h\prime - d1
g\prime - a
\biggr) \prime
. (5.16)
Twice integration in (5.16) leads to
h =
H0
1 + a2
(g - ay)2 + d1y + d3(g - ay) + d4. (5.17)
On the other hand, by (5.1) and (5.17) we deduce (5.15). This completes the proof of the item (1) of
the theorem.
Case 2: f \prime \prime \not = 0. By taking partial derivative of (5.9) with respect to x, we obtain (5.11) again.
It means that next steps are similar to those of Theorem 5.2. Thus, we have (5.13), namely,
f(x) = c1 \mathrm{e}\mathrm{x}\mathrm{p}(c2x) + d1x+ d2 (5.18)
and \bigl[
1 + (g\prime )2
\bigr]
(g\prime - a)c2 = -
\bigl(
1 + a2
\bigr)
g\prime \prime . (5.19)
Substituting (5.18) and (5.19) into (5.9), we conclude
2H0
(1 + a2)
(g\prime - a) =
- d1g
\prime \prime
c2(g\prime - a)2
+
\biggl(
h\prime
g\prime - a
\biggr) \prime
. (5.20)
Twice integration in (5.20) gives
h =
H0
(1 + a2)
(g - ay)2 - d1
c2
y + d3(g - ay) + d4. (5.21)
By (5.1) and (5.21), we deduce (5.15) again.
Theorem 5.3 is proved.
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 301
6. Surfaces of Type III. Let \alpha and \beta denote the non-isotropic planar and space curves given
by, respectively,
\alpha (x) = (x, f(x), ax) , \beta (y) = (y, g(y), h(y)) ,
where a \in \BbbR . Then the torsion of \beta is non-vanishing, namely, (2.2) implies
g\prime \prime h\prime \prime \prime - g\prime \prime \prime h\prime \prime \not = 0, (6.1)
where g\prime =
dg
dy
, h\prime =
dh
dy
and so on. Hence, the surface obtained by a sum of \alpha and \beta belongs to
Type III and has the form
r(x, y) = (x+ y, f(x) + g(y), ax+ h(y)) . (6.2)
It implies from (6.1) that the surface is admissible, i.e., g\prime - f \prime \not = 0, f \prime =
df
dx
. By a calculation, the
isotropic Gaussian curvature K is
K = - f \prime \prime (h\prime - a) [h\prime \prime (g\prime - f \prime ) - g\prime \prime (h\prime - a)]
(g\prime - f \prime )4
. (6.3)
Theorem 6.1. A translation surface in \BbbI 3 of the form (6.2) with CIGC (K0) is a generalized
cylinder with non-isotropic rulings, namely, K0 = 0.
Proof. Assume that K is a nonzero constant K0 . Then we have f \prime \prime \not = 0 and the partial
derivative of (6.3) with respect to x yields
4(g\prime - f \prime )3
f \prime \prime +
(g\prime - f \prime )4f \prime \prime \prime
(f \prime \prime )3
=
(h\prime - a)h\prime \prime
K0
. (6.4)
We have two cases:
Case 1: f \prime \prime \prime = 0, f \prime \prime = c1. Then from (6.4), we get
4K0(g
\prime - f \prime )3 = c1(h
\prime - a)h\prime \prime . (6.5)
The partial derivative of (6.5) with respect to x gives f \prime \prime = 0, which is not possible.
Case 2: f \prime \prime \prime \not = 0. Taking partial derivative of (6.4) with respect to x and after dividing with
(g\prime - f \prime )2 gives
- 12 - 8(g\prime - f \prime )
f \prime \prime \prime
(f \prime \prime )2
+ (g\prime - f \prime )2
\biggl(
f \prime \prime \prime
(f \prime \prime )3
\biggr) \prime
= 0. (6.6)
This is a polynomial equation on g\prime and the leading coefficient
\biggl(
f \prime \prime \prime
(f \prime \prime )3
\biggr) \prime
coming from (g\prime )2 has to
vanish. Thereby, (6.6) reduces to
- 12 - 8(g\prime - f \prime )
f \prime \prime \prime
(f \prime \prime )2
= 0. (6.7)
Taking partial derivative of (6.7) with respect to y implies f \prime \prime \prime = 0, which is not our case.
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302 M. E. AYDIN
The discussion above yields K0 = 0. In that case, because \beta is a space curve, only possibility in
(6.3) is f \prime \prime = 0, namely, \alpha is a non-isotropic line.
Theorem 6.1 is proved.
By a direct calculation, the isotropic mean curvature is
2H =
\bigl[
1 + (f \prime )2
\bigr]
[h\prime \prime (g\prime - f \prime ) - g\prime \prime (h\prime - a)] -
\bigl[
1 + (g\prime )2
\bigr]
(h\prime - a)f \prime \prime
(g\prime - f \prime )3
. (6.8)
Theorem 6.2. A translation surface in \BbbI 3 of the form (6.2) cannot be isotropic minimal.
Proof. We prove by contradiction. If the surface is isotropic minimal, then (6.8) reduces to\bigl[
1 + (f \prime )2
\bigr] \bigl[
h\prime \prime (g\prime - f \prime ) - g\prime \prime (h\prime - a)
\bigr]
-
\bigl[
1 + (g\prime )2
\bigr]
(h\prime - a)f \prime \prime = 0. (6.9)
We have two cases:
Case 1: f \prime \prime = 0, f = d1x+ d2. Then (6.9) reduces to
h\prime \prime
h\prime - a
=
g\prime \prime
g\prime - d1
.
Solving last equation leads to a contradiction due to (6.1).
Case 2: f \prime \prime \not = 0. By dividing (6.9) with
\bigl[
1 + (g\prime )2
\bigr] \bigl[
1 + (f \prime )2
\bigr]
(h\prime - a), we derive
h\prime \prime (g\prime - f \prime )
(h\prime - a) [1 + (g\prime )2]
+
f \prime \prime
1 + (f \prime )2
- g\prime \prime
1 + (g\prime )2
= 0. (6.10)
The partial derivatives of (6.10) with respect to x and y yield
h\prime \prime
(h\prime - a) [1 + (g\prime )2]
= c1. (6.11)
Substituting (6.11) into (6.10) gives
f \prime \prime
1 + (f \prime )2
- c1f
\prime = d1 and
g\prime \prime
1 + (g\prime )2
- c1g
\prime = d1. (6.12)
By considering the second equality in (6.12) into (6.11), we obtain
h\prime \prime
h\prime - a
=
c1g
\prime \prime
c1g\prime + d1
. (6.13)
Solving (6.13) implies a contradiction due to (6.1).
Theorem 6.2 is proved.
Theorem 6.3. For a translation surface in \BbbI 3 of the form (6.2) with nonzero CIMC (H0), we
have \alpha (x) = (x, ax, d1x+ d2) and
\beta (y) =
\biggl(
y, g(y),
H0
1 + d21
(g - d1y)
2 + ay + d3(g - d1y) + d4
\biggr)
,
where g = g(y) is a non-linear function and
g - d1y \not = - 1
c1
\sqrt{}
- 2c1y + d5 + d6. (6.14)
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 303
Proof. We divide the proof into two cases:
Case 1: f \prime \prime = 0. Then f(x) = d1x+ d2 and (6.8) reduces to
2H
1 + d21
(g\prime - d1) =
\biggl(
h\prime - a
g\prime - d1
\biggr) \prime
. (6.15)
After twice integration of (6.15), we obtain
h =
H0
1 + d21
(g - d1y)
2 + ay + d3 (g - d1y) + d4. (6.16)
By (6.1) and (6.16), we conclude (6.14) and, thus, the hypothesis of the theorem is obtained.
Case 2: f \prime \prime \not = 0. By producting (6.8) with
(g\prime - f \prime )3
1 + (f \prime )2
and taking partial derivatives with respect
to x and y, we have
12H0
\biggl[
(g\prime - f \prime )f \prime \prime g\prime \prime
1 + (f \prime )2
+
(g\prime - f \prime )2f \prime f \prime \prime g\prime \prime
[1 + (f \prime )2]2
\biggr]
=
= h\prime \prime \prime f \prime \prime +
\bigl\{
2g\prime g\prime \prime (h\prime - a) +
\bigl[
1 + (g\prime )2
\bigr]
h\prime \prime
\bigr\} \biggl( f \prime \prime
1 + (f \prime )2
\biggr) \prime
. (6.17)
By dividing (6.17) with 12H0f
\prime \prime g\prime \prime and putting
A(y) = 2g\prime (h\prime - a) +
1 + (g\prime )2
g\prime \prime
h\prime \prime , B(x) =
\bigl\{
f \prime \prime /
\bigl[
1 + (f \prime )2
\bigr] \bigr\} \prime
f \prime \prime ,
we conclude
g\prime - f \prime
1 + (f \prime )2
+
(g\prime - f \prime )2f \prime
[1 + (f \prime )2]2
=
1
12H0
\biggl(
h\prime \prime \prime
g\prime \prime
+AB
\biggr)
. (6.18)
Taking partial derivative of (6.18) with respect to y gives
1
1 + (f \prime )2
+
2(g\prime - f \prime )f \prime
[1 + (f \prime )2]2
=
1
12H0
\biggl[
(h\prime \prime \prime /g\prime \prime )\prime
g\prime \prime
+
A\prime
g\prime \prime
B
\biggr]
. (6.19)
Again taking partial derivative of (6.19) with respect to y and putting
C(y) =
\biggl[
(h\prime \prime \prime /g\prime \prime )\prime
g\prime \prime
\biggr] \prime
, (6.20)
we deduce
f \prime
[1 + (f \prime )2]2
=
1
24H0
\biggl[
C
g\prime \prime
+
(A\prime /g\prime \prime )\prime
g\prime \prime
B
\biggr]
. (6.21)
The partial derivatives of (6.21) with respect to x and y yield
0 =
\biggl[
(A\prime /g\prime \prime )\prime
g\prime \prime
\biggr] \prime
B\prime . (6.22)
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304 M. E. AYDIN
From (6.22), we have two possibilities:
(1) B = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}. Taking partial derivative of (6.19) with respect to x and producting with\bigl[
1 + (f \prime )2
\bigr] 3
f \prime \prime implies the following polynomial equation on f \prime :
(f \prime )3 - 3g\prime (f \prime )2 - 3f \prime + g\prime = 0
which yields a contradiction.
(2) B \not = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}. Then, from (6.22), we derive
A = d1(g
\prime )2 + d2g
\prime + d3 (6.23)
and, from (6.21), C = d4g
\prime \prime . Hence, by (6.20), we deduce
h\prime \prime \prime
g\prime \prime
= d4(g
\prime )2 + d5g
\prime + d6. (6.24)
Substituting (6.23) and (6.24) into (6.18) yields the polynomial equation on g\prime :\Biggl\{
f \prime
[1 + (f \prime )2]2
- d1B + d4
12H0
\Biggr\}
(g\prime )2 +
\Biggl\{
1 - (f \prime )2
[1 + (f \prime )2]2
- d2B + d5
12H0
\Biggr\}
g\prime +
+
f \prime
[1 + (f \prime )2]2
- d3B + d6
12H0
= 0,
in which the coefficients must be zero, namely,
f \prime
[1 + (f \prime )2]2
=
d4 + d1B
12H0
,
1 - (f \prime )2
[1 + (f \prime )2]2
=
d5 + d2B
12H0
, (6.25)
- f \prime
[1 + (f \prime )2]2
=
d6 + d3B
12H0
.
Because f \prime \prime \not = 0 none of d1, d2, d3 can vanish. By using the first and the second equations in (6.25),
we obtain the polynomial equation on f \prime :
d1 - d1(f
\prime )2 - d2f
\prime =
d1d5 - d2d4
12H0
\bigl[
1 + (f \prime )2
\bigr] 2
,
which yields a contradiction.
Theorem 6.3 is proved.
7. Several remarks. 1. By considering the obtained results above it can be stated that there do
not exist:
surfaces of Types I.3, II and III with non-zero CIGC;
isotropic minimal surfaces of Types II and III.
2. Isotropic minimal translation surfaces belong to the family of isotropic Scherk surfaces. When
the translating curves lie in orthogonal planes, the members of this family are locally given by [36]
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CLASSIFICATIONS OF TRANSLATION SURFACES IN ISOTROPIC GEOMETRY . . . 305
r(x, y) =
\bigl(
x, y, c
\bigl[
x2 - y2
\bigr] \bigr)
,
r(x, z) =
\biggl(
x,
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| cz
\mathrm{c}\mathrm{o}\mathrm{s} (cx)
\bigm| \bigm| \bigm| \bigm| , z\biggr) ,
r(y, z) =
1
2
\biggl(
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \mathrm{c}\mathrm{o}\mathrm{s}(cz)\mathrm{c}\mathrm{o}\mathrm{s}(cy)
\bigm| \bigm| \bigm| \bigm| , y - z + \pi , y + z
\biggr)
, c \in \BbbR - 0.
When the translating curves are in arbitrary planes, the isotropic Scherk surfaces can be described in
the following explicit forms:
z(x, y) = c
\biggl[
(a11x+ a12y)
2 - a211 + a212
a221 + a222
(a21x+ a22z)
2
\biggr]
(see [2]),
y(x, z) =
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
cx
a11
\biggr)
c(a21x+ a22z)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| ,
y(x, z) =
1
c
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
ca22
| aij |
[a11x+ a12z]
\biggr)
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
ca12
| aij |
[a21x+ a22z]
\biggr)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| , c \in \BbbR - 0.
3. To classify surfaces of Type IV with arbitrary CIGC and CIMC is somewhat complicated, but
still it could be a challenging open problem.
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| id | umjimathkievua-article-505 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:56Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-5052020-07-02T03:56:29Z Classifications of translation surfaces in isotropic geometry with constant curvature Classifications of translation surfaces in isotropic geometry with constant curvature Classifications of translation surfaces in isotropic geometry with constant curvature Aydin , M. E. Aydin, M. E. UDC 515.12 We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvatures under the condition that at least one of translating curves lies in a plane. УДК 515.12 Запропоновано класифiкацiю трансляцiйних поверхонь в iзотропнiй геометрiї з довiльною сталою iзотропною гауссовою та середньою кривиною за умови, що принаймнi одна з трансляцiйних кривих лежить у площинi. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/505 10.37863/umzh.v72i3.505 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 291-306 Український математичний журнал; Том 72 № 3 (2020); 291-306 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/505/8657 |
| spellingShingle | Aydin , M. E. Aydin, M. E. Classifications of translation surfaces in isotropic geometry with constant curvature |
| title | Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_alt | Classifications of translation surfaces in isotropic geometry with constant curvature Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_full | Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_fullStr | Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_full_unstemmed | Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_short | Classifications of translation surfaces in isotropic geometry with constant curvature |
| title_sort | classifications of translation surfaces in isotropic geometry with constant curvature |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/505 |
| work_keys_str_mv | AT aydinme classificationsoftranslationsurfacesinisotropicgeometrywithconstantcurvature AT aydinme classificationsoftranslationsurfacesinisotropicgeometrywithconstantcurvature |