abstract: In this paper, ruled surfaces of type II in a three-dimensional PseudoGalilean space are given. By studying its Gauss map and Laplacian operator, we obtain a classification of ruled surfaces of type II in a three-dimensional PseudoGalilean space

Alper Osman

Mahmut Ergüt

Öğrenmiş

CiteSeerX

Bol. Soc. Paran. Mat. (3s.) v. 31 1 (2013): 145–152.
c©SPM –ISSN-2175-1188 on line ISSN-00378712 in press
SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v31i1.15406
On the Gauss Map of Ruled Surfaces of Type II in 3-Dimensional
Pseudo-Galilean Space
Alper Osman Öğrenmiş and Mahmut Ergüt
abstract: In this paper, ruled surfaces of type II in a three-dimensional Pseudo-
Galilean space are given. By studying its Gauss map and Laplacian operator, we
obtain a classification of ruled surfaces of type II in a three-dimensional Pseudo-
Galilean space.
Key Words: Pseudo-Galilean Space, Ruled Surfaces, Gauss Map.
Contents
1 Introduction 145
2 Ruled Surfaces 146
3 Cylindrical Ruled Surfaces 148
1. Introduction
The classification of submanifold in Euclidean or Non-Euclidean spaces has been
of particular interest for Geometers. In 1983, the authors classified spacelike ruled
surfaces in a three-dimensional Minkowski space R31 [13], and De Woestijne [11]
extended it to the Lorentz version in 1998. In late 1970s Chen [4,5] introduced the
notion of Euclidean immersions of finite type. In a framework of theory of finite
type, Chen and Piccinni [6] characterized the submanifold satisfying the condition
∆G = λG (λ ∈ R), where ∆ in the Laplacian of the induced metric and G the Gauss
map for submanifold. Submanifold of Euclidean and pseudo-Euclidean spaces with
finite type Gauss map also studied by following geometers. (cf. [7,9,12,15,16], etc.)
On the otherhand, for the Gauss map of a surface in a three-dimensional Euclidean
space R3 the following theorem is proved by Boikoussis and Blair [8].
Theorem 1.1. The only ruled surfaces in R3 whose Gauss map ξ satisfies
∆ξ = Aξ, A ∈ Mat(3, R) (1.1)
are locally the plane and the circular cylinder.
Also, for the Lorentz version Choi [15] investigated ruled surfaces with non-null
base curve satisfying the condition (1.1) in a three-dimensional Minkowski space
R
3
1.
2000 Mathematics Subject Classification: 53B30,53A35
145
Typeset by BSP
M
style.
c© Soc. Paran. de Mat.
146 Alper Osman Öğrenmiş and Mahmut Ergüt
It seems to be interesting to investigate the Pseudo-Galilean version of the
above theorem. Now, let G13 be a three-dimensional Pseudo-Galilean space with
standard coordinate system {XA}. Let S
1
m (resp. Hm) be an m-dimensional de
Sitter space (resp. a hyperbolic space) in a m + 1−dimensional Pseudo-Galilean
space G1m+1. We denote by Xm(ε) a de Sitter space S
1
m(1) or a hyperbolic space
Hm(−1), according as ε = 1 or ε = −1. Let X be a surface in G
1
3 and ξ be a
unit vector field normal to X. Then, for any point ρ in X, we can regard ξ(ρ) as a
point in H2(−1) or S
1
2(1) by translating parallelly to the origin in ambient space
G13, according as ε = ∓1. The map ξ of X into X2(ε) is called a Gauss map of X
in G13. Then we prove the following
Theorem 1.2. The only ruled surfaces of type II in G13 whose Gauss map ξ :
X → X2(ε) satisfies (1.1) are locally the following spaces:
1. G12 and S
1
1 × R if ε = −1
2. G12 and H1 × R if ε = −1.
The theorem is proved in section 3.
2. Ruled Surfaces
First of all, we recall fundamental properties in three-dimensional Pseudo-
Galilean space.
Differential geometry of the Galilean space G3 has been largely developed in
O. Röschel’s paper [14].
The Pseudo-Galilean geometry is one of the real Cayley-Klein geometries
(of projective signature (0, 0,+,−), explained in [10]. The absolute of the Pseudo-
Galilean geometry is an ordered triple {w, f, I} where w is the ideal (absolute)
plane, f is line in w and I is the fixed hyperbolic involution of points of f.
The group of motions of G13 is a six-parameter group given (in affine coordinates)
by
x = a + x
y = b + cx + ychϕ + zshϕ
z = d + ex + yshϕ + zchϕ.
As in [1], Pseudo-Galilean scalar product g can be written as
g(v1, v2) =
{
x1x2 if x1 6= 0 ∨ x2 6= 0
y1y2 − z1z2 if x1 = 0 ∧ x2 = 0
(2.1)
where v1 = (x1, y1, z1), v2 = (x2, y2, z2).
It leaves invariant the Pseudo-Galilean norm of the vector v = (x, y, z) defined
by
‖v‖ =
{
x, x 6= 0
√
|y2 − z2|, x = 0 [2].
(2.2)
On the Gauss Map of Ruled Surfaces of Type II 147
A vector v = (x, y, z) is said to be non-isotropic if x 6= 0. All unit non-isotropic
vectors are the form (1, y, z). For isotropic vectors x = 0 holds. There are four
types of isotropic vectors: spacelike (y2 − z2 > 0), timelike (y2 − z2 < 0) and two
types of lightlike (y = ±z) vectors. A non-lightlike isotropic vector is a unit vector
if y2 − z2 = ±1 [3].
A trihedron (T0; e1, e2, e3), with a proper origin
T0(x0, y0, z0) ∼ (1 : x0 : y0 : z0),
is orthonormal in Pseudo-Galilean sense if the vectors e1, e2, e3 are of the following
form:
e1 = (1, y1, z1), e2 = (0, y2, z2), e3 = (0, εz2, εy2),
with y22 − z
2
2 = δ, where ε, δ is +1 or −1 [3].
Such trihedron (T0; e1, e2, e3) is called positively oriented if for its vectors
det(e1, e2, e3) = 1 holds, i.e. if y
2
2 − z
2
2 = ε [3].
Let X be a ruled surface, r ≥ 1, in G13 given by its parametrization
X(u, v) = r(u) + va(u), v ∈ R (2.3)
where r is the directrix (parametrized by the Pseudo-Galilean arc length) and a is
a unit generator vector field. according to the position of the striction curve with
respect to the absolute figure, we have three types of skew ruled surfaces in G13.
Throughout this paper, we assume that all objects are smooth unless otherwise
mentioned. Now, we define a ruled surface of type II in G13.
A ruled surface of type II in G13 is a surface parametrized by
X(u, v) = r(u) + va(u)
y, z, a2, a3 ∈ C
2, u ∈ I ⊆ R, v ∈ R (2.4)
∣
∣
∣
y
′
2 − z
′
2
∣
∣
∣
= 1, y
′
a
′
2 − z
′
a
′
3 = 0.
It is a surface whose striction curve r(u) = (0, y(u), z(u)) lies in pseudo-Euclidean
plane, u is the arc-lenght on the striction curve, and whose generators
a(u) = (1, a2(u), a3(u)) are non-isotropic. In particular, if a(u) is constant, then
it said to be cylindrical, and if it is not so, then the surface is said to be non-
cylindrical. Since our discussion is local, we may assume that we always have
a
′
(u) 6= 0 in the non-cylindrical case. That is, the direction of the rulings is always
changing. [3]
The natural basis {Xu,Xv} along the coordinate curves are given by
Xu = dx(
∂
∂u
) = r
′
+ va
′
Xv = dx(
∂
∂v
) = a.
148 Alper Osman Öğrenmiş and Mahmut Ergüt
Accordingly we see
g(Xu,Xu) = g(r
′
, r
′
) + 2vg(r
′
,a
′
) + v2g(a
′
,a
′
)
g(Xu,Xv) = 0
g(Xv,Xv) = g(a,a).
For the components gij of the Pseudo-Galilean metric g we denote (g
ij) (resp. g)
the inverse matrix (resp. the determinant) of matrix (gij). Then the Laplacian ∆
is given by
∆ = −
1
√
|g|
∑ ∂
∂xi
(
√
|g|gij
∂
∂xj
). (2.5)
3. Cylindrical Ruled Surfaces
In this section we are concerned with cylindrical ruled surfaces. Let X be a
cylindrical ruled surface swept out by the vector field a along the base curve r
in G13. That is, r = r(u) is isotropic curve and a = a(u) is a non-isotropic unit
constant vector along orthogonal to r. Then the cylindrical ruled surface X is only
or type II. And X is parametrized by
X(u, v) = r(u) + va(u), u ∈ I ⊆ R, v ∈ R.
Then we get g(ξ, ξ) = ε (= ∓1). Let X2(ε) be a two-dimensional space form as
follows:
X2(ε) =
{
S2(1) in G
1
3 if ε = −1
H2(−1) in G
1
3 if ε = 1.
Then, for any point x in X, ξ(x) can be regarded as a point in X2(ε) and the
map ξ : X → X2(ε) is the Gauss map of X into X2(ε).
We give here theorem of ruled surface of type II whose Gauss map satisfies
∆ξ = Aξ, A ∈ Mat(3, R). (3.1)
Theorem 3.1. The only cylindrical ruled surface of type II in G13 whose Gauss
map satisfies the condition (3.1) are locally the plane and the hyperbolic cylinder
(resp. the Pseudo-Galilean plane and Pseudo-Galilean circular cylinder).
Proof: Let X be a cylindrical ruled surface of type II in G13 parametrized by
X(u, v) = r(u) + va(u)
y, z, a2, a3 ∈ C
2, u ∈ I ⊆ R, v ∈ R (3.2)
∣
∣
∣
y
′
2 − z
′
2
∣
∣
∣
= 1, y
′
a
′
2 − z
′
a
′
3 = 0
On the Gauss Map of Ruled Surfaces of Type II 149
where a is a unit constant vector along the curve r orthogonal to it. That is, it
satisfies g(r
′
,a) = 0, g(a,a) = 1. Acting a Pseudo-Galilean transformation, we
may assume that a = (1, 0, 0) without loss of generality. Then r may be regarded
as the plane curve r(u) = (0, y(u), z(u)) parametrized by arc length:
g(r
′
, r
′
) = y
′
2 − z
′
2 = ε (ε = ∓1). (3.3)
The Gauss map ξ is given by ξ = (0, εz
′
(u), εy
′
(u)). It is unit normal to X.
Since the induced Pseudo-Galilean metric g is given by g11 = ε, g12 = 0 and
g22 = 1, the Laplacian of ξ is given by ∆ξ = (0, εz
′′′
, εy
′′′
) from (2.5). Thus, from
the condition (3.1) we have the following system of differential equation:
0 = a12z
′
+ a13y
′
εz
′′′
= a22z
′
+ a23y
′
(3.4)
εy
′′′
= a32z
′
+ a33y
′
.
where A = (aij) is the constant matrix.
Now, in order to prove this theorem we may solve this equation and obtain the
solution y and z. First we consider that ε = −1. So we get g(r
′
, r
′
) = y
′
2−z
′
2 = −1.
Accordingly we can parametrize as follows:
y
′
= sinh θ, z
′
= cosh θ (3.5)
where θ = θ(u). Differentiating (3.5), we obtain
y
′′
= θ
′
cosh θ, y
′′′
= θ
′′
cosh θ + (θ
′
)2 sinh θ
z
′′
= θ
′
sinh θ, z
′′′
= θ
′′
sinh θ + (θ
′
)2 cosh θ (3.6)
By (3.4), (3.5) and (3.6) we have
−θ
′′
sinh θ − (θ
′
)2 cosh θ = a22 cosh θ + a23 sinh θ
−θ
′′
cosh θ − (θ
′
)2 sinh θ = a32 cosh θ + a33 sinh θ,
which give
θ
′′
= (a22 − a33) cosh θ sinh θ + a23 sinh
2 θ − a32 cosh
2 θ (3.7)
(θ
′
)2 = (a32 − a23) cosh θ sinh θ + a33 sinh
2 θ − a22 cosh
2 θ. (3.8)
Differentiating (3.8), we get
2θ
′
θ
′′
= θ
′ {
(a22 − a33)(cosh
2 θ + sinh2 θ) + 2(a33 − a22) sinh θ cosh θ
}
.
Substituting (3.7) into this equation, we get
150 Alper Osman Öğrenmiş and Mahmut Ergüt
θ
′ {
4(a22 − a33) cosh θ sinh θ + (a23 − 3a32) cosh
2 θ + (3a23 − a32) sinh
2 θ
}
= 0.
(3.9)
We suppose that θ
′
6= 0. By (3.4) and (3.9) we get
a22 = a33 and a12 = a13 = a23 = a32 = 0, (3.10)
because sinh θ cosh θ, sinh2 θ and cosh2 θ are linearly independent functions of
θ = θ(u). Combining the above equations with (3.8) gives
θ = ∓
1
r
u + c0, r〉0, c0 ∈ R
where − 1
r2
= a22 = a33. Accordingly we have
y = ∓r cosh θ + c1, c1 ∈ R
z = ∓r sinh θ + c2, c2 ∈ R.
This representation gives us to
(y − c1)
2 − (z − c2)
2 = r2, r〉0.
We denote by S11 (r,(c1, c2)) the pseudo-circle centered at (c1, c2) with radius r in
the Pseudo-Galilean plane G12 (the yz−plane). By the above equation the curve
r is contained in S11 (r,(c1, c2)) and hence the ruled surface X is contained in the
Pseudo-Galilean circular cylinder S11 × R.
Now, we consider that ε = 1. So we get g(r
′
, r
′
) = y
′
2 − z
′
2 = 1. Accordingly
we can parametrize as follows:
y
′
= cosh θ, z
′
= sinh θ
where θ = θ(u). By similar discussion to that of the above ε = −1 we can get
θ
′ {
4(a22 − a33) cosh θ sinh θ + (3a23 − a32) cosh
2
θ + (a23 − 3a32) sinh
2
θ
}
= 0. (3.11)
We suppose that θ
′
6= 0. By (3.4) and (3.11) we get
a22 = a33 and a12 = a13 = a23 = a32 = 0,
which yields that
θ = ∓
1
r
u + c0, r〉0, c0 ∈ R
where 1
r2
= a22 = a33. Accordingly we have
y = ∓r sinh θ + c1, c1 ∈ R
z = ∓r cosh θ + c2, c2 ∈ R.
On the Gauss Map of Ruled Surfaces of Type II 151
This representation gives us to
(y − c1)
2 − (z − c2)
2 = −r2, r〉0.
We denote by H1 (r,(c1, c2)) the hyperbolic circle centered at (c1, c2) with radius
r in the Pseudo-Galilean plane G12 (the yz−plane). By the above equation the curve
r is contained in H1 (r,(c1, c2)) and hence the ruled surface X is contained in the
hyperbolic cylinder H1 × R.
Example 3.2. A Pseudo-Galilean circular cylinder
S11 × R =
{
(x, y, z) ∈ G13 : y
2 − z2 = r2, r〉0
}
is a cylindrical ruled surface of type II with base curve r(u) = (0, r sinh u
r
, r cosh u
r
)
and director curve a(u) = (1, 0, 0). The Gauss map is given by
ξ = (0, cosh
u
r
, sinh
u
r
)
and the Laplacian ∆ξ of the Gauss map ξ can be expressed as
∆ξ = −
1
r2
ξ.
Hence the Pseudo-Galilean circular cylinder satisfies (3.1) with
A =


a11 0 0
a21 −
1
r2
0
a31 0 −
1
r2

 .
Example 3.3. A hyperbolic cylinder
H1 × R =
{
(x, y, z) ∈ G13 : y
2 − z2 = −r2, r〉0
}
is a cylindrical ruled surface of type II with base curve r(u) = (0, r sinh u
r
, r cosh u
r
)
and director curve a(u) = (1, 0, 0). The Gauss map is given by
ξ = (0, sinh
u
r
, cosh
u
r
)
and the Laplacian ∆ξ of the Gauss map ξ can be expressed as
∆ξ =
1
r2
ξ.
Hence the hyperbolic cylinder satisfies (3.1) with
A =


a11 0 0
a21
1
r2
0
a31 0
1
r2

 .
152 Alper Osman Öğrenmiş and Mahmut Ergüt
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Alper Osman Öğrenmiş
Fırat University,
Faculty of Science,
Department of Mathematics,
23119 Elazığ, Turkey
E-mail address: ogrenmisalper@gmail.com
and
Mahmut Ergüt
Fırat University,
Faculty of Science,
Department of Mathematics,
23119 Elazığ, Turkey
E-mail address: mergut@firat.edu.tr


On the Gauss Map of Ruled Surfaces of Type II in 3-Dimensional Pseudo-Galilean Space

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On the Gauss Map of Ruled Surfaces of Type II in 3-Dimensional Pseudo-Galilean Space

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