# On Pseudospherical Smarandache Curves in Minkowski 3-Space

Authors: Esra Betul Koc Ozturk; Ufuk Ozturk; Kazim Ilarslan; Emilija Nešović
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curveα, then α has constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves.


---

## Body

## 1. Introduction

It is known that a Smarandache geometry is a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied, if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. Smarandache curves are the objects of Smarandache geometry. By definition, if the position vector of a curve β is composed by the Frenet frame’s vectors of another curve α, then the curve β is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–7]. The curves lying on a pseudosphere S
1
2 in Minkowski 3-space E
1
3 are characterized in [8].In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski  3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curveα, then α has nonzero constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves in Minkowski 3-space.


## 2. Basic Concepts

The Minkowski 3-spaceℝ
1
3 is the Euclidean 3-space ℝ
3 provided with the standard flat metric given by
(1)
〈
·
,
·
〉
=
-
d
x
1
2
+
d
x
2
2
+
d
x
3
2
,
where (
x
1
,
x
2
,
x
3
) is a rectangular coordinate system of ℝ
1
3. Since g is an indefinite metric, recall that a nonzero vector x
→
∈
ℝ
1
3 can have one of three Lorentzian causal characters: it can be spacelike if 〈
x
→
,
x
→
〉
>
0, timelike if 〈
x
→
,
x
→
〉
<
0, and null (lightlike) if 〈
x
→
,
x
→
〉
=
0. In particular, the norm (length) of a vector x
→
∈
ℝ
1
3 is given by ∥
x
→
∥
=
|
〈
x
→
,
x
→
〉
| and two vectors x
→ and y
→ are said to be orthogonal if 〈
x
→
,
y
→
〉
=
0. Next, recall that an arbitrary curve α
=
α
(
s
) in 𝔼
1
3 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors α
′
(
s
) are, respectively, spacelike, timelike, or null (lightlike) for all s
∈
I [9]. A spacelike or a timelike curve α is parameterized by arclength parameter s if 〈
α
′
(
s
)
,
α
′
(
s
)
〉
=
1 or 〈
α
′
(
s
)
,
α
′
(
s
)
〉
=
-
1, respectively. For any two vectors x
→
=
(
x
1
,
x
2
,
x
3
) and y
→
=
(
y
1
,
y
2
,
y
3
) in the space ℝ
1
3, the pseudovector product of x
→ and y
→ is defined by
(2)
x
→
×
y
→
=
|
-
e
→
1
e
→
2
e
→
3
x
1
x
2
x
3
y
1
y
2
y
3
|
=
(
-
x
2
y
3
+
x
3
y
2
,
x
3
y
1
-
x
1
y
3
,
x
1
y
2
-
x
2
y
1
)
.Lemma 1.
Letx
→, y
→, and z
→ be vectors in ℝ
1
3. Then,(i)
〈
x
→
×
y
→
,
z
→
〉
=
det
⁡
(
x
→
,
y
→
,
z
→
),
(ii)
x
→
×
(
y
→
×
z
→
)
=
-
〈
x
→
,
z
→
〉
y
→
+
〈
x
→
,
y
→
〉
z
→,
(iii)
〈
x
→
×
y
→
,
x
→
×
y
→
〉
=
-
〈
x
→
,
x
→
〉
〈
y
→
,
y
→
〉
+
〈
x
→
,
y
→
〉
2,where x is the pseudovector product in ℝ
1
3.Lemma 2.
In the Minkowski 3-spaceℝ
1
3, the following properties are satisfied [9]:(i)
two timelike vectors are never orthogonal;
(ii)
two null vectors are orthogonal if and only if they are linearly dependent;
(iii)
timelike vector is never orthogonal to a null vector.Thepseudospherewith center at the origin and of radius r
=
1 in the Minkowski 3-space ℝ
1
3 is a quadric defined by
(3)
S
1
2
=
{
x
→
∈
ℝ
1
3
∣
-
x
1
2
+
x
2
2
+
x
3
2
=
1
}
.Letα
:
I
⊂
ℝ
→
S
1
2 be a curve lying fully in pseudosphere S
1
2 in ℝ
1
3. Then its position vector α is a spacelike, which means that the tangent vector T
=
α
′ can be a spacelike, a timelike, or a null. Depending on the causal character of T, we distinguish the following three cases.Case 1 (T is a unit spacelike vector).
Then we have orthonormal Sabban frame{
α
(
s
)
,
T
(
s
)
,
ξ
(
s
)
} along the curve α, where ξ
(
s
)
=
-
α
(
s
)
×
T
(
s
) is the unit timelike vector. The corresponding Frenet formulae of α, according to the Sabban frame, read
(4)
[
α
′
T
′
ξ
′
]
=
[
0
1
0
-
1
0
-
k
g
(
s
)
0
-
k
g
(
s
)
0
]
[
α
T
ξ
]
,
where k
g
(
s
)
=
det
⁡
(
α
(
s
)
,
T
(
s
)
,
T
′
(
s
)
) is the geodesic curvature of α and s is the arclength parameter of α. In particular, the following relations hold:
(5)
α
×
T
=
-
ξ
,
T
×
ξ
=
α
,
ξ
×
α
=
T
.Case 2 (T is a unit timelike vector).
Hence, we have orthonormal Sabban frame{
α
(
s
)
,
T
(
s
)
,
ξ
(
s
)
} along the curve α, where ξ
(
s
)
=
α
(
s
)
×
T
(
s
) is the unit spacelike vector. The corresponding Frenet formulae of α, according to the Sabban frame, read
(6)
[
α
′
T
′
ξ
′
]
=
[
0
1
0
1
0
k
g
(
s
)
0
k
g
(
s
)
0
]
[
α
T
ξ
]
,
where k
g
(
s
)
=
det
⁡
(
α
(
s
)
,
T
(
s
)
,
T
′
(
s
)
) is the geodesic curvature of α and s is the arclength parameter of α. In particular, the following relations hold:
(7)
α
×
T
=
ξ
,
T
×
ξ
=
α
,
ξ
×
α
=
-
T
.Case 3 (T is a null vector).
It is known that the only null curves lying on pseudosphereS
1
2 are the null straight lines, which are the null geodesics.


## 3. Spacelike and Timelike Pseudospherical Smarandache Curves in Minkowski 3-Space

In this section, we consider spacelike pseudospherical curveα and define its spacelike and timelike pseudospherical Smarandache curves according to the Sabban frame of α in Mikowski 3-space. Let α
=
α
(
s
) be a unit speed spacelike curve with the Sabban frame {
α
,
T
,
ξ
}, lying fully on pseudosphere S
1
2 in ℝ
1
3. Denote by β
=
β
(
s
*
) arbitrary nonnull curve lying on pseudosphere, where s
* is the arclength parameter of β. Then we have the following definitions of special pseudospherical Smarandache curves of α.Definition 3.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully on pseudosphere S
1
2. The nonnull pseudospherical α
ξ-Smarandache curve β of α is defined by
(8)
β
(
s
*
(
s
)
)
=
1
2
(
a
α
(
s
)
+
b
ξ
(
s
)
)
,
where s
* is arclength parameter of β, a
,
b
∈
R
0, and a
2
-
b
2
=
2.Definition 4.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully on pseudosphere S
1
2. The nonnull pseudospherical α
T-Smarandache curve β of α is defined by
(9)
β
(
s
*
(
s
)
)
=
1
2
(
a
α
(
s
)
+
b
T
(
s
)
)
,
where s
* is arclength parameter of β,  a
,
b
∈
R
0, and a
2
+
b
2
=
2.Definition 5.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully on pseudosphere S
1
2. The nonnull pseudospherical T
ξ-Smarandache curve β of α is defined by
(10)
β
(
s
*
(
s
)
)
=
1
2
(
a
T
(
s
)
+
b
ξ
(
s
)
)
,
where s
* is arclength parameter of β,  a
,
b
∈
R
0, and a
2
-
b
2
=
2.Definition 6.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully on pseudosphere S
1
2. The nonnull pseudospherical α
T
ξ-Smarandache curve β of α is defined by
(11)
β
(
s
*
(
s
)
)
=
1
3
(
a
α
(
s
)
+
b
T
(
s
)
+
c
ξ
(
s
)
)
,
where s
* is arclength parameter of β,  a
,
b
,
c
∈
R
0, and a
2
+
b
2
-
c
2
=
3.Note that ifα is a timelike pseudospherical curve, the corresponding nonnull pseudospherical Smarandache curves according to the Sabban frame of α can be defined in analogous way. In particular, if α is a null pseudospherical curve, then it is a null straight line, so the vectors α
×
T and T are linearly dependent. Thus in this case we do not have the orthonormal Sabban frame of α.Next we obtain the Sabban frame{
β
,
T
β
,
ξ
β
} and the geodesic curvature κ
g
β of some special spacelike and timelike pseudospherical Smarandache curves of α. We consider the following two cases: (i) β is a spacelike curve and (ii) β is a timelike curve.Case 4 (β is a spacelike curve).
Then, we have the following theorem.Theorem 7.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a spacelike pseudospherical α
ξ-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by
(12)
[
β
T
β
ξ
β
]
=
[
a
2
0
b
2
0
ϵ
0
ϵ
b
2
0
ϵ
a
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(13)
k
g
β
=
b
-
a
k
g
|
a
-
b
k
g
|
,
where a
,
b
∈
R
0, a
2
-
b
2
=
2, and ϵ
=
±
1.Proof.
Differentiating (8) with respect to s and using (4) we obtain
(14)
β
′
(
s
)
=
d
β
d
s
*
d
s
*
d
s
=
a
-
b
k
g
2
T
,
and hence
(15)
T
β
d
s
*
d
s
=
a
-
b
k
g
2
T
,
where
(16)
d
s
*
d
s
=
〈
β
′
(
s
)
,
β
′
(
s
)
〉
=
|
a
-
b
k
g
|
2
.
Therefore, the unit spacelike tangent vector of the curve β is given by
(17)
T
β
=
ϵ
T
,
where ϵ
=
+
1 if a
-
b
k
g
>
0 for all s and ϵ
=
-
1 if a
-
b
k
g
<
0 for all s.
Differentiating (17) with respect to s and using (4) we find
(18)
d
T
β
d
s
*
d
s
*
d
s
=
ϵ
(
-
α
-
k
g
ξ
)
.
In particular, from (16) and (18) we get
(19)
T
β
′
=
2
ϵ
|
a
-
b
k
g
|
(
-
α
-
k
g
ξ
)
.
Sinceβ and T
β are spacelike vectors, from (8) and (17) we obtain that the unit timelike vector ξ
β is given by
(20)
ξ
β
=
-
β
×
T
β
=
ϵ
b
2
α
+
ϵ
a
2
ξ
.
Consequently, the geodesic curvature k
g
β of β is given by
(21)
k
g
β
=
det
⁡
(
β
,
T
β
,
T
β
′
)
=
b
-
a
k
g
|
a
-
b
k
g
|
.Theorem 8.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a spacelike pseudospherical α
T-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by
(22)
[
β
T
β
ξ
β
]
=
[
a
2
b
2
0
-
b
2
-
(
b
k
g
)
2
a
2
-
(
b
k
g
)
2
-
b
k
g
2
-
(
b
k
g
)
2
b
2
k
g
4
-
2
(
b
k
g
)
2
-
a
b
k
g
4
-
2
(
b
k
g
)
2
2
4
-
2
(
b
k
g
)
2
]
×
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(23)
k
g
β
=
-
k
g
b
2
ε
1
+
a
k
g
b
ε
2
+
2
ε
3
(
2
-
b
2
k
g
2
)
5
/
2
,
where
(24)
ε
1
=
-
b
3
k
g
′
k
g
+
a
b
2
k
g
2
-
2
a
,
ε
2
=
a
b
2
k
g
′
k
g
-
b
3
k
g
4
+
2
b
k
g
2
+
b
3
k
g
2
-
2
b
,
ε
3
=
-
2
b
k
g
′
+
a
b
2
k
g
3
-
2
a
k
g
,
a
,
b
∈
R
0,  a
2
+
b
2
=
2, and b
2
k
g
2
-
2
<
0 for all s.Proof.
Differentiating (9) with respect to s and using (4) we obtain
(25)
β
′
(
s
)
=
d
β
d
s
*
d
s
*
d
s
=
1
2
(
-
b
α
+
a
T
-
b
k
g
ξ
)
,
and consequently
(26)
T
β
d
s
*
d
s
=
1
2
(
-
b
α
+
a
T
-
b
k
g
ξ
)
,
where
(27)
d
s
*
d
s
=
2
-
(
b
k
g
)
2
2
.
Therefore, the unit spacelike tangent vector of the curve β is given by
(28)
T
β
=
1
2
-
(
b
k
g
)
2
(
-
b
α
+
a
T
-
b
k
g
ξ
)
,
where 2
-
(
b
k
g
)
2
>
0 for all s.
Differentiating (28) with respect to s, it follows that
(29)
d
T
β
d
s
*
d
s
*
d
s
=
1
(
2
-
(
b
k
g
)
2
)
3
/
2
(
ε
1
α
+
ε
2
T
+
ε
3
ξ
)
,
where
(30)
ε
1
=
-
b
3
k
g
′
k
g
+
a
b
2
k
g
2
-
2
a
,
ε
2
=
a
b
2
k
g
′
k
g
-
b
3
k
g
4
+
2
b
k
g
2
+
b
3
k
g
2
-
2
b
,
ε
3
=
-
2
b
k
g
′
+
a
b
2
k
g
3
-
2
a
k
g
.
From (27) and (29) we get
(31)
T
β
′
=
1
(
2
-
(
b
k
g
)
2
)
2
(
ε
1
α
+
ε
2
T
+
ε
3
ξ
)
.
On the other hand, from (9) and (28) it can be easily seen that the unit timelike vector ξ
β is given by
(32)
ξ
β
=
-
β
×
T
β
=
b
2
k
g
4
-
2
(
b
k
g
)
2
α
-
a
b
k
g
4
-
2
(
b
k
g
)
2
T
+
2
4
-
2
(
b
k
g
)
2
ξ
.
Therefore, the geodesic curvature k
g
β of β is given by
(33)
k
g
β
=
det
⁡
(
β
,
T
β
,
T
β
′
)
=
-
k
g
b
2
ε
1
+
a
k
g
b
ε
2
+
2
ε
3
(
2
-
b
2
k
g
2
)
5
/
2
.Theorem 9.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a spacelike pseudospherical T
ξ-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by
(34)
[
β
T
β
ξ
β
]
=
[
0
a
2
b
2
-
a
a
2
-
2
k
g
2
-
b
k
g
a
2
-
2
k
g
2
-
a
k
g
a
2
-
2
k
g
2
2
k
g
2
a
2
-
4
k
g
2
a
b
2
a
2
-
4
k
g
2
a
2
2
a
2
-
4
k
g
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(35)
k
g
β
=
-
2
k
g
ε
1
-
b
a
ε
2
+
ε
3
a
2
(
a
2
-
2
k
g
2
)
5
/
2
,
where
(36)
ε
1
=
-
2
a
k
g
′
k
g
-
2
b
k
g
3
+
a
2
b
k
g
,
ε
2
=
-
2
a
k
g
4
+
(
a
3
+
2
a
)
k
g
2
-
a
3
-
a
2
b
k
g
′
,
ε
3
=
-
a
3
k
g
′
-
2
b
k
g
4
+
a
2
b
k
g
2
,
a
,
b
∈
R
0, a
2
-
b
2
=
2, and a
2
-
2
k
g
2
>
0 for all s.Proof.
Differentiating (10) with respect to s and using (4) we obtain
(37)
β
′
(
s
)
=
d
β
d
s
*
d
s
*
d
s
=
1
2
(
-
a
α
-
b
k
g
T
-
a
k
g
ξ
)
,
and consequently
(38)
T
β
d
s
*
d
s
=
1
2
(
-
a
α
-
b
k
g
T
-
a
k
g
ξ
)
,
where
(39)
d
s
*
d
s
=
a
2
-
2
k
g
2
2
.
It follows that the unit spacelike tangent vector of the curve β is given by
(40)
T
β
=
1
a
2
-
2
k
g
2
(
-
a
α
-
b
k
g
T
-
a
k
g
ξ
)
,
where a
2
-
2
k
g
2
>
0 for all s. Differentiating (40) with respect to s, we find
(41)
d
T
β
d
s
*
d
s
*
d
s
=
1
(
a
2
-
2
k
g
2
)
3
/
2
(
ε
1
α
+
ε
2
T
+
ε
3
ξ
)
,
where
(42)
ε
1
=
-
2
a
k
g
′
k
g
-
2
b
k
g
3
+
a
2
b
k
g
,
ε
2
=
-
2
a
k
g
4
+
(
a
3
+
2
a
)
k
g
2
-
a
3
-
a
2
b
k
g
′
,
ε
3
=
-
a
3
k
g
′
-
2
b
k
g
4
+
a
2
b
k
g
2
.
From (39) and (41) we get
(43)
T
β
′
=
1
(
a
2
-
2
k
g
2
)
2
(
ε
1
α
+
ε
2
T
+
ε
3
ξ
)
.
Equations (10) and (40) imply
(44)
ξ
β
=
-
β
×
T
β
=
2
k
g
2
a
2
-
4
k
g
2
α
+
a
b
2
a
2
-
4
k
g
2
T
+
a
2
2
a
2
-
4
k
g
2
ξ
.
Finally, the geodesic curvature k
g
β of the curve β is given by
(45)
k
g
β
=
det
⁡
(
β
,
T
β
,
T
β
′
)
=
-
2
k
g
ε
1
-
b
a
ε
2
+
ε
3
a
2
(
a
2
-
2
k
g
2
)
5
/
2
.Theorem 10.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a spacelike pseudospherical α
T
ξ-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by(46)
[
β
T
β
ξ
β
]
=
[
a
3
b
3
c
3
-
b
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
a
-
c
k
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
-
b
k
g
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
c
(
a
-
c
k
g
)
+
b
2
k
g
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
-
a
b
k
g
+
b
c
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
a
(
a
-
c
k
g
)
+
b
2
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
]
[
α
T
ξ
]and the corresponding geodesic curvature k
g
β reads
(47)
k
g
β
=
(
(
(
a
2
-
3
)
k
g
-
a
c
)
ε
1
+
(
a
b
k
g
-
b
c
)
ε
2
+
(
-
a
c
k
g
+
c
2
+
3
)
ε
3
)
×
(
(
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
)
5
/
2
)
-
1
,
where
(48)
ε
1
=
(
a
2
-
3
)
b
k
g
′
k
g
-
a
b
c
k
g
′
+
(
a
2
-
3
)
c
k
g
3
+
(
a
b
2
-
3
a
c
2
)
k
g
2
+
(
3
a
2
c
+
b
2
c
)
k
g
-
a
(
c
2
+
3
)
,
ε
2
=
(
a
2
-
3
)
c
k
g
′
k
g
2
+
(
a
b
2
-
2
a
c
2
)
k
g
′
k
g
+
a
2
c
k
g
′
+
b
(
a
2
-
3
)
k
g
4
-
2
a
b
c
k
g
3
+
(
b
3
+
3
b
)
k
g
2
+
2
a
b
c
k
g
-
b
(
c
2
+
3
)
,
ε
3
=
2
b
(
a
2
-
3
)
k
g
′
k
g
2
-
3
a
b
c
k
g
′
k
g
+
b
(
c
2
+
3
)
k
g
′
+
c
(
a
2
-
3
)
k
g
4
+
(
a
b
2
-
3
a
c
2
)
k
g
3
+
(
3
a
2
c
+
b
2
c
)
k
g
2
-
a
(
c
2
+
3
)
k
g
,
a
,
b
,
c
∈
R
0, a
2
+
b
2
-
c
2
=
3, and (
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
>
0 for all s.Proof.
Differentiating (11) with respect to s and by using (4) we find
(49)
β
′
(
s
)
=
d
β
d
s
*
d
s
*
d
s
=
1
3
(
-
b
α
+
(
a
-
c
k
g
)
T
-
b
k
g
ξ
)
,
and thus
(50)
T
β
d
s
*
d
s
=
1
3
(
-
b
α
+
(
a
-
c
k
g
)
T
-
b
k
g
ξ
)
,
where
(51)
d
s
*
d
s
=
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
3
.
Therefore, the unit spacelike tangent vector of the curve β is given by
(52)
T
β
=
1
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
×
(
-
b
α
+
(
a
-
c
k
g
)
T
-
b
k
g
ξ
)
,
where (
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
>
0 for all s.
Differentiating (52) with respect to s and using (51), it follows that
(53)
T
β
′
=
1
(
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
)
2
(
ε
1
α
+
ε
2
T
+
ε
3
ξ
)
,
where
(54)
ε
1
=
(
a
2
-
3
)
b
k
g
′
k
g
-
a
b
c
k
g
′
+
(
a
2
-
3
)
c
k
g
3
+
(
a
b
2
-
3
a
c
2
)
k
g
2
+
(
3
a
2
c
+
b
2
c
)
k
g
-
a
(
c
2
+
3
)
,
ε
2
=
(
a
2
-
3
)
c
k
g
′
k
g
2
+
(
a
b
2
-
2
a
c
2
)
k
g
′
k
g
+
a
2
c
k
g
′
+
b
(
a
2
-
3
)
k
g
4
-
2
a
b
c
k
g
3
+
(
b
3
+
3
b
)
k
g
2
+
2
a
b
c
k
g
-
b
(
c
2
+
3
)
,
ε
3
=
2
b
(
a
2
-
3
)
k
g
′
k
g
2
-
3
a
b
c
k
g
′
k
g
+
b
(
c
2
+
3
)
k
g
′
+
c
(
a
2
-
3
)
k
g
4
+
(
a
b
2
-
3
a
c
2
)
k
g
3
+
(
3
a
2
c
+
b
2
c
)
k
g
2
-
a
(
c
2
+
3
)
k
g
.
From (11) and (52) we get
(55)
ξ
β
=
-
β
×
T
β
=
c
(
a
-
c
k
g
)
+
b
2
k
g
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
α
+
-
a
b
k
g
+
b
c
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
T
+
a
(
a
-
c
k
g
)
+
b
2
3
(
a
-
c
k
g
)
2
+
3
b
2
(
1
-
k
g
2
)
ξ
.
Consequently, the geodesic curvature k
g
β of β reads
(56)
k
g
β
=
det
⁡
(
β
,
T
β
,
T
β
′
)
=
(
(
(
a
2
-
3
)
k
g
-
a
c
)
ε
1
+
(
a
b
k
g
-
b
c
)
ε
2
h
+
(
-
a
c
k
g
+
c
2
+
3
)
ε
3
)
×
(
(
(
a
-
c
k
g
)
2
+
b
2
(
1
-
k
g
2
)
)
5
/
2
)
-
1
.Case 5 (β is a timelike curve).
Then, we have the following theorem.Theorem 11.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. Then the timelike pseudospherical α
ξ-Smarandache curve of α does not exist.Proof.
Assume that there exists a timelike pseudosphericalα
ξ-Smarandache curve β of α. Differentiating (8) with respect to s and using (4) we obtain
(57)
β
′
(
s
)
=
d
β
d
s
*
d
s
*
d
s
=
a
-
b
k
g
2
T
,
where s
* is the acrlength parameter of β. The previous equation implies
(58)
T
β
d
s
*
d
s
=
a
-
b
k
g
2
T
.
This means that a timelike vector T
β is collinear with a spacelike vector T, which is a contradiction.In the theorems which follow, in a similar way as in the Case4, we obtain the Sabban frame {
β
,
T
β
,
ξ
β
} and geodesic curvature κ
g
β of a timelike pseudospherical Smarandache curve β. We omit the proofs of Theorems 11, 12, and 13, since they are analogous to the proofs of Theorems 8, 9, and 10.Theorem 12.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a timelike pseudospherical α
T-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by
(59)
[
β
T
β
ξ
β
]
=
[
a
2
b
2
0
-
b
(
b
k
g
)
2
-
2
a
(
b
k
g
)
2
-
2
-
b
k
g
(
b
k
g
)
2
-
2
-
b
2
k
g
2
(
b
k
g
)
2
-
4
a
b
k
g
2
(
b
k
g
)
2
-
4
-
2
2
(
b
k
g
)
2
-
4
]
×
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(60)
k
g
β
=
-
k
g
b
2
ε
1
+
a
k
g
b
ε
2
+
2
ε
3
(
(
b
k
g
)
2
-
2
)
5
/
2
,
where
(61)
ε
1
=
b
3
k
g
′
k
g
-
a
b
2
k
g
2
+
2
a
,
ε
2
=
-
a
b
2
k
g
′
k
g
+
b
3
k
g
4
-
(
b
3
+
2
b
)
k
g
2
+
2
b
,
ε
3
=
2
b
k
g
′
-
a
b
2
k
g
3
+
2
a
k
g
,
a
,
b
∈
R
0, a
2
+
b
2
=
2, and (
b
k
g
)
2
-
2
>
0 for all s.Theorem 13.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a timelike pseudospherical T
ξ-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by
(62)
[
β
T
β
ξ
β
]
=
[
0
a
2
b
2
-
a
2
k
g
2
-
a
2
-
b
k
g
2
k
g
2
-
a
2
-
a
k
g
2
k
g
2
-
a
2
-
2
k
g
4
k
g
2
-
2
a
2
-
a
b
4
k
g
2
-
2
a
2
-
a
2
4
k
g
2
-
2
a
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(63)
k
g
β
=
-
2
k
g
ε
1
-
b
a
ε
2
+
a
2
ε
3
(
2
k
g
2
-
a
2
)
5
/
2
,
where
(64)
ε
1
=
2
a
k
g
′
k
g
+
2
b
k
g
3
-
a
2
b
k
g
,
ε
2
=
2
a
k
g
4
-
(
a
3
+
2
a
)
k
g
2
+
a
3
+
a
2
b
k
g
′
,
ε
3
=
a
3
k
g
′
+
2
b
k
g
4
-
a
2
b
k
g
2
,
a
,
b
∈
R
0, a
2
-
b
2
=
2, and 2
k
g
2
-
a
2
>
0 for all s.Theorem 14.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β
:
I
⊂
R
↦
S
1
2 is a timelike pseudospherical α
T
ξ-Smarandache curve of α, then its frame {
β
,
T
β
,
ξ
β
} is given by(65)
[
β
T
β
ξ
β
]
=
[
a
3
b
3
c
3
-
b
b
2
(
k
g
2
-
1
)
-
(
a
-
c
k
g
)
2
a
-
c
k
g
b
2
(
k
g
2
-
1
)
-
(
a
-
c
k
g
)
2
-
b
k
g
b
2
(
k
g
2
-
1
)
-
(
a
-
c
k
g
)
2
-
b
2
k
g
-
(
a
-
c
k
g
)
c
3
b
2
(
k
g
2
-
1
)
-
3
(
a
-
c
k
g
)
2
a
b
k
g
-
b
c
3
b
2
(
k
g
2
-
1
)
-
3
(
a
-
c
k
g
)
2
-
(
a
-
c
k
g
)
a
-
b
2
3
b
2
(
k
g
2
-
1
)
-
3
(
a
-
c
k
g
)
2
]
[
α
T
ξ
]and the corresponding geodesic curvature k
g
β reads
(66)
k
g
β
=
(
(
(
a
2
-
3
)
k
g
-
a
c
)
ε
1
+
(
-
b
c
+
a
b
k
g
)
ε
2
h
+
(
c
2
+
3
-
a
c
k
g
)
ε
3
)
×
(
(
b
2
(
k
g
2
-
1
)
-
(
a
-
c
k
g
)
2
)
5
/
2
)
-
1
,
where
(67)
ε
1
=
(
3
-
a
2
)
b
k
g
′
k
g
+
a
b
c
k
g
′
+
(
a
2
-
3
)
c
k
g
3
+
(
a
b
2
-
3
a
c
2
)
k
g
2
+
(
b
2
c
-
3
a
2
c
)
k
g
+
a
(
c
2
-
3
)
,
ε
2
=
(
3
-
a
2
)
c
k
g
′
k
g
2
+
(
2
a
c
2
-
a
b
2
)
k
g
′
k
g
+
a
2
c
k
g
′
+
(
a
2
-
3
)
b
k
g
4
-
2
a
b
c
k
g
3
+
(
b
3
-
3
b
)
k
g
2
-
2
a
b
c
k
g
+
b
(
c
2
-
3
)
,
ε
3
=
-
a
b
c
k
g
′
k
g
+
(
c
2
-
3
)
b
k
g
′
+
(
a
2
-
3
)
c
k
g
4
+
(
a
b
2
-
3
a
c
2
)
k
g
3
+
(
b
2
c
-
3
a
2
c
)
k
g
2
+
(
c
2
-
3
)
a
k
g
,
a
,
b
,
c
∈
R
0, a
2
+
b
2
-
c
2
=
3, and b
2
(
k
g
2
-
1
)
-
(
a
-
c
k
g
)
2
>
0 for all s.Corollary 15.
Ifα is a spacelike geodesic curve on pseudosphere S
1
2 in Minkowski 3-space E
1
3, then(1)
the spacelike and timelike pseudosphericalα
T-Smarandache curves are also geodesic on S
1
2;
(2)
the spacelike and timelike pseudosphericalT
ξ and α
T
ξ-Smarandache curves have constant geodesic curvatures on S
1
2;
(3)
the spacelike pseudosphericalα
ξ-Smarandache curve has constant geodesic curvature on S
1
2.


## 4. Null Pseudospherical Smarandache Curves in Minkowski 3-Space

In this section, we give definitions of null pseudospherical Smarandache curves which are analogous to the definitions of nonnull pseudospherical Smarandache curves ofα, given in Section 3.Definition 16.
Letα
:
I
⊂
R
↦
S
1
2 and β
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike and a null curve, respectively, lying fully in pseudosphere S
1
2. The curve β is pseudospherical α
ξ-Smarandache curve of α, if it is given by
(68)
β
(
s
)
=
1
2
(
a
α
(
s
)
+
b
ξ
(
s
)
)
,
where a
,
b
∈
R
0, and a
2
-
b
2
=
2.Definition 17.
Letα
:
I
⊂
R
↦
S
1
2 and β
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere S
1
2. The curve β is pseudospherical α
T-Smarandache curve of α, if it is given by
(69)
β
(
s
)
=
1
2
(
a
α
(
s
)
+
b
T
(
s
)
)
,
where a
,
b
∈
R
0, and a
2
+
b
2
=
2.Definition 18.
Letα
:
I
⊂
R
↦
S
1
2 and β
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere S
1
2. The curve β is pseudospherical T
ξ-Smarandache curve of α, if it is given by
(70)
β
(
s
)
=
1
2
(
a
T
(
s
)
+
b
ξ
(
s
)
)
,
where a
,
b
∈
R
0, and a
2
-
b
2
=
2.Definition 19.
Letα
:
I
⊂
R
↦
S
1
2 and β
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere S
1
2. The curve β is pseudospherical α
T
ξ-Smarandache curve of α, if it is given by
(71)
β
(
s
)
=
1
3
(
a
α
(
s
)
+
b
T
(
s
)
+
c
ξ
(
s
)
)
,
where a
,
b
,
c
∈
R
0, and a
2
+
b
2
-
c
2
=
3.Theorem 20.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. Then the null pseudospherical α
ξ-Smarandache curve of α does not exist.Proof.
Assume that there exists null pseudosphericalα
ξ-Smarandache curve of α. Differentiating (68) with respect to s and using (4) we obtain
(72)
β
′
(
s
)
=
a
-
b
k
g
2
T
.
This means that a null vector T
β
=
β
′ is collinear with a spacelike vector T, which is a contradiction.Theorem 21.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β is a null pseudospherical α
T-Smarandache curve of α, then α has constant geodesic curvature given by
(73)
k
g
2
=
2
2
-
a
2
,
a
∈
R
0
,
a
2
<
2
.Proof.
Differentiating (69) with respect to s and using (4) we obtain
(74)
β
′
(
s
)
=
1
2
(
-
b
α
+
a
T
-
b
k
g
ξ
)
,
and consequently
(75)
T
β
=
1
2
(
-
b
α
+
a
T
-
b
k
g
ξ
)
,
where a
,
b
∈
R
0, and a
2
+
b
2
=
2. The condition 〈
T
β
,
T
β
〉
=
0 implies
(76)
k
g
2
=
2
2
-
a
2
,
a
∈
R
0
,
a
2
<
2
,
which proves the theorem.The next two theorems can be proved in a similar way, so we omit their proofs.Theorem 22.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β is a null pseudospherical T
ξ-Smarandache curve of α, then α has constant geodesic curvature given by
(77)
k
g
2
=
a
2
2
,
a
∈
R
0
,
a
2
>
2
.Theorem 23.
Letα
:
I
⊂
R
↦
S
1
2 be a unit speed spacelike curve lying fully in S
1
2 with the Sabban frame {
α
,
T
,
ξ
} and the geodesic curvature k
g. If β is a null pseudospherical α
T
ξ-Smarandache curve of α, then α has constant geodesic curvature given by
(78)
k
g
=
2
a
c
±
4
a
2
c
2
-
4
(
c
2
-
b
2
)
(
a
2
+
b
2
)
2
(
a
2
-
3
)
,
where a
,
b
,
c
∈
R
0, a
2
+
b
2
-
c
2
=
3, and a
2
≠
3.Corollary 24.
There are no spacelike pseudospherical geodesic curves whose pseudosphericalα
T, T
ξ, and α
T
ξ-Smarandache curves are the null curves.


## 5. Examples

Example 1.
Letα be a unit speed spacelike curve lying on pseudosphere S
1
2 in the Minkowski 3-space 𝔼
1
3 with parameter equation (see Figure 1)
(79)
α
(
s
)
=
(
s
2
2
,
s
,
1
-
s
2
2
)
.
The orthonormal Sabban frame {
α
,
T
,
ξ
} along the curve α is given by
(80)
α
(
s
)
=
(
s
2
2
,
s
,
1
-
s
2
2
)
,
T
(
s
)
=
α
′
(
s
)
=
(
s
,
1
,
-
s
)
,
ξ
(
s
)
=
-
α
(
s
)
×
T
(
s
)
=
(
-
1
-
s
2
2
,
-
s
,
s
2
2
)
.
In particular, the geodesic curvature k
g of the curve α has the form
(81)
k
g
(
s
)
=
det
⁡
(
α
(
s
)
,
T
(
s
)
,
T
′
(
s
)
)
=
-
1
.Figure 1
The curveα on S
1
2.Case 1.
Takinga
=
3 and b
=
1 and using (8), we obtain that pseudospherical α
ξ-Smarandache curve β is given by (see Figure 2)
(82)
β
(
s
)
=
2
2
(
(
3
-
1
)
2
s
2
-
1
,
(
3
-
1
)
s
,
3
-
(
3
-
1
)
2
s
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
>
0, which means that β is a spacelike curve. According to Theorem 7, its Sabban frame {
β
,
T
β
,
ξ
β
} is given by
(83)
[
β
T
β
ξ
β
]
=
[
6
2
0
2
2
0
1
0
2
2
0
6
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(84)
k
g
β
=
b
-
a
k
g
|
a
-
b
k
g
|
=
1
.Figure 2
The spacelikeα
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 2.
Takinga
=
1 and b
=
-
1 and using (9) we get that pseudospherical α
T-Smarandache curve is given by (see Figure 3)
(85)
β
(
s
*
(
s
)
)
=
2
2
(
s
2
-
2
s
2
,
s
-
1
,
-
s
2
2
+
s
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
>
0, which means that β is a spacelike curve. According to Theorem 8, its frame {
β
,
T
β
,
ξ
β
} is given by
(86)
[
β
T
β
ξ
β
]
=
[
2
2
-
2
2
0
1
1
1
-
2
2
-
2
2
-
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(87)
k
g
β
=
-
2
.Figure 3
The spacelikeα
T-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 3.
Takinga
=
2 and b
=
2, from (10) we find that the pseudospherical T
ξ-Smarandache curve is given by (see Figure 4)
(88)
β
(
s
*
(
s
)
)
=
2
2
(
-
2
2
s
2
+
2
s
-
2
,
2
-
2
s
,
2
2
s
2
-
2
s
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
>
0, which means that β is a spacelike curve. According to Theorem 9, its frame {
β
,
T
β
,
ξ
β
} is given by
(89)
[
β
T
β
ξ
β
]
=
[
0
2
1
-
2
-
1
-
2
-
1
-
2
-
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β has the following form:
(90)
k
g
β
=
1
.Figure 4
The spacelikeT
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 4.
Takinga
=
2, b
=
2, and c
=
1 and using (11), we find that the pseudospherical α
T
ξ-Smarandache curve β has parameter equation (see Figure 5):
(91)
β
(
s
*
(
s
)
)
=
1
3
(
(
2
+
1
)
2
s
2
+
2
s
+
1
,
(
1
+
2
)
s
+
2
,
-
(
1
+
2
)
2
s
2
-
2
s
+
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
>
0, which means that β is a spacelike curve. By Theorem 10, its frame {
β
,
T
β
,
ξ
β
} is given by
(92)
[
β
T
β
ξ
β
]
=
[
6
3
6
3
3
3
-
2
-
2
1
-
2
-
2
-
2
6
+
3
3
3
6
3
-
3
6
+
2
3
3
]
[
α
T
ξ
]
,
while the corresponding geodesic curvature k
g
β has the following form:
(93)
k
g
β
=
-
1200
2
-
1697
.
Special spacelike Smarandache curves ofα and the curve α on S
1
2 are shown in Figure 6.Figure 5
The spacelikeα
T
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Figure 6
Special spacelike pseudospherical Smarandache curves ofα and the curve α on S
1
2.Example 2.
Let us consider a unit speed spacelike circleα lying on pseudosphere S
1
2 in the Minkowski 3-space 𝔼
1
3 with parameter equation (see Figure 7):
(94)
α
(
s
)
=
(
cosh
⁡
s
,
sinh
⁡
s
,
2
)
.
The orthonormal Sabban frame{
α
,
T
,
ξ
} along the curve α is given by
(95)
α
(
s
)
=
(
cosh
⁡
s
,
sinh
⁡
s
,
2
)
,
T
(
s
)
=
α
′
(
s
)
=
(
sinh
⁡
s
,
cosh
⁡
s
,
0
)
,
ξ
(
s
)
=
-
α
(
s
)
×
T
(
s
)
=
(
-
2
cosh
⁡
s
,
-
2
sinh
⁡
s
,
-
1
)
.
In particular, the geodesic curvature k
g of α reads
(96)
k
g
(
s
)
=
det
⁡
(
α
(
s
)
,
T
(
s
)
,
T
′
(
s
)
)
=
-
2
.Figure 7
The spacelike circleα on S
1
2.Case 1.
Takinga
=
(
2
/
2
) and b
=
(
6
/
2
) and using (9), we obtain that pseudospherical α
T-Smarandache curve β is given by (see Figure 8)
(97)
β
(
s
)
=
(
1
2
cosh
⁡
s
+
3
2
sinh
⁡
s
,
1
2
sinh
⁡
s
+
3
2
cosh
⁡
s
,
2
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
<
0, which means that β is a timelike curve. According to Theorem 12, its Sabban frame {
β
,
T
β
,
ξ
β
} is given by
(98)
[
β
T
β
ξ
β
]
=
[
1
2
3
2
0
-
6
2
2
2
3
3
2
3
2
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(99)
k
g
β
=
-
k
g
b
2
ε
1
+
a
k
g
b
ε
2
+
2
ε
3
(
(
b
k
g
)
2
-
2
)
5
/
2
=
-
10
.Figure 8
The timelikeα
T-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 2.
Takinga
=
3 and b
=
1 and using (10), we obtain that pseudospherical T
ξ-Smarandache curve β is given by (see Figure 9)
(100)
β
(
s
)
=
(
3
sinh
⁡
s
-
2
cosh
⁡
s
,
3
cosh
⁡
s
-
2
sinh
⁡
s
,
-
1
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
<
0, which means that β is a timelike curve. According to Theorem 13, its Sabban frame {
β
,
T
β
,
ξ
β
} is given by
(101)
[
β
T
β
ξ
β
]
=
[
0
6
2
2
2
-
3
2
6
2
2
-
6
2
-
3
2
2
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(102)
k
g
β
=
11
.Figure 9
The timelikeT
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 3.
Takinga
=
-
2, b
=
2, and c
=
1 and using (11), we obtain that pseudospherical α
T
ξ-Smarandache curve β is given by (see Figure 10)
(103)
β
(
s
)
=
(
2
sinh
⁡
s
-
2
2
cosh
⁡
s
,
2
cosh
⁡
s
-
2
2
sinh
⁡
s
,
-
3
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
<
0, which means that β is a timelike curve. According to Theorem 14, its Sabban frame {
β
,
T
β
,
ξ
β
} is given by
(104)
[
β
T
β
ξ
β
]
=
[
-
6
3
6
3
3
3
-
1
0
-
2
2
3
3
3
3
-
6
3
]
[
α
T
ξ
]
,
and the corresponding geodesic curvature k
g
β reads
(105)
k
g
β
=
-
5
2
.
Special timelike Smarandache curves ofα and the curve α on S
1
2 are shown in Figure 11.Figure 10
The timelikeα
T
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Figure 11
Special timelike pseudospherical Smarandache curves ofα and the curve α on S
1
2.Example 3.
Let us consider a unit speed spacelike circleα lying on pseudosphere S
1
2 in the Minkowski 3-space 𝔼
1
3 with parameter (94). Then its geodesic curvature is given by (96).Case 1.
Ifβ is null pseudospherical α
T-Smarandache curve of α, then according to Theorem 21 the curve α has nonzero constant geodesic curvature given by
(106)
k
g
2
=
2
2
-
a
2
,
a
∈
R
0
,
a
2
<
2
.
The last relation together with (96) implies a
=
1. By Definition 17 there holds a
2
+
b
2
=
2 and therefore we can take b
=
1. Finally, by using (69) we obtain that pseudospherical α
T-Smarandache curve β of α is given by (see Figure 12)
(107)
β
(
s
)
=
2
2
(
sinh
⁡
s
+
cosh
⁡
s
,
cosh
⁡
s
+
sinh
⁡
s
,
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
=
0, which means that β is a null straight line.Figure 12
The nullT
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 2.
Ifβ is null pseudospherical T
ξ-Smarandache curve of α, then according to Theorem 22 the curve α has nonzero constant geodesic curvature given by
(108)
k
g
2
=
a
2
2
,
a
∈
R
0
,
a
2
>
2
.
The last relation together with (96) implies a
=
2. By Definition 18 there holds a
2
-
b
2
=
2 and therefore we can take b
=
2. Finally, by using (70) we obtain that pseudospherical T
ξ-Smarandache curve β of α is given by (see Figure 13)
(109)
β
(
s
)
=
(
2
sinh
⁡
s
-
2
cosh
⁡
s
,
2
cosh
⁡
s
-
2
sinh
⁡
s
,
-
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
=
0, which means that β is a null straight line.Figure 13
The nullT
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.Case 3.
Ifβ is null pseudospherical α
T
ξ-Smarandache curve of α, then according to Theorem 23 the curve α has nonzero constant geodesic curvature given by
(110)
k
g
=
2
a
c
±
4
a
2
c
2
-
4
(
c
2
-
b
2
)
(
a
2
+
b
2
)
2
(
a
2
-
3
)
.
The last relation together with (96) implies a
=
2, b
=
3. By Definition 19 there holds a
2
+
b
2
-
c
2
=
3 and therefore we can take c
=
-
2. Finally, by using (71) we obtain that pseudospherical α
T
ξ-Smarandache curve β of α is given by (see Figure 14)
(111)
β
(
s
)
=
(
(
2
+
2
2
)
cosh
⁡
s
+
3
sinh
⁡
s
,
(
2
+
2
2
)
sinh
⁡
s
+
3
cosh
⁡
s
,
2
2
+
2
)
.
It can be easily checked that 〈
β
′
(
s
)
,
β
′
(
s
)
〉
=
0, which means that β is a null straight line.Figure 14
The nullα
T
ξ-pseudospherical Smarandache curve β and the curve α on S
1
2.



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*Source: 404521-2014-02-24.xml*