1 Introduction
In classical differential geometry, the problem of
obtaining Gaussian and mean curvatures of a sur-
face in Euclidean space and other spaces is one
of the most important problems, so we are inter-
ested here to study such a problem for a surface
known as affine factorable surface in the three-
dimensional pseudo-Galilean space G1
3.
The geometry of Galilean Relativity acts like
a “bridge” from Euclidean geometry to special
Relativity. The Galilean space which can be de-
fined in three-dimensional projective space P3(R)
is the space of Galilean Relativity, [1]. The ge-
ometries of Galilean and pseudo-Galilean spaces
have similarities, but, of course, are different. In
the Galilean and pseudo Galilean spaces, some
special surfaces such as surfaces of revolution,
ruled surfaces, translation surfaces and tubular
surfaces have been studied in [2], [3], [4], [5], [6],
[7], [8], [9], [10]. For further study of surfaces
in the pseudo-Galilean space, we refer the reader
to [9]. Recall that the graph surfaces are also
known as Monge surfaces, [11]. In this work, we
are interested here in studying a special type of
Monge surface, namely the factorable surface of
the second kind that is a graph of the function
y(x, z) = f(x)g(z). Such surfaces with non-zero
constant Gaussian and mean curvatures in vari-
ous ambient spaces have been classified (see, [12],
[13], [14], [15], [16]). Our purpose is to analyze the
factorable surfaces in the pseudo-Galilean space
G1
3that is one of real Cayley-Klein spaces (for
more details see, [17], [18], [19]). There exist three
different kinds of factorable surfaces, explicitly, a
Monge surface in G1
3is said to be factorable (so-
called a homothetic) if it is given in one of the fol-
lowing forms: Φ1:z(x, y) = f(x)g(y) is the first
kind, Φ2:y(x, z) = f(x)g(z) the second kind, and
Φ3:x(y, z) = f(y)g(z) the third kind where f,g
are smooth functions, [14]. These surfaces have
different geometric structures in different spaces
such as metric, curvatures, etc. We hope that this
work will be useful for the specialists in this field.
2 Basic concepts
The pseudo-Galilean space G1
3is one of the
Cayley-Klein spaces with absolute figure that con-
sists of the ordered triple {ω, f, I}, where ωis
the absolute plane given by xo= 0,in the three-
dimensional real projective space P3(R), fthe ab-
solute line in ωgiven by xo=x1= 0 and Ithe
fixed hyperbolic involution of points of fand rep-
resented by (0 : 0 : x2:x3)→(0 : 0 : x3:x2),
which is equivalent to the requirement that the
conic x2
2−x2
3= 0 is the absolute conic. The metric
Investigation of Affine Factorable Surfaces in Pseudo-Galilean Space
MOHAMED SAAD1*, HOSSAM ABDEL-AZIZ2, HAYTHAM ALI2
1Mathematics Department, Faculty of Science, Islamic University of Madinah, Al-Madinah,
SAUDI ARABIA
2Mathematics Department, Faculty of Science, Sohag University, EGYPT
Abstract: In this paper, we investigate affine factorable surfaces of the second kind in the three-dimensional
pseudo-Galilean space G1 3. We use the invariant theory and theory of diffeerential equations to study the
geometric properties of these surfaces, namely, the first and second fundamental forms, Gaussian and mean
curvatures. Also, we present some special cases by changing the partial diffeerential equation into the ordinary
diffeerential equation to simplify our special cases. Furthermore, we give some theorems according to zero and
non-zero Gaussian and mean curvatures of the meant surfaces. Finally, we give some examples to confifirm and
demonstrate our results.
KeyWords: Affine factorable surfaces, minimal surfaces, Gaussian and mean curvatures, pseudo-Galilean
space.
Received: April 29, 2023. Revised: August 2, 2023. Accepted: August 23, 2023. Published: September 26, 2023.
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connections in G1
3are introduced with respect to
the absolute figure. In terms of the affine coordi-
nates given by (xo:x1:x2:x3) = (1 : x:y:z),
the distance between the points p= (p1, p2, p3)
and q= (q1, q2, q3) is defined by (see for instance,
[9], [18])
d(p, q) = (|q1−p1|, if p16=q1,
p|(q2−p2)2−(q3−p3)2|, if p1=q1.
The pseudo-Galilean scalar product of the
vectors X= (x1, x2, x3) and Y= (y1, y2, y3) is
given by
hX, Y iG1
3=(x1y1, if x16= 0 or y16= 0,
x2y2−x3y3, if x1= 0 and y1= 0.
In this sense, the pseudo-Galilean norm of a
vector Xis kXk=p|X.X|. A vector X=
(x1, x2, x3) is called isotropic (non-isotropic) if
x1= 0 (x16= 0). All unit non-isotropic vectors
are of the form (1, x2, x3). The isotropic vector
X= (0, x2, x3) is called spacelike, timelike and
lightlike if x2
2−x2
3>0, x2
2−x2
3<0 and x2=±x3,
respectively. The pseudo-Galilean cross product
of Xand Yon G1
3is given as follows
X∧G1
3Y=
0−e2e3
x1x2x3
y1y2y3
,
where e2and e3are canonical basis.
Let Mbe a connected, oriented 2-dimensional
manifold and φ:M→G1
3be a surface in G1
3with
parameters (u, v). The surface parametrization φ
is expressed as
φ(u, v)=(x(u, v), y(u, v), z(u, v)).
On the other hand, we denote by E,F,G
and L,M,Nthe coefficients of the first and sec-
ond fundamental forms of φ, respectively. The
Gaussian curvature Kand mean curvature Hare
expressed as
K=LN −M2
EG −F2, H =EN +GL −2F M
2|EG −F2|,
(1)
where
E=φ0
u.φ0
u, F =φ0
u.φ0
v, G =φ0
v.φ0
v,
L=φ00
uu.n, M =φ00
uv.n, N =φ00
vv.n,
where the normal surface is given by
n=φ0
u∧φ0
v
|φ0
u∧φ0
v|.
3 Factorable surfaces in pseudo-
Galilean space G1
3
In what follows, we consider the factorable surface
of second kind in G1
3which can be locally written
as
φ(x, z)=(x, f(x)g(z), z).(2)
Definition 1 An affine factorable surface in
pseudo-Galilean space G1
3is defined as a parame-
ter surface φ(u, v)and can be written as
φ(u, v)=(x(u, v), y(u, v), z(u, v))
= (u, f(u)g(v+au), v)
= (x, f(x)g(z+ax), z),(3)
for non zero constant a, and functions f(x)and
g(z+ax), [19].
Now, from Eq. (3) by a straightforward cal-
culation, the first fundamental form with its co-
efficients of φis given by
I=Edx2+ 2F dxdy +Gdy2,
E= 1, F = 0, G = (fg0)2−1,
g0=dg(z+ax)
d(z+ax).
Also, the second fundamental form of φis
II =Ldx2+ 2Mdxdy +Ndy2,
L=f00g+ 2af0g0+a2fg00
D,
M=(f0g0+afg00 )
D, N =fg00
D,
where
D(x, z) = p1−(fg0)2.
In addition, the Gaussian and mean curvature of
φcan be obtained
K=f02g02−f00fg00g
(1 −(fg0)2)2,(4)
H=Ω(x, z)
2 (1 −(f g0)2)
3
2
,(5)
such that
Ω(x, z) = (1 −a2)fg00 −f00 g−2af 0g0
+f2f00g02g+ 2af0f2g03+a2f3g02g00.
A surface in G1
3is said to be an isotropic minimal
(resp. flat) if H(resp. K) vanishes identically.
Further, it is said to have constant an isotropic
mean (resp. Gaussian) curvature if H(resp. K)
is a constant function on a whole surface.
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4 Affine factorable surfaces with
zero curvatures
In this section, if the Gaussian and mean curva-
tures of Eq. (3) are vanished, then we get the
following result.
Theorem 2 Let φ:I⊂R→G1
3be an affine
factorable surface of second kind given in the form
φ(x, z) = (x, f(x)g(z+ax), z),
if its Gaussian curvature is zero, then the surface
is one of the following forms:
(1) y(x, z) = fog(z+ax),
(2) y(x, z) = gof(x),
(3) y(x, z) = cec5x+c4z,
(4) y(x, z) = [(1 −k)(c6x+c7)]
1
1−k
Theorem 3 .k−1
k(c8(z+ax) + c9)
k
k−1.
Proof. If the Gaussian curvature of φis zero,
then from Eq. (4), we have
f02g02−f00fg00g= 0.(6)
To solve this equation we have the following cases:
Case 1. if f0= 0,then f00 = 0, f =fo=const.,
then y(x, z) = fog(z+ax).
Case 2. if g0= 0,then g00 = 0, g =go=const.,
then y(x, z) = gof(x).
Case 3. if f06= 0 and g06= 0,and let
(u=x,
v=z+ax,
where ∂(u, v)/∂(x, z)6= 0. Then Eq. (7) can be
written as
f2
ug2
v−ffuuggvv = 0,
or
df
du2dg
dv 2
=fdfu
df
df
dugdgv
dg
dg
du.(7)
From Eq. (8), we find
df
du
dg
dv =fdfu
df gdgv
dg .
Since, df
du
dg
dv 6= 0 and gdgv
dg 6= 0,then
fdfu
df
fu!= gv
gdgv
dg !,(8)
let’s rewrite the last equation as follows:
fdfu
df
fu!= gv
gdgv
dg !=k;k=const. (9)
(a) If k= 1,then from Eq. (10), we have
dfu
fu
=df
f,dgv
gv
=dg
g,(10)
it leads to
f=c1ec2u, g =c3ec4v,
where c1, c2, c3, c4are constants. And then
y(x, z) = f(x)g(z+ax) = c1ec2xc3ec4(z+ax)
=c5ec6x+c4z,
where c5=c1c3and c6=c2+ac4are constants.
(b) When k6= 1,then from Eq. (10), we get
fdfu
df =kfu, kg dgv
dg =gv,
which has the solution
f(x) = [(1 −k)(c7x+c8)]
1
1−k,
g(z+ax) = k−1
k(c9(z+ax) + c10)
k
k−1
.
Therefore, we have
y(x, z) = [(1 −k)(c7x+c8)]
1
1−k
.k−1
k(c9(z+ax) + c10)
k
k−1
,
where c7, c8, c9and c10 are constants.
Theorem 4 For given affine factorable surface
of second kind in a three-dimensional pseudo-
Galilean space in the form
φ(x, z)=(x, f(x)g(z+ax), z).
Let its mean curvature be zero, then this surface
will be one of the following forms:
(1) y(x, z) = fo(b1(z+ax) + b2),or y(x, z) =
foqa2−1
a2f2
o(z+ax) + b3,
(2) y(x, z) = go(b4x+b5),
(3) y(x, z) = b8(b6x+b7),or y(x, z)=(b6x+
b7)(b9(z+ax) + b10),
(4) y(x, z) = (b12x+b13)(b11(z+ax) + b12),or
y(x, z) = 1
b11 (b11(z+ax) + b12).
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Proof. If H= 0, then from Eq. (6), we find
(1 −a2)fg00 −f00 g−2af 0g0
+f2f00g02g+ 2af0f2g03+a2f3g02g00 = 0.(11)
This equation can be solved with the aid of the
following:
(1) If f0=f00 = 0, then f=fo=const., and
(4.6) becomes
(1 −a2)fg00 +a2f3g02g00 = 0,
it can be written in a simple form as
g00 = 0 or g0=sa2−1
a2f2
o
,
which has the solution
g=b1(z+ax)+b2or g=sa2−1
a2f2
o
(z+ax)+b3,
it leads to
y(x, z) = fo(b1(z+ax) + b2),
and then, we get
y(x, z) = fo sa2−1
a2f2
o
(z+ax) + b3!,
where b1, b2, and b3are constants.
(2) When g0=g00 = 0, then g=go=const., and
Eq. (12) becomes
f00g= 0,
it has the solution
f=b4x+b5.
Using what we got from solutions, we can write
y(x, z) = go(b4x+b5),
where b4, b5are constants.
(3) When f00 = 0, this leads to f0=b6
which gives f=b6x+b7.From Eq. (12), we have
(1 −a2)fg00 −2af0g0+ 2af0f2g03+a2f3g02g00 = 0,
which can be written as
(1−a2)f gvv −2afugv+2afuf2g3
v+a2f3g2
vgvv = 0,
therefore, by differentiating this equation three
times with respect to u, we obtain
g2
vgvv = 0,
which gives
gv= 0 →g=b8,
and so
gvv = 0 →g=b9(z+ax) + b10,
in light of this, we get
y(x, z) = b8(b6x+b7),
and then, we have
y(x, z)=(b6x+b7)(b9(z+ax) + b10),
where b6, b7, b8, b9and b10 are constants.
(4) If g00 = 0, it means that g0=b11 →g=
b11(z+ax) + b12 and then from Eq. (12), we
obtain
f00g+ 2af0g0−f2f00g02g−2af0f2g03= 0,
which can be written as
fuug+ 2afugv−f2fuug2
vg−2afuf2g3
v= 0.
Differentiate this equation with respect to v, we
find
b11fuu −b3
11f2fuu = 0,
fuu = 0 →f=b12x+b13,
it leads to
f=1
b11
,
Therefore, we get
y(x, z)=(b12x+b13)(b11(z+ax) + b12),
it follows that
y(x, z) = 1
b11
(b11(z+ax) + b12).
Taking into consideration that b11, b12 and b13 are
constants. Thus, this completes the proof.
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5 Affine factorable surfaces with
non-zero curvatures
In this section, we describe the affine factorable
surfaces of the second kind in G1
3with non-zero
constant Gaussian and mean curvatures.
Theorem 5 Let φ:I⊂R→G1
3be an affine
factorable surface of the second kind in G1
3, and it
has a non-zero constant Gaussian curvature, then
this surface takes the form:
y(x, z)=(go(z+ax) + λ2)
.±1
go
tanh hpKox∓goλ1i, λ1, λ2∈R.
Proof. Let Kobe a non-zero constant Gaussian
curvature. Hence, we get
Ko=f02g02−f00fg00g
(1 −(fg0)2)2,(12)
Since, Kovanishes identically when for gis a
constant function. Then fand gmust be non-
constant functions. So, we can distinguish two
cases for Eq. (13), as follows:
Case 1. f0=fo, fo∈R− {0},then from Eq.
(13), we get a polynomial equation in (g0):
Ko−(2Kof2+f2
o)g02+Kof4g04= 0,
which it yields a contradiction.
Case 2. If g0=go;go∈R− {0}.Then, Eq. (13)
leads to
f0=±pKo−2Kog2
of2+Kog4
of4
go
,
therefore, it has the solution:
f(x) = ±1
go
tanh hgopKox∓goλ1i, λ1∈R.
Case 3. If f00 6= 0; g00 6= 0. Then, Eq. (13) leads
to
Ko=f02g02−f00fg00g
(1 −(fg0)2)2,
So, using u=x, v =z+ax and ∂(u, v)/∂(x, y)6=
0, we can obtain
Ko=f2
ug2
v−fuufgvvg
(1 −(fgv)2)2,(13)
it leads to
f0
f2f00 +3f0f2
f00 g04= 0,(14)
which means that all coefficients must vanish,
therefore the contradiction f0= 0 is obtained.
Thus the proof is completed.
Theorem 6 For given affine factorable surface
of the second kind in G1
3which has a non-zero
constant mean curvature Ho. Then
y(x, z) = fo p9H2
o−a4f2
oλ2
3
3foHo
(z+ax) + λ4!,
=−2Ho
go
x2+cx +cgo.
Proof. From Eq. (6), we have
Ho= (1 −a2)fg00 −f00 g−2af 0g0
+f2f00g02g+ 2af0f2g03+a2f3g02g00 !
2 (1 −(f g0)2)3/2,
Solving this equation leads to the following two
cases:
Case 1. If f=fo,g00 =λ3=const., we
obtain
2Ho1−(fg0)23/2= (1 −a2)fg00 +a2f3g02g00 ,
and using u=x, v =z+ax and
∂(u, v)/∂(x, y)6= 0, we have
2Ho1−(fgv)23/2= (1 −a2)fgvv +a2f3g2
vgvv,
(15)
it leads to
gv=p9H2
o−a4f2
oλ2
3
3foHo
,
it has the solution:
g=±p9H2
o−a4f2
oλ2
3
3foHo
(z+ax) + λ4;λ4∈R,
and then we get
y(x, z) = fo p9H2
o−a4f2
oλ2
3
3foHo
(z+ax) + λ4!.
Case 2. If g=go, we have
2Ho=−f00g,
it leads to
f=−Ho
go
x2+λ5x+λ6,
where λ5, λ6∈R. Hence, the result is clear.
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Proposition 7 Let φ:I⊂R→G1
3be an affine
factorable surface in G1
3. Then, the relation be-
tween its Gaussian and mean curvatures is given
by
H=A(x, z)K, (16)
where A(x, z) = D3(a2fg00 +2af0g0+f00 g)−fg00 D
f00 fg00 g−f02g02;
D=p1−(fg0)2. Further, if D= 0,then φ
is an isotropic minimal affine factorable surface
of the second kind.
6 Examples
In this section, we present some examples of the
affine factorable surfaces of the second kind. So,
let us consider the affine factorable surfaces of the
second kind in G1
3given as follows:
(1) φ:y(x, z)=8e6x+z; (x, z)∈[−1,1] ×[0,2π]
(an isotropic flat; K= 0, see Fig. 1),
(2) φ:y(x, z) = q3
4(2x+z)+9; (x, z)∈[0,15]×
[−1,30] (an isotropic minimal; H= 0, see
Fig. 2),
(3) φ:y(x, z) = (10x+z) tanh[x]; (x, z)∈[−1,1]
(K=constant, see Fig. 3),
(4) φ:y(x, z) = −x2+ 2x+ 1; (x, z)∈[−1,1]
(H=constant, see Fig. 4).
-1.0
-0.5
0.0
0.5
1.0
0
2
4
6
0
5000
10000
15000
20000
Figure 1: The isotropic flat surface of the second
kind.
0
5
10
15
0
10
20
30
20
40
60
Figure 2: The isotropic minimal surface of the
second kind.
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
0
2
4
6
8
Figure 3: The affine factorable surface of the sec-
ond kind with K=constant.
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-2
-1
0
1
2
Figure 4: The affine factorable surface of the sec-
ond kind with H=constant.
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7 Concluding Remarks
In the surface theory, especially factorable
surfaces, there are three kinds of these surfaces
known as first, second and third kinds. In this
paper, the factorable surface of the second kind
which has an affine form in the three-dimensional
pseudo-Galilean space G1
3has been studied. The
classification of these surfaces with zero and
non-zero Gaussian and mean curvatures has
been investigated. Also, an essential relation
between the curvatures of these surface has been
obtained. Finally, some computational examples
to support our findings are given and plotted.
In future works, we plan to study the factorable
surfaces in Lorentz-Minkowski space for different
queries and further improve the results in this
paper, combined with the techniques and results
in [20], [21], [22].
Acknowledgments: We gratefully acknowl-
edge the constructive comments from the editor
and the anonymous referees. Also, the author (M.
Khalifa Saad) would like to express his gratitude
to the Islamic University of Madinah.
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