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September 4, 2017 14:57 WSPC/S0219-8878
IJGMMP-J043
1750140
International Journal of Geometric Methods in Modern Physics
Vol. 14, No. 10 (2017) 1750140 (10 pages)
c World Scientific Publishing Company
DOI: 10.1142/S0219887817501407
Characterizations of special time-like curves
in Lorentzian plane L2
Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com
by TUFTS UNIVERSITY on 10/28/17. For personal use only.
Abdullah Mağden∗,§ , Süha Yılmaz†,¶ and Yasin Ünlütürk‡,
∗Department
†Buca
of Mathematics, Atatürk University
25400 Erzurum, Turkey
Faculty of Education, Dokuz Eylül University
35150 Buca-Izmir, Turkey
‡Department
of Mathematics, Kırklareli University
39100 Kırklareli, Turkey
§[email protected][email protected]
[email protected]
Received 28 March 2017
Accepted 17 May 2017
Published 4 July 2017
In this paper, we first obtain the differential equation characterizing position vector
of time-like curve in Lorentzian plane L2 . Then we study the special curves such as
Smarandache curves, circular indicatrices, and curves of constant breadth in Lorentzian
plane L2 . We give some characterizations of these special curves in L2 .
Keywords: Lorentzian plane; time-like curve; circular indicatrices; Smarandache curves;
curves of constant breadth.
Mathematics Subject Classification 2010: 53A35, 53A40, 53B25
1. Introduction
There are lots of interesting and important problems in the theory of curves at
differential geometry. One of the interesting problems is the problem of characterization of a regular curve in the theory of curves in the Euclidean and Minkowski
spaces, see [1, 2]. Also, there are special curves which are obtained under some
definitions such as Smarandache curves, spherical indicatrices, curves of constant
breadth, etc.
Smarandache curves are regular curves whose position vectors are obtained by
Frénet frame vectors on another regular curve. These curves were examined by
Turgut and Yılmaz in [3]. Then many researches occurred about the different characterizations of Smarandache curves in Euclidean and Minkowsi spaces, see [4, 5].
Corresponding
author.
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A. Mağden, S. Yılmaz & Y. Ünlütürk
Spherical indicatrix is the locus of a point whose position vector is equal to
one of the frame field vectors such as the unit tangent T, the principal normal
vector N, and the principal binormal vector B at any point of a given curve in the
space. If this vector is chosen as the unit tangent vector field T of a given curve
in the space, then the spherical indicatrix is said to be tangent spherical indicatrix.
The principal normal and binormal spherical indicatrices are also obtained by using
the position vectors of the principal normal vector N, and the principal binormal
vector B, respectively. Spherical indicatrices as special curves have been studied
in many works, see [6, 7]. As you know, spherical indicatrix turns into circular
indicatrix in the plane. Curves of constant breadth as another special curve, were
first introduced by Euler [8]. Then in chronological order, it was studied in [9–12].
Our motivation was to see the corresponding results of the special curves mentioned above in Lorentzian plane L2 . As much as we look at the classical differential
geometry literature of the works in Lorentzian plane L2 , the works were rare, see
[13–16]. First, we obtain the differential equation characterizing position vector of
time-like curve in Lorentzian plane L2 . Then we study the special curves mentioned
above in Lorentzian plane L2 . We give some characterizations of these special curves
in L2 .
2. Preliminaries
Let L2 be the Lorentzian plane with metric
g = dx21 − dx22 ,
(1)
2
where x1 and x2 are rectangular coordinate system. A vector r of L is said to be
space-like if g(r, r) > 0, or r = 0, time-like if g(r, r) < 0 and null if g(r, r) = 0 for
r = 0 [6].
A curve x is a smooth mapping
x : I → L2 ,
from an open interval I onto L2 . Let s be an arbitrary parameter of α, then we
denote the orthogonal coordinate representation of α as α = (α1 (s), α2 (s)) and also
the vector
dα1 dα2
dα
=
,
=t
(2)
ds
ds ds
is called the tangent vector field of the curve α = α(s). If tangent vector field of α(s)
is a space-like, time-like or null then, the curve α(s) is called space-like, time-like
or null, respectively [13].
In the rest of the paper, we shall consider time-like curves. While the tangent
vector field T is time-like vector so N is space-like one. We can have the arc-length
parameter s and the Frénet formula as
0 κ
T
T
=
·
,
(3)
N
κ 0
N
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Characterizations of special time-like curves
where κ = κ(s) is the curvature of the unit speed curve α = α(s) [15]. We should
also indicate that the space-like vector field N is the normal vector field of the curve
α(s), the tangent vector field T is a time-like vector, and also T, N = 0. Given
φ(s) is the slope angle of the curve, then as in [13] we have
dφ
= κ(s).
ds
(4)
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3. Position Vector of a Curve in L2
Let α = α(s) be an unit speed time-like curve on the plane L2 . Then we can write
the position vector of α(s) with respect to Frénet frame as
α = α(s) = λ1 T + λ2 N,
(5)
where λ1 and λ2 are arbitrary functions of s. Differentiating (5) and using Frénet
equtaions we have a system of ordinary differential equations as follows:

dλ1



 ds + λ2 κ − 1 = 0,
(6)

dλ2



+ λ1 κ = 0.
ds
Using (6)1 in (6)2 we obtain
d 1
dλ1
+ 1 + λ1 κ = 0.
−
ds κ
ds
(7)
The differential equation of second order, according
s to λ1 , is a characterization
for the curve α = α(s). Using change of variable θ = 0 κ ds in (7), we arrive
d2 λ1
dρ
,
− λ1 =
2
dθ
dθ
(8)
where κ = 1ρ .
By the method of variation of parameters and solution of (8), we have
θ
θ
θ
θ
−θ
−θ
λ1 = e A −
κe dθ + e
κe dθ ,
B+
0
0
where A, B ∈ R. Rewriting the change of variable, we get
θ
Rθ
Rθ
κ
ds
θ
−
κ
ds
λ1 = e 0
κe dθ + e 0
A−
B+
0
θ
κe
−θ
dθ .
(9)
0
Denoting differentiation of Eq. (9) as
dλ1
ds
= ξ(s), and using (6), we have
1
[ξ(s) − 1].
κ
Hence, we give the following theorem:
λ2 =
(10)
Theorem 3.1. Let α = α(s) be an arbitrary unit speed time-like curve in
Lorentzian plane L2 . Position vector of the curve α = α(s) with respect to Frénet
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A. Mağden, S. Yılmaz & Y. Ünlütürk
frame can be composed by the following equation:
θ
θ
1
θ
θ
−θ
−θ
[ξ(s) − 1] N.
α = α(s) = e A −
κe dθ + e
κe dθ
T+
B+
κ
0
0
Int. J. Geom. Methods Mod. Phys. 2017.14. Downloaded from www.worldscientific.com
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3.1. Differential equation of the third-order characterizing
Lorentzian plane curves
Theorem 3.2. Let α = α(s) be an arbitrary unit speed time-like curve in Lorentzian plane L2 . Position vector and curvature of it is to satisfy the differential equations of third order
d 1 d2 α
dα
= 0.
−κ
ds κ ds2
ds
Proof. Let α = α(s) be an arbitrary unit speed time-like curve in Lorentzian plane
L2 . Then Frénet derivative formula holds (3)1 in (3)2 , we easily have
d 1 dT
− κT = 0.
(11)
ds κ ds
Let
dα
ds
= t = α̇. So, expression of (11) can be written as follows:
dα
d 1 d2 α
= 0,
−κ
ds κ ds2
ds
(12)
formula (12) completes the proof.
Here, we try to solve the differential equation (11) with respect to t. We know
that
t = (t1 , t2 ) = (α̇1 , α̇2 ),
s
using the change of variable θ = 0 κ ds in (12) we get
d2 t
− θ = 0,
dθ2
(13)
or in parametric form it is
d2 t1
d2 t2
−
θ
=
0,
− θ = 0,
dθ2
dθ2
as the solution of (14), we obtain
t1 = ψ1 eθ + ψ2 e−θ ,
t2 = ψ3 eθ + ψ4 e−θ ,
(14)
(15)
where ψi ∈ R for 1 ≤ i ≤ 4.
4. Special Curves in L2
In this section, we will study some special curves such as Smarandache curves,
circular indicatrices, and curves of constant breadth in Lorentzian plane L2 .
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Characterizations of special time-like curves
4.1. Smarandache curves
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A regular curve in Lorentzian plane L2 whose position vector is composed of Frénet
frame vectors on another regular curve is called a Smarandache curve. In this
section, we will study TN –Smarandache curve as the only Smarandache curve of
Lorentzian plane L2 .
Definition 4.1 (TN–Smarandache curves). Let α = α(s) be a unit speed timelike curve in L2 and {T α , N α } be its moving Frénet frame, here T α , N α are the
tangent and principal normal vectors of the Smarandache curve of the curve α. The
curve α = α(s) is said to be TN –Smarandache curve whose form is
−1
(16)
β(s∗ ) = √ (T α − N α ).
2
We can investigate Frénet invariants of TN –Smarandache curves according to α =
α(s). Differentiating (16) with respect to s gives us
−1
dβ ds∗
= √ (καN α − κα T α ).
∗
ds ds
2
Rearranging of this expression we get
−1
ds∗
= √ (καN α − κα T α ),
Tβ
ds
2
by (18) we have
β̇ =
(17)
(18)
ds∗
= κα ,
(19)
ds
hence using (18) and (19) we find the tangent vector of the curve β as follows:
(T α − N α )
√
2
and differentiating (20) with respect to s, we have
Tβ =
dTβ ds∗
(−κα T α + καN α )
√
=
.
ds∗ ds
2
(20)
(21)
Substituting (19) in (21), we obtain
Tβ =
The curvature of the curve β is
Tβ = κβ =
−(T α − N α )
√
.
2
(T α )2 − (N α )2
= 0.
2
4.2. Circular indicatrices
Circular indicatrix is the locus of a point whose position vector is equal to the
unit tangent T or the principal normal vector N at any point of a given curve.
We will characterize tangent and normal circular indicatrices of time-like curves in
Lorentzian plane L2 .
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A. Mağden, S. Yılmaz & Y. Ünlütürk
4.2.1. The tangent circular indicatrices of time-like curves in L2
Let ε = ε(s) be a time-like curve in Lorentzian plane L2 . If we translate the first
vector field of Frénet frame to the center of the unit Lorentzian circle S 1 , then we
have the tangent circular indicatrix δ = δ(sδ ) in Lorentzian plane L2 .
Here, we shall denote differentiation according to s by a dash and differentiation
according to sδ by a dot. By using the Frénet frame, we obtain the tangent vector
of δ = δ(sδ ) as
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δ =
dsδ
dδ dsδ
= Tδ
= κN,
dsδ ds
ds
(22)
dsδ
= κ(s).
ds
(23)
where
Tδ = N
and
From (22), we have
Ṫδ = κN + κ2 T
and we also arrive at
κδ = Ṫδ =
(−(κ )2 + κ4 ).
(24)
Therefore, we have the principal normal of the curve δ = δ(sδ ) as
−κ2 T + κN
.
Nδ = √
−κ2 + κ4
(25)
4.2.2. The principal normal circular indicatrices of time-like curves in L2
Let ε = ε(s) be a time-like curve in Lorentzian plane L2 . If we translate the second
vector field of Frénet frame to the center of the unit Lorentzian circle S 1 , then we
have the principal normal circular indicatrix φ = φ(sφ ) in Lorentzian plane L2 .
We will follow the procedure used to determine the tangent circular indicatrix
for the principal normal circular indicatrices of time-like curves in L2 . Let us first
define
dφ dsφ
φ =
= κT,
dsφ ds
where
Tφ = N
and
dsφ
ds
(26)
= κ(s).
Differentiating (26), we obtain
Tφ = Ṫφ
dsφ
= −κN,
ds
(27)
or in other words,
Ṫφ = N.
1750140-6
(28)
September 4, 2017 14:57 WSPC/S0219-8878
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Characterizations of special time-like curves
Using (27) and (28) we have the first curvature and the principal normal vector
of the principal normal circular indicatrix φ = φ(sφ ) as
κφ = Ṫφ = 1 and Nφ = N.
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4.3. Curves of constant breadth
Let ϕ = ϕ(s) and ϕ∗ = ϕ∗ (s) be simple closed time-like curve in Lorentzian plane
L2 . These curves will be denoted by C and C ∗ . The normal plane at every point
p on the curve meets the curve at a single point q other than p. We call the point
q as the opposite point of p. We consider curves in the class Γ as in Fujivara [17]
having parallel tangents T and T ∗ in opposite directions at the opposite points ϕ
and ϕ∗ of the curve.
A simple closed curve of constant breadth having parallel tangents in opposite
directions at opposite points can be represented with respect to Frénet frame by
the following equation:
ϕ∗ = ϕ + m1 T + m2 N,
(29)
∗
where mi (s), 1 ≤ i ≤ 2 are arbitrary functions of s and ϕ and ϕ are opposite
points.
The vector
d = ϕ∗ − ϕ
is called “the distance vector” between the opposite points of C and C ∗ . Differentiating (29), and considering Frénet derivative equations (3), we have
dm1
dϕ∗
ds∗
dm2
= T∗
=T +
T + m1 κN +
N + m2 κT.
ds
ds
ds
ds
Since
dϕ∗
dϕ
= T and
= T ∗,
ds
ds∗
and using Frénet derivative formulas, we get
∗
dm1
dm2
∗ ds
= 1+
+ m2 κ T + m1 κ +
T
N.
ds
ds
ds
(30)
Since the definition of curve of constant breadth, there is the relation
T ∗ = −T,
(31)
between the tangent vectors of the curves, and also using (31) in (30), we obtain
ds∗
dm1
dm2
= −m2 κ −
− 1 and m1 κ +
= 0.
(32)
ds
ds
ds
Let θ be the angle between the tangent vector T at a point α(s) of an oval and
a fixed direction, then we have
1
ds
=ρ=
dθ
κ
and
ds∗
1
= ρ∗ = ∗ .
dθ
κ
1750140-7
(33)
September 4, 2017 14:57 WSPC/S0219-8878
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1750140
A. Mağden, S. Yılmaz & Y. Ünlütürk
Substituting (33) into (32), Eq. (32) turns into

dm1

∗


−m2 − dθ = ρ + ρ = f (θ),
(34)

dm2



= −m1 ,
dθ
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eliminating m1 in (34) we obtain the linear differential equation of the second
order as
d2 m2
− m2 = f (θ),
dθ2
where f (θ) = ρ + ρ∗ .
By general solution of Eq. (35) we find
m2 = e
θ
−θ
t
f (t)e dt
0
+ ε2 − e
θ
(35)
θ
f (t)e
−t
dt
0
+ ε1 .
2
Using m1 = − dm
dθ in (34) we obtain the value of m1 as
θ
θ
f (t) cos t dt + ε2 − e−θ
f (t) sin t dt + ε1 .
m1 = −eθ
0
0
Hence using (29) the position vector of the curve ϕ
∗ is given as follows:
θ
θ
∗
θ
−θ
ϕ = ϕ + −e
f (t) cos t dt + ε2 − e
f (t) sin t dt + ε1
T
+ e
−θ
0
θ
t
f (t)e dt
0
+ ε2 − e
θ
0
θ
f (t)e
−t
dt
0
+ ε1 N.
If the distance between the opposite points of C and C ∗ is constant, then we
can write that
ϕ∗ − ϕ
= −m21 + m22 = const.
(36)
and differentiating (36) we have
dm1
dm2
+ m2
=0
(37)
dθ
dθ
and also taking the system (34) and (37) together into consideration, we obtain
dm1
m1
+ m2 = 0,
(38)
dθ
−m1
so we arrive at
dm1
= −m2 .
dθ
Due to the cases in (39), we will study the following conditions:
m1 = 0
or
1750140-8
(39)
September 4, 2017 14:57 WSPC/S0219-8878
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Characterizations of special time-like curves
Case 1: If
m1 = 0,
then from (34) we find that
f (θ) = const. and m2 = const.
If
dm1
= −m2 ,
dθ
m1 = 0 = const. and also
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then we obtain
m2 = 0.
If
m1 = k1
(k1 ∈ R),
then Eq. (29) turns into
ϕ∗ = ϕ + k1 T.
(40)
Case 2: If
dm1
= −m2 ,
dθ
then from (34) we have
f (θ) = 0
and m1 = −
0
θ
m2 dθ.
If
dm1
= −m2 = 0 = k2 = const.,
dθ
then from (34) we obtain
f (θ) = 0
and m1 = 0.
Hence, Eq. (29) becomes as follows:
ϕ∗ = ϕ + k2 N.
(41)
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Math. Combin. 3 (2008) 51–55.
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