Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 658670, 7 pages
http://dx.doi.org/10.1155/2013/658670
Research Article
On Pseudohyperbolical Smarandache Curves in
Minkowski 3-Space
Esra Betul Koc Ozturk,1,2 Ufuk Ozturk,1,2 Kazim Elarslan,3 and Emilija NešoviT4
1
Department of Mathematics, Faculty of Sciences, University of Cankiri Karatekin, Cankiri 18100, Turkey
School of Mathematics & Statistical Sciences, Arizona State University, Room PSA442, Tempe, AZ 85287-1804, USA
3
Department of Mathematics, Faculty of Sciences and Art, University of Kırıkkale, Kırıkkale 71450, Turkey
4
University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Kragujevac 34000, Serbia
2
Correspondence should be addressed to Ufuk Ozturk; uuzturk@asu.edu
Received 10 March 2013; Accepted 7 May 2013
Academic Editor: B. N. Mandal
Copyright © 2013 Esra Betul Koc Ozturk et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic
curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some
examples of such curves.
1. Introduction
In the theory of curves in the Euclidean and Minkowski
spaces, one of the interesting problems is the problem of
characterization of a regular curve. In the solution of the
problem, the curvature functions 𝜅 and 𝜏 of a regular curve
have an effective role. It is known that we can determine
the shape and size of a regular curve by using its curvatures
𝜅 and 𝜏. Another approach to the solution of the problem
is to consider the relationship between the corresponding
Frenet vectors of two curves. For instance, Bertrand curves
and Mannheim curves arise from this relationship. Another
example is the Smarandache curves. They are the objects
of Smarandache geometry, that 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.
If the position vector of a regular curve 𝛼 is composed
by the Frenet frame vectors of another regular curve 𝛽,
then the curve 𝛼 is called a Smarandache curve [2]. Special
Smarandache curves in Euclidean and Minkowski spaces
are studied by some authors [3–8]. The curves lying on
a pseudohyperbolic space 𝐻02 in Minkowski 3-space E31 are
characterized in [9].
In this paper, we define pseudohyperbolical Smarandache
curves according to the Sabban frame {𝛼, 𝑇, 𝜉} in Minkowski
3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that
special 𝑇𝜉-pseudohyperbolical Smarandache curves do not
exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space.
2. Basic Concepts
The Minkowski 3-space R31 is the Euclidean 3-space R3
provided with the standard flat metric given by
⟨⋅, ⋅⟩ = −𝑑𝑥12 + 𝑑𝑥22 + 𝑑𝑥32 ,
(1)
where (𝑥1 , 𝑥2 , 𝑥3 ) is a rectangular Cartesian coordinate system of R31 . Since 𝑔 is an indefinite metric, recall that a nonzero
vector 𝑥⃗ ∈ R31 can have one of the 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
⃗ and
(length) of a vector 𝑥⃗ ∈ R31 is given by ‖𝑥‖⃗ = √|⟨𝑥,⃗ 𝑥⟩|
2
International Journal of Mathematics and Mathematical Sciences
two vectors 𝑥⃗ and 𝑦⃗ are said to be orthogonal if ⟨𝑥,⃗ 𝑦⟩⃗ = 0.
Next, recall that an arbitrary curve 𝛼 = 𝛼(𝑠) in E31 can locally
be spacelike, timelike, or null (lightlike) if all of its velocity
vectors 𝛼󸀠 (𝑠) are, respectively, spacelike, timelike, or null
(lightlike) for every 𝑠 ∈ 𝐼 [10]. If ‖𝛼󸀠 (𝑠)‖ ≠ 0 for every 𝑠 ∈ 𝐼,
then 𝛼 is a regular curve in R31 . A spacelike (timelike) regular
curve 𝛼 is parameterized by pseudo-arclength parameter 𝑠
which is given by 𝛼 : 𝐼 ⊂ R → R31 , and then the tangent
vector 𝛼󸀠 (𝑠) along 𝛼 has unit length, that is, ⟨𝛼󸀠 (𝑠), 𝛼󸀠 (𝑠)⟩ =
1(⟨𝛼󸀠 (𝑠), 𝛼󸀠 (𝑠)⟩ = −1) for all 𝑠 ∈ 𝐼, respectively.
For any 𝑥⃗ = (𝑥1 , 𝑥2 , 𝑥3 ) and 𝑦⃗ = (𝑦1 , 𝑦2 , 𝑦3 ) in the space
R31 , the pseudovector product of 𝑥⃗ and 𝑦⃗ is defined by
𝑥⃗ × 𝑦⃗ = (−𝑥2 𝑦3 + 𝑥3 𝑦2 , 𝑥3 𝑦1 − 𝑥1 𝑦3 , 𝑥1 𝑦2 − 𝑥2 𝑦1 ) .
(2)
Remark 1. Let 𝑥⃗ = (𝑥1 , 𝑥2 , 𝑥3 ), 𝑦⃗ = (𝑦1 , 𝑦2 , 𝑦3 ), and 𝑧⃗ =
(𝑧1 , 𝑧2 , 𝑧3 ) be vectors in R31 . Then,
󵄨󵄨 𝑥1 𝑥2 𝑥3 󵄨󵄨
(i) ⟨𝑥⃗ × 𝑦,⃗ 𝑧⟩⃗ = 󵄨󵄨󵄨 𝑦𝑧1 𝑦𝑧2 𝑦𝑧3 󵄨󵄨󵄨,
󵄨 1 2 3󵄨
(ii) 𝑥⃗ × (𝑦⃗ × 𝑧)⃗ = −⟨𝑥,⃗ 𝑧⟩⃗ 𝑦⃗ + ⟨𝑥,⃗ 𝑦⟩⃗ 𝑧,⃗
Definition 3. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit
speed curve lying fully in 𝐻02 . Then 𝛼𝜉-pseudohyperbolical
Smarandache curve 𝛽 : 𝐼 ⊂ R 󳨃→ 𝐻02 of 𝛼 is defined by
𝛽 (𝑠⋆ (𝑠)) =
1
(𝑎𝛼 (𝑠) + 𝑏𝜉 (𝑠)) ,
√2
(6)
Definition 4. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit
speed curve lying fully in 𝐻02 .Then 𝛼𝑇-pseudohyperbolical
Smarandache curve 𝛽 : 𝐼 ⊂ R 󳨃→ 𝐻02 of 𝛼 is defined by
⃗ 𝑦,⃗ 𝑦⟩⃗ + ⟨𝑥,⃗ 𝑦⟩⃗ ,
(iii) ⟨𝑥⃗ × 𝑦,⃗ 𝑥⃗ × 𝑦⟩⃗ = −⟨𝑥,⃗ 𝑥⟩⟨
where × is the pseudovector product in the space R31 .
Lemma 2. In the Minkowski 3-space R31 , the following properties are satisfied [10]:
(ii) two null vectors are orthogonal if and only if they are
linearly dependent;
(iii) timelike vector is never orthogonal to a null vector.
Pseudohyperbolic space in the Minkowski 3-space R31 is a
quadric defined by
𝐻02 = {𝑥⃗ ∈ R31 | −𝑥12 + 𝑥22 + 𝑥32 = −1} .
(3)
Let 𝛼 : 𝐼 ⊂ R → 𝐻02 be a regular unit speed curve lying
fully in 𝐻02 in R31 . Then its position vector 𝛼 is timelike vector,
which implies that tangent vector 𝑇⃗ = 𝛼󸀠 is the unit spacelike
vector for all 𝑠 ∈ 𝐼. 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
󸀠
0
1
0
𝛼
𝛼
[𝑇󸀠 ] = [1
0
𝑘𝑔 (𝑠)] [𝑇] ,
󸀠
[ 𝜉 ] [0 −𝑘𝑔 (𝑠) 0 ] [ 𝜉 ]
(4)
where 𝑘𝑔 (𝑠) = det(𝛼(𝑠), 𝑇(𝑠), 𝑇󸀠 (𝑠)) is the geodesic curvature
of 𝛼 on 𝐻02 in R31 and 𝑠 is arc length parameter of 𝑎𝑙𝑝ℎ𝑎. In
particular, the following relations hold:
𝜉 × 𝛼 = 𝑇.
𝛽 (𝑠⋆ (𝑠)) =
1
(𝑎𝛼 (𝑠) + 𝑏𝑇 (𝑠)) ,
√2
(7)
where 𝑎, 𝑏 ∈ 𝑅0 and 𝑎2 − 𝑏2 = 2.
(i) two timelike vectors are never orthogonal;
𝑇 × 𝜉 = −𝛼,
In this section, we define and investigate pseudohyperbolical
Smarandache curves in Minkowski 3-space according to the
Sabban frame.
Let 𝛼 = 𝛼(𝑠) and 𝛽 = 𝛽(𝑠∗ ) be two regular unit speed
curves lying fully in pseudohyperbolic space 𝐻02 in R31 and let
{𝛼, 𝑇, 𝜉} and {𝛽, 𝑇𝛽 , 𝜉𝛽 } be the moving Sabban frames of these
curves, respectively. Then we have the following definitions
of pseudohyperbolical Smarandache curves.
where 𝑎, 𝑏 ∈ R0 and 𝑎2 − 𝑏2 = 2.
2
𝛼 × 𝑇 = 𝜉,
3. Pseudohyperbolical Smarandache Curves in
Minkowski 3-Space
(5)
Definition 5. Let 𝛼 : 𝐼 ⊂ 𝑅 󳨃→ 𝐻02 be a regular unit
speed curve lying fully in 𝐻02 . Then 𝛼𝑇𝜉 -pseudohyperbolical
Smarandache curve 𝛽 : 𝐼 ⊂ 𝑅 󳨃→ 𝐻02 of 𝛼 is defined by
𝛽 (𝑠⋆ (𝑠)) =
1
(𝑎𝛼 (𝑠) + 𝑏𝑇 (𝑠) + 𝑐𝜉 (𝑠)) ,
√3
(8)
where 𝑎, 𝑏, 𝑐 ∈ R0 and 𝑎2 − 𝑏2 − 𝑐2 = 3.
Theorem 6. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit speed curve
lying fully in 𝐻02 . Then 𝑇𝜉-pseudohyperbolical Smarandache
curve 𝛽 : 𝐼 ⊂ R 󳨃→ 𝐻02 of 𝛼 does not exist.
Proof. Assume that there exists 𝑇𝜉-pseudohyperbolical
Smarandache curve of 𝛼. Then it can be written as
𝛽 (𝑠⋆ (𝑠)) =
1
(𝑎𝑇 (𝑠) + 𝑏𝜉 (𝑠)) ,
√2
(9)
where 𝑎2 + 𝑏2 = −2, which is a contradiction.
In the theorems which follow, we obtain Sabban frame
{𝛽, 𝑇𝛽 , 𝜉𝛽 } and geodesic curvature 𝜅𝑔𝛽 of pseudohyperbolical
Smarandache curve 𝛽.
Theorem 7. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit speed
curve lying fully in 𝐻02 with the Sabban frame {𝛼, 𝑇, 𝜉} and
International Journal of Mathematics and Mathematical Sciences
the geodesic curvature 𝑘𝑔 . If 𝛽 : 𝐼 ⊂ R →
󳨃
𝐻02 is 𝛼𝜉pseudohyperbolical Smarandache curve of 𝛼, then its frame
{𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by
𝑏
𝑎
[ √2 0 √2 ] 𝛼
𝛽
[
]
[𝑇𝛽 ] = [ 0 𝜖 0 ] [𝑇] ,
[
]
[ 𝑏
𝑎 ] [𝜉]
[ 𝜉𝛽 ]
0 𝜖
𝜖
√2 ]
[ √2
(10)
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
𝑎𝑘𝑔 − 𝑏
𝑘𝑔𝛽 = 󵄨󵄨
󵄨,
󵄨󵄨𝑎 − 𝑏𝑘𝑔 󵄨󵄨󵄨
󵄨
󵄨
𝑎, 𝑏 ∈ R0 ,
(11)
where 𝑎2 − 𝑏2 = 2 and 𝜖 = ±1.
Proof. Differentiating (6) with respect to 𝑠 and using (4), we
obtain
𝑑𝛽 𝑑𝑠∗ 𝑎 − 𝑏𝑘𝑔
𝑇,
𝛽󸀠 (𝑠) = ∗
=
√2
𝑑𝑠 𝑑𝑠
(12)
𝑑𝑠∗ 𝑎 − 𝑏𝑘𝑔
𝑇𝛽
𝑇,
=
√2
𝑑𝑠
where
󵄨󵄨
󵄨󵄨
𝑑𝑠∗ 󵄨󵄨󵄨𝑎 − 𝑏𝑘𝑔 󵄨󵄨󵄨
(13)
.
=
√2
𝑑𝑠
Therefore, the unit spacelike tangent vector of the curve 𝛽 is
given by
𝑇𝛽 = 𝜖𝑇,
(14)
3
Theorem 8. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit speed
curve lying fully in 𝐻02 with the Sabban frame {𝛼, 𝑇, 𝜉} and
the geodesic curvature 𝑘𝑔 . If 𝛽 : 𝐼 ⊂ R 󳨃→ 𝐻02 is 𝛼𝑇pseudohyperbolical Smarandache curve of 𝛼, then its frame
{𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by
𝛽
[𝑇𝛽 ]
[ 𝜉𝛽 ]
𝑏
𝑎
0
[
]
√
√
2
2
[
]
𝑏𝑘𝑔
[
]
𝑏
𝑎
[
]
[
]
2
2
2]
[ √
√ 2 + (𝑏𝑘𝑔 )
√ 2 + (𝑏𝑘𝑔 ) ]
= [ 2 + (𝑏𝑘𝑔 )
[
]
[
]
𝑏2 𝑘𝑔
𝑎𝑏𝑘𝑔
√2
[
]
[−
]
−
[
]
2
2
2
√ 4 + 2(𝑏𝑘𝑔 )
√ 4 + 2(𝑏𝑘𝑔 ) √ 2 + (𝑏𝑘𝑔 )
[
]
𝛼
× [𝑇] ,
[𝜉]
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
𝑘𝑔𝛽
𝑑𝑠∗
= 𝜖 (𝛼 + 𝑘𝑔 𝜉) ,
𝑑𝑠
and from (13) and (15) we get
𝑎
𝑏
𝛼+𝜖 𝜉
√2
√2
5/2
,
𝑎, 𝑏 ∈ R0 ,
(20)
2
2
𝜀2 = −𝑎𝑏2 𝑘𝑔 𝑘𝑔󸀠 + (𝑏 − 𝑏𝑘𝑔2 ) (2 + (𝑏𝑘𝑔 ) ) ,
(21)
2
𝜀3 = −𝑏3 𝑘𝑔2 𝑘𝑔󸀠 + (𝑎𝑘𝑔 + 𝑏𝑘𝑔󸀠 ) (2 + (𝑏𝑘𝑔 ) ) ,
and 𝑎2 − 𝑏2 = 2.
Proof. Differentiating (7) with respect to 𝑠 and using (4), we
obtain
𝛽󸀠 (𝑠) =
𝑑𝛽 𝑑𝑠∗
1
(𝑏𝛼 + 𝑎𝑇 + 𝑏𝑘𝑔 𝜉) ,
=
∗
√2
𝑑𝑠 𝑑𝑠
𝑑𝑠∗
1
𝑇𝛽
(𝑏𝛼 + 𝑎𝑇 + 𝑏𝑘𝑔 𝜉) ,
=
√2
𝑑𝑠
(17)
(22)
where
is a unit spacelike vector.
Consequently, the geodesic curvature 𝑘𝑔𝛽 of the curve 𝛽 =
∗
𝛽(𝑠 ) is given by
𝑘𝑔𝛽 = det (𝛽, 𝑇𝛽 , 𝑇𝛽󸀠 )
𝑎𝑘𝑔 − 𝑏
= 󵄨󵄨
󵄨.
󵄨󵄨𝑎 − 𝑏𝑘𝑔 󵄨󵄨󵄨
󵄨
󵄨
(2 + 𝑏2 𝑘𝑔2 )
𝜀1 = −𝑏3 𝑘𝑔 𝑘𝑔󸀠 + 𝑎 (2 + (𝑏𝑘𝑔 ) ) ,
(15)
√2𝜖
𝑇𝛽󸀠 = 󵄨󵄨
(16)
󵄨 (𝛼 + 𝑘𝑔 𝜉) .
󵄨󵄨𝑎 − 𝑏𝑘𝑔 󵄨󵄨󵄨
󵄨
󵄨
On the other hand, from (6) and (14) it can be easily seen
that
𝜉𝛽 = 𝛽 × 𝑇𝛽
=𝜖
=
𝑏2 𝑘𝑔 𝜀1 − 𝑎𝑏𝑘𝑔 𝜀2 + 2𝜀3
where
where 𝜖 = +1 if 𝑎 − 𝑏𝑘𝑔 > 0 for all 𝑠 and 𝜖 = −1 if 𝑎 − 𝑏𝑘𝑔 < 0
for all 𝑠.
Differentiating (14) with respect to 𝑠, we find
𝑑𝑇𝛽 𝑑𝑠∗
(19)
(18)
2
𝑑𝑠∗ √ 2 + (𝑏𝑘𝑔 )
=
.
𝑑𝑠
2
(23)
Therefore, the unit spacelike tangent vector of the curve
𝛽 is given by
𝑇𝛽 =
1
2
√ 2 + (𝑏𝑘𝑔 )
(𝑏𝛼 + 𝑎𝑇 + 𝑏𝑘𝑔 𝜉) .
(24)
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International Journal of Mathematics and Mathematical Sciences
Differentiating (24) with respect to 𝑠, it follows that
𝑑𝑇𝛽 𝑑𝑠∗
𝑑𝑠∗ 𝑑𝑠
𝜉𝛽 = 𝛽 × 𝑇𝛽
1
=
On the other hand, from (7) and (24), it can be easily seen
that
2 3/2
(2 + (𝑏𝑘𝑔 ) )
(𝜀1 𝛼 + 𝜀2 𝑇 + 𝜀3 𝜉) .
(25)
=−
𝑏2 𝑘𝑔
√ 4 + 2(𝑏𝑘𝑔 )
2
𝛼−
𝑎𝑏𝑘𝑔
√ 4 + 2(𝑏𝑘𝑔 )
2
√2
𝑇+
√2
2 2
(2 + (𝑏𝑘𝑔 ) )
𝜉.
(28)
From (4) and (23), we get
𝑇𝛽󸀠 =
2
√ 2 + (𝑏𝑘𝑔 )
(𝜀1 𝛼 + 𝜀2 𝑇 + 𝜀3 𝜉) ,
(26)
Hence 𝜉𝛽 is a unit spacelike vector.
Therefore, the geodesic curvature 𝑘𝑔𝛽 of the curve 𝛽 =
∗
𝛽(𝑠 ) is given by
𝑘𝑔𝛽 = det (𝛽, 𝑇𝛽 , 𝑇𝛽󸀠 )
=
where
𝑏2 𝑘𝑔 𝜀1 − 𝑎𝑏𝑘𝑔 𝜀2 + 2𝜀3
(2 + 𝑏2 𝑘𝑔2 )
5/2
.
(29)
2
𝜀1 = −𝑏3 𝑘𝑔 𝑘𝑔󸀠 + 𝑎 (2 + (𝑏𝑘𝑔 ) ) ,
2
𝜀2 = −𝑎𝑏
𝑘𝑔 𝑘𝑔󸀠
+ (𝑏 −
𝑏𝑘𝑔2 ) (2
2
+ (𝑏𝑘𝑔 ) ) ,
(27)
2
𝜀3 = −𝑏3 𝑘𝑔2 𝑘𝑔󸀠 + (𝑎𝑘𝑔 + 𝑏𝑘𝑔󸀠 ) (2 + (𝑏𝑘𝑔 ) ) .
𝑎
[
√3
[
[
[
[
𝑏
[
𝛽
[ 󵄨
󵄨
2
[𝑇𝛽 ] = [
[ √ 󵄨󵄨󵄨(𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3󵄨󵄨󵄨
󵄨󵄨
󵄨
[
󵄨
[ 𝜉𝛽 ] [
[
[
− (𝑎2 − 3) 𝑘𝑔 + 𝑎𝑐
[
[
[ 󵄨󵄨
󵄨󵄨
2
Theorem 9. Let 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 be a regular unit speed
curve lying fully in 𝐻02 with the Sabban frame {𝛼, 𝑇, 𝜉} and
the geodesic curvature 𝑘𝑔 . If 𝛽 : 𝐼 ⊂ R 󳨃→ 𝐻02 is 𝛼𝑇𝜉pseudohyperbolical Smarandache curve of 𝛼, then its frame
{𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by
𝑏
√3
𝑐
√3
𝑎 − 𝑐𝑘𝑔
󵄨
󵄨
2
√ 󵄨󵄨󵄨󵄨(𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3󵄨󵄨󵄨󵄨
󵄨
󵄨
𝑏𝑐 − 𝑎𝑏𝑘𝑔
]
]
]
]
𝑏𝑘𝑔
]
]
𝛼
󵄨󵄨 ]
󵄨󵄨
2
] [𝑇 ]
2
√ 󵄨󵄨󵄨(𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔 + 3󵄨󵄨󵄨 ]
󵄨]
󵄨
] [𝜉]
]
2
]
3 + 𝑐 − 𝑎𝑐𝑘𝑔
]
]
󵄨󵄨 ]
󵄨󵄨
2
(30)
󵄨
󵄨
2
√ 󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨 √ 󵄨󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨󵄨 √ 󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨
󵄨
󵄨
󵄨]
󵄨
󵄨
[ 󵄨
𝜀2 = − ((𝑎𝑘𝑔 − 𝑐) 𝑎𝑘𝑔󸀠 − 3𝑘𝑔󸀠 𝑘𝑔 ) (𝑎 − 𝑐𝑘𝑔 )
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
2
+ ((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3) (𝑏 − 𝑐𝑘𝑔󸀠 − 𝑏𝑘𝑔2 ) ,
𝑘𝑔𝛽 = (− (𝑎𝑐 − (𝑎2 − 3) 𝑘𝑔 ) 𝜀1
+ (𝑏𝑐 − 𝑎𝑏𝑘𝑔 ) 𝜀2 + (3 + 𝑐2 − 𝑎𝑐𝑘𝑔 ) 𝜀3 )
2
× (((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3)
5/2 −1
) ,
𝑎, 𝑏, 𝑐 ∈ R0 ,
where
𝜀1 = − ((𝑎𝑘𝑔 −
𝑐) 𝑎𝑘𝑔󸀠
2
−
3𝑘𝑔󸀠 𝑘𝑔 ) 𝑏
+ ((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3) (𝑎 − 𝑐𝑘𝑔 ) ,
𝜀3 = − ((𝑎𝑘𝑔 − 𝑐) 𝑎𝑘𝑔󸀠 − 3𝑘𝑔󸀠 𝑘𝑔 ) 𝑏𝑘𝑔
(31)
2
+ ((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3) ((𝑎 − 𝑐𝑘𝑔 ) 𝑘𝑔 + 𝑏𝑘𝑔󸀠 )
(32)
and 𝑎2 − 𝑏2 − 𝑐2 = 3.
Proof. Differentiating (8) with respect to 𝑠 and by using (4),
we find
𝑑𝛽 𝑑𝑠∗
1
𝛽󸀠 (𝑠) = ∗
(𝑏𝛼 + (𝑎 − 𝑐𝑘𝑔 ) 𝑇 + 𝑏𝑘𝑔 𝜉) , (33)
=
√3
𝑑𝑠 𝑑𝑠
International Journal of Mathematics and Mathematical Sciences
5
and thus
𝑇𝛽
𝑑𝑠∗
1
(𝑏𝛼 + (𝑎 − 𝑐𝑘𝑔 ) 𝑇 + 𝑏𝑘𝑔 𝜉) ,
=
√3
𝑑𝑠
(34)
where
󵄨󵄨
󵄨󵄨
2
2
󵄨
󵄨
𝑑𝑠∗ √ 󵄨󵄨󵄨(𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔 + 3󵄨󵄨󵄨
(35)
=
.
𝑑𝑠
3
±
The
geodesic
curvature
𝑘𝑔 ≠ (𝑎𝑐
√𝑎2 𝑐2 − (𝑎2 − 3)(𝑐2 + 3))/(𝑎2 − 3) for all 𝑠, since 𝛽 = 𝛽(𝑠∗ ) is
a unit speed regular curve in R31 .
Therefore, the unit spacelike tangent vector of the curve
𝛽 is given by
1
(𝑏𝛼 + (𝑎 − 𝑐𝑘𝑔 ) 𝑇 + 𝑏𝑘𝑔 𝜉) .
𝑇𝛽 =
󵄨󵄨
󵄨
2
√ 󵄨󵄨󵄨(𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3󵄨󵄨󵄨󵄨
󵄨
󵄨
(36)
Differentiating (36) with respect to 𝑠 and from (4) and
(35), it follows that
𝑇𝛽󸀠 =
√3
2
((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3)
2
(𝜀1 𝛼 + 𝜀2 𝑇 + 𝜀3 𝜉) ,
𝑘𝑔𝛽 = det (𝛽, 𝑇𝛽 , 𝑇𝛽󸀠 )
= (− (𝑎𝑐 − (𝑎2 − 3) 𝑘𝑔 ) 𝜀1 + (𝑏𝑐 − 𝑎𝑏𝑘𝑔 ) 𝜀2
𝜀1 = − ((𝑎𝑘𝑔 − 𝑐) 𝑎𝑘𝑔󸀠 − 3𝑘𝑔󸀠 𝑘𝑔 ) 𝑏
+ ((𝑎𝑘𝑔 − 𝑐) −
3𝑘𝑔2
+ ((𝑎𝑘𝑔 − 𝑐) −
3𝑘𝑔2
+ 3) (𝑎 − 𝑐𝑘𝑔 ) ,
+ 3) (𝑏 −
𝑐𝑘𝑔󸀠
−
(40)
+ (3 + 𝑐2 − 𝑎𝑐𝑘𝑔 ) 𝜀3 )
2
× (((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3)
𝜀2 = − ((𝑎𝑘𝑔 − 𝑐) 𝑎𝑘𝑔󸀠 − 3𝑘𝑔󸀠 𝑘𝑔 ) (𝑎 − 𝑐𝑘𝑔 )
2
Therefore, the geodesic curvature 𝑘𝑔𝛽 of the curve 𝛽 =
𝛽(𝑠 ) is given by
∗
(37)
where
2
Figure 1: The curve 𝛼 on 𝐻02 (1).
𝑏𝑘𝑔2 ) ,
5/2 −1
) .
(38)
Corollary 10. If 𝛼 : 𝐼 ⊂ R 󳨃→ 𝐻02 is a geodesic curve on 𝐻02 in
Minkowski 3-space E31 , then
𝜀3 = − ((𝑎𝑘𝑔 − 𝑐) 𝑎𝑘𝑔󸀠 − 3𝑘𝑔󸀠 𝑘𝑔 ) 𝑏𝑘𝑔
2
+ ((𝑎𝑘𝑔 − 𝑐) − 3𝑘𝑔2 + 3) ((𝑎 − 𝑐𝑘𝑔 ) 𝑘𝑔 + 𝑏𝑘𝑔󸀠 ) .
On the other hand, from (8) and (36) it can be easily seen
that
𝜉𝛽 = 𝛽 × 𝑇𝛽
=
− (𝑎2 − 3) 𝑘𝑔 + 𝑎𝑐
󵄨
󵄨
2
√ 󵄨󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨󵄨
󵄨
󵄨
+
+
(2) 𝛼𝜉-pseudohyperbolic and 𝛼𝑇𝜉-pseudohyperbolic Smarandache curves have constant geodesic curvatures on
𝐻02 .
𝛼
𝑏𝑐 − 𝑎𝑏𝑘𝑔
󵄨
󵄨
2
√ 󵄨󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨󵄨
󵄨
󵄨
3 + 𝑐2 − 𝑎𝑐𝑘𝑔
󵄨
󵄨
2
√ 󵄨󵄨󵄨󵄨3(𝑎𝑘𝑔 − 𝑐) − 9𝑘𝑔2 + 9󵄨󵄨󵄨󵄨
󵄨
󵄨
𝑇
4. Examples
𝜉.
Example 1. Let 𝛼 be a unit speed curve lying in pseudohyperbolic space 𝐻02 (1) in the Minkowski 3-space E31 with
parameter equation (see Figure 1)
(39)
Hence 𝜉𝛽 is a unit spacelike vector.
(1) 𝛼𝑇-pseudohyperbolic Smarandache curve is also geodesic on 𝐻02 ;
𝛼 (𝑠) = (
𝑠2
𝑠2
+ 1, , 𝑠) .
2
2
(41)
6
International Journal of Mathematics and Mathematical Sciences
Figure 2: The 𝛼𝜉-pseudohyperbolical Smarandache curve 𝛽 and the
curve 𝛼 on 𝐻02 (1).
Figure 4: The 𝛼𝑇𝜉-pseudohyperbolical Smarandache curve 𝛽 and
the curve 𝛼 on 𝐻02 (1).
From Theorem 7, its frame {𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by
√2 0 1
𝛽
𝛼
[𝑇𝛽 ] = [ 0 1 0 ] [𝑇] ,
[ 𝜉𝛽 ] [ −1 0 √2] [ 𝜉 ]
(45)
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
𝑘𝑔𝛽 = −1.
(46)
Case 2. If we take 𝑎 = √3, 𝑏 = −1, then from (7) the
𝛼𝑇-pseudohyperbolical Smarandache curve is given by (see
Figure 3)
Figure 3: The 𝛼𝑇-pseudohyperbolical Smarandache curve 𝛽 and the
curve 𝛼 on 𝐻02 (1).
The orthonormal Sabban frame {𝛼(𝑠), 𝑇(𝑠), 𝜉(𝑠)} along the
curve 𝛼 is given by
𝑠2
𝑠2
𝛼 (𝑠) = ( + 1, , 𝑠) ,
2
2
󸀠
𝑇 (𝑠) = 𝛼 (𝑠) = (𝑠, 𝑠, 1) ,
𝜉 (𝑠) = 𝛼 (𝑠) × 𝑇 (𝑠) = (
(42)
𝑠2 𝑠2
, − 1, 𝑠) .
2 2
𝛽 (𝑠∗ (𝑠)) =
√2
(√3𝑠2 − 2𝑠 + 2√3, √3𝑠2 − 2𝑠, 2√3𝑠 − 2) .
4
(47)
According to Theorem 8, its frame {𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by
√2
√6
−
0 ]
[
2
2
𝛽
𝛼
[ √
√3 ]
] [𝑇]
3
[𝑇𝛽 ] = [
,
]
[−
1
] 𝜉
[ 3
3
𝜉
]
[
[ 𝛽]
√2 √6 [ ]
√6
−
[ 6
2
3 ]
(48)
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
In particular, the geodesic curvature 𝑘𝑔 of the curve 𝛼 has the
form
𝑘𝑔𝛽 = −1.
𝑘𝑔 (𝑠) = −1.
Case 3. If 𝑎 = 3, 𝑏 = √3, and 𝑐 = √3, then from (7) the 𝛼𝑇𝜉pseudohyperbolical Smarandache curve 𝛽 is given by (see
Figure 4)
(43)
Case 1. If we take 𝑎 = 2, 𝑏 = √2, then from (6) the 𝛼𝜉pseudohyperbolical Smarandache curve 𝛽 is given by (see
Figure 2)
𝛽 (𝑠∗ (𝑠)) =
1 √
(( 2 + 1) 𝑠2 + 2√2,
2
2
(√2 + 1) 𝑠 − 2, (2√2 + 2) 𝑠) .
(44)
𝛽 (𝑠∗ (𝑠)) =
(49)
1 √
(( 3 + 1) 𝑠2 + 2𝑠 + 2√3,
2
(√3 + 1) 𝑠2 +2𝑠 − 2, (2√3 + 2) 𝑠 + 2) .
(50)
International Journal of Mathematics and Mathematical Sciences
Figure 5: Pseudohyperbolical Smarandache curves of 𝛼 and the
curve 𝛼 on 𝐻02 (1).
By using Theorem 9, it follows that the frame {𝛽, 𝑇𝛽 , 𝜉𝛽 } is
given by (see Figure 4)
√3 1
1
]
[
]
[
]
[√
𝛽
[ 3−1
1 − √3 ] 𝛼
[𝑇𝛽 ] = [
] [𝑇] ,
1
[ 2
2 ]
] 𝜉
[
𝜉
[ 𝛽] [
][ ]
[ √3 + 1
√3 + 1 ]
1
[ 2
2 ]
(51)
and the corresponding geodesic curvature 𝑘𝑔𝛽 reads
𝑘𝑔𝛽 = −6 (2 + √3) (3 + √3)
1/2
.
(52)
Also, Pseudohyperbolical Smarandache curves of 𝛼 and
the curve𝛼 on 𝐻02 (1) with Figure 5 are shown.
Acknowledgment
The authors would like to thank the referees for their helpful
suggestions and comments which significantly improved the
first version of the paper.
References
[1] C. Ashbacher, “Smarandache geometries,” Smarandache
Notions Journal, vol. 8, no. 1–3, pp. 212–215, 1997.
[2] M. Turgut and S. Yılmaz, “Smarandache curves in Minkowski
space-time,” International Journal of Mathematical Combinatorics, vol. 3, pp. 51–55, 2008.
[3] A. T. Ali, “Special smarandache curves in the euclidean space,”
International Journal of Mathematical Combinatorics, vol. 2, pp.
30–36, 2010.
[4] K. Taşköprü and M. Tosun, “Smarandache curves on 𝑆2 ,”
Boletim da Sociedade Paranaense de Matemática, vol. 32, no. 1,
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[5] N. Bayrak, O. Bektas, and S. Yuce, “Special Smarandache a
Curves in E31 ,” In press, http://arxiv.org/abs/1204.5656.
7
[6] T. Körpinar and E. Turhan, “A new approach on timelike biharmonic slant helices according to Bishop frame in Lorentzian
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[7] T. Körpinar and E. Turhan, “Characterization of Smarandache 𝑀1 𝑀2 -curves of spacelike biharmonic 𝐵-slant helices
according to Bishop frame in E(1, 1),” Advanced Modeling and
Optimization, vol. 14, no. 2, pp. 327–333, 2012.
[8] Y. Yayli and E. Ziplar, “Frenet-serret motion and ruled surfaces
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[9] M. Petrović-Torgašev and E. Šućurović, “Some characterizations of the spacelike, the timelike and the null curves on
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York, NY, USA, 1983.
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