on mannheim partner curves in dual lorentzian space

on mannheim partner curves in dual lorentzian space

Hacettepe Journal of Mathematics and Statistics
Volume 40 (5) (2011), 649 – 661
ON MANNHEIM PARTNER CURVES
IN DUAL LORENTZIAN SPACE
Sıddıka Özkaldı∗†‡, Kazım İlarslan§ and Yusuf Yaylı∗
Received 28 : 12 : 2009 : Accepted 09 : 03 : 2011
Abstract
In this paper we define non-null Mannheim partner curves in three
dimensional dual Lorentzian space D31 , and obtain necessary and sufficient conditions for the existence of non-null Mannheim partner curves
in dual Lorentzian space D31 .
Keywords:
curve.
Mannheim partner curve, Dual Lorentzian space, Dual Lorentzian space
2000 AMS Classification: 53 A 25, 53 A 40, 53 B 30.
1. Introduction
In the differential geometry of a regular curve in Euclidean 3-space E3 , it is well
known that one of the important problems is the characterization of a regular curve.
The curvature functions k1 (curvature κ) and k2 (torsion τ ) of a regular curve play an
important role in determining the shape and size of the curve [4, 8]. For example: If
k1 = k2 = 0, then the curve is a geodesic. If k1 6= 0 (constant) k2 = 0, then the curve is
a circle with radius 1/k1 . If k1 6= 0 (constant) and k2 6= 0 (constant), then the curve is
a helix in the space, etc.
Another route to the classification and characterization of curves is the relationship
between the Frenet vectors of the curves. For example Saint Venant, in 1845, proposed
the question whether upon the surface generated by the principal normal of a curve, a
second curve can exist which has for its principal normal the principal normal of the
given curve. This question was answered by Bertrand in 1850, when he showed that a
necessary and sufficient condition for the existence of such a second curve is that a linear
∗
Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, Ankara,
Turkey. E-mail:
(S. Özkaldı) [email protected] (Y. Yaylı) [email protected]
†
Current Address: Department of Mathematics, Faculty of Science and Arts, Bilecik University, Bilecik, Turkey. E-mail: [email protected]
‡
Corresponding Author.
§
Department of Mathematics, Faculty of Science and Arts, Kırıkkale University, Kırıkkale,
Turkey. E-mail: [email protected]
650
S. Özkaldı, K. İlarslan, Y. Yaylı
relationship with constant coefficients shall exist between the first and second curvatures
of the given original curve. The pairs of curves of this kind have been called Conjugate
Bertrand Curves, or more commonly Bertrand Curves [4, 8, 13]. There are many works
related with Bertrand curves in Euclidean space and Minkowski space. Associated curves
of another kind, called Mannheim curves and Mannheim partner curves occur if there
exists a relationship between the space curves α and β such that, at the corresponding
points of the curves, the principal normal lines of α coincide with the binormal lines of β.
Then, α is called a Mannheim curve, and β the Mannheim partner curve of α. Mannheim
partner curves were studied by Liu and Wang [9] in Euclidean 3-space and in Minkowski
3-space.
Dual numbers had been introduced by William Kingdon Clifford (1845–1879) as a
tool for his geometrical investigations. After him Eduard Study (1862–1930) used dual
numbers and dual vectors in his research on line geometry and kinematics. He devoted
special attention to the representation of oriented lines by dual unit vectors and defined
the famous mapping: The set of oriented lines in a Euclidean three-dimensional space
E3 is one to one correspondence with the points of a dual space D3 of triples of dual
numbers [5].
Differential Geometric properties of regular curves in a dual space D3 , as well as
in a dual Lorentzian space D31 have been studied in many papers. For example see
[16, 3, 7, 17, 18, 1, 2, 11, 15, 14]. Mannheim curves in a dual space were studied by the
authors in [12]
In this paper we study non-null Mannheim partner curves in a dual Lorentzian space
D31 .
2. Preliminary
Dual numbers were introduced by W. K. Clifford (1845–1879) as a tool for his geometrical investigations. After him, E. Study used dual numbers and dual vectors in his
research on the geometry of lines and kinematics. He devoted special attention to the
representation of directed lines by dual unit vectors, and defined the mapping that is
known by his name. Namely, there exists one-to-one correspondence between the points
of dual unit sphere S2 and the directed lines in R3 [5].
If we take the Minkowski 3-space R31 instead of R3 the E. Study (1862–1930) mapping can be stated as follows: The dual timelike and spacelike unit vectors of the dual
hyperbolic and Lorentzian unit spheres H20 and S21 in the dual Lorentzian space D31 are in
one-to-one correspondence with the directed timelike and spacelike lines in R31 , respectively. Then a differentiable curve on H20 corresponds to a timelike ruled surface at R31 .
Similarly, the timelike (resp. spacelike) curve on S21 corresponds to any spacelike (resp.
timelike) ruled surface in R31 .
We will survey briefly various fundamental concepts and properties in Lorentzian
space. We refer mainly to O’Neill [10].
Let R31 be the 3-dimensional Lorentzian space with Lorentzian metric h · , · i : −dx21 +
dx22 +dx23 . It is known that in R31 there are three categories of curves and vectors, namely,
→
spacelike, timelike and null, depending on their causal character. Let −
x be a tangent
→
−
→
−
→
−
→
−
→
vector of Lorentzian space. Then x is said to be spacelike if h x , x i > 0 or −
x = 0,
→
−
→
→
→
→
→
timelike if h−
x,−
x i < 0, null (lightlike) if h−
x,−
x i = 0 and −
x 6= 0 .
Let α : I ⊂ R −→ R31 be a regular curve in R31 . Then, the curve α is spacelike if all
its velocity vectors are spacelike. Similarly, it is called timelike and a null curve if all its
velocity vectors are timelike and null vectors, respectively.
On Mannheim Partner Curves in Dual Lorentzian Space
651
A dual number x
b has the form x + εx∗ with the properties
ε 6= 0,
0ε = ε0 = 0,
1ε = ε1 = ε,
ε2 = 0,
where x and x∗ are real numbers and ε is the dual unit (for the properties of dual
numbers, see [16]). An ordered triple of dual numbers (b
x1 , x
b2 , x
b3 ) is called a dual vector
and the set of dual vectors is denoted by
D3 = D × D × D
−
→ −
→
= x
b | x
b = (x1 + εx∗1 , x2 + εx∗2 , x3 + εx∗3 )
→ −
−
→
={x
b | x
b = (x1 , x2 , x3 ) + ε (x∗1 , x∗2 , x∗3 )}
−
−
→ → −
→
→ −
→ −
= x
b | x
b =→
x + εx ∗ , −
x , x ∗ ∈ R3
−
→ −
→
−
→ −
−
→ →
For any x
b =→
x + εx∗ , yb = −
y + εy ∗ ∈ D3 , if the Lorentzian inner product of the dual
→
−
→
−
vectors x
b and yb is defined by
−
→ −
→ −
→ −
→ → → −
→
x
b , yb := −
x,→
y +ε −
x , y ∗ + x∗ , −
y ,
then the dual space D3 together with this Lorentzian inner product is called the dual
→
−
Lorentzian space, and it is shown by D31 . A dual vector x
b in D3 is said to be spacelike,
→
−
timelike and lightlike (null) if the vector x
b is spacelike, timelike and lightlike (null),
→
−
→
−
respectively. The Lorentzian vector product of dual vectors x
b = (b
x1 , x
b2 , x
b3 ) and x
b =
3
(b
x1 , x
b2 , x
b3 ) in D1 is defined by
→ −
−
→
x
b ∧ yb := (b
x3 yb2 − x
b2 yb3 , x
b3 yb1 − x
b1 yb3 , x
b1 yb2 − x
b2 yb1 ) .
−
−
→
→ −
−
→
→
b =→
x + εx∗ is defined by
If −
x 6= 0, the norm x
b of x
r
−
→ −
→
−
→
b, x
b .
b := x
x
−
→
−
A dual vector →
x with norm 1 is called a dual unit vector. Let x
b = (b
x1 , x
b2 , x
b3 ) ∈ D31 .
Then,
i) The set
S21 =
n−
o
−
→ −
→
→ −
→
−
−
−
x
b =→
x + εx ∗ x
b = (1, 0); →
x , x∗ ∈ R31 and the vector →
x is spacelike
→
−
is called the pseudo dual sphere with center O in D31 .
ii) The set
H20 =
n−
o
−
→ −
→
→ −
→
−
−
−
x
b =→
x + εx ∗ x
b = (1, 0); →
x , x∗ ∈ R31 and the vector →
x is timelike
is called the pseudo dual hyperbolic space in D31 [15]
If all the real valued functions x1 (t) and x∗1 , 1 ≤ i ≤ 3, are differentiable, the dual
Lorentzian curve
x
b : I ⊂ R → D31
→
−
t 7→ x
b (t) = (x1 (t) + εx∗1 (t), x2 (t) + εx∗2 (t), x3 (t) + εx∗3 (t))
−
→
→
=−
x (t) + εx∗ (t)
652
S. Özkaldı, K. İlarslan, Y. Yaylı
−
→
→
in D31 is differentiable. We call the real part −
x (t) the indicatrix of x
b (t). The dual arc
→
−
length of the curve x
b (t) from t1 to t is defined as
sb :=
Zt Zt
Z t D ′ E
→ ′ →
→ ′
→ −
−
−
b (t) dt = −
x (t) dt + ε
t , x∗
= s + εs∗ ,
x
t1
t1
t1
−
→
−
where t is a unit tangent vector of →
x (t). From now on we will take the arc length s of
→
−
x (t) as the parameter instead of t.
→
→ −
−
→ −
The equalities relative to the derivatives of the dual Frenet vectors b
t,n
b, b
b throughout the dual space curve are written in the matrix form as
−
→
→ 
 −
b
t
t
0
κ
b
0 b


d −
→ 
→
−


(2.1)
=
−ε
ε
κ
b
0
τ
b
n
b
n
b ,
1 2
−
db
s −
→
→
0
−ε
ε
τ
b
0
2
3
b
b
b
b
where κ
b = κ + εκ∗ is the nowhere pure dual curvature, τb = τ + ετ ∗ is the nowhere pure
dual torsion and
D−
D−
D−
D−
→ −
→E
→E D−
→E
→ −
→ −
→ −
→E
→ −
→E
→E D−
→ −
b
t, b
t = ε1 , n
b, n
b = ε2 , b
b, b
b = ε3 , b
t,n
b = b
t, b
b = n
b, b
b = 0.
The formulae (2.1) are called the Frenet formulae of the dual curve in dual Lorentzian
n−
n−
→o n −
→o
→ −
→ −
→o
→ −
space. The planes spanned by b
t,b
b , b
t,n
b and n
b, b
b at each point of the dual
Lorentzian curve are called the rectifying plane, the osculating plane, and the normal
plane, respectively [1, 2, 11, 14].
3. Mannheim partner curves in D31
In this section, we define Mannheim partner curves in the dual space D31 and we give
characterizations for Mannheim partner curves in the same space.
3.1. Definition. Let D31 be dual Lorentzian space with the Lorentzian inner product
h·, ·i. If there exists a correspondence between the dual Lorentzian space curves α
b and
βb such that, at the corresponding points of the dual Lorentzian curves, the principal
b then α
normal lines of α
b coincides with the binormal lines of β,
b is called a dual Lorentzian
b
Mannheim curve, and βb a dual Lorentzian Mannheim partner curve of α
b. The pair {b
α, β}
is said to be a dual Lorentzian Mannheim pair.
Let α
b : x
b(b
s) be a dual Lorentzian Mannheim curve in D31 parameterized by its arc
length sb and βb : x
b1 (b
s1 ) the dual Lorentzian Mannheim
partner curve with an arc length
→ −
→
−
→
−
b
parameter b
s1 . Denote by
t (b
s), n
b (b
s), b
b (b
s) the dual Lorentzian Frenet frame field
along α
b:x
b(b
s).
In the following theorems, we give necessary and sufficient conditions for a dual space
curve to be a dual Lorentzian Mannheim curve.
→
−
→
−
→
−
3.2. Theorem. Let α
b:x
b(b
s) be a Lorentzian curve in D31 and b
t (b
s), n
b (b
s) and b
b (b
s) the
tangent, principal normal and binormal vector fields of α
b:x
b(b
s), respectively.
→
−
→
−
→
−
i) In the case when b
t (b
s) is a timelike vector, n
b (b
s) and b
b (b
s) are spacelike vectors,
then α
b:x
b(b
s) is a dual Lorentzian Mannheim curve if and only if its curvature κ
b
b τ2 − κ
b is a never pure dual
and torsion τb satisfy the formula κ
b = λ(b
b2 ), where λ
constant.
On Mannheim Partner Curves in Dual Lorentzian Space
653
→
−
→
−
→
−
ii) In the case when b
t (b
s) and n
b (b
s) are spacelike vectors, b
b (b
s) a timelike vector,
then α
b:x
b(b
s) is a dual Lorentzian Mannheim curve if and only if its curvature κ
b
b κ2 − τb2 ), where λ
b is a never pure dual
and torsion τb satisfy the formula κ
b = λ(b
constant.
→
−
→
−
→
−
iii) In the case when b
t (b
s) and b
b (b
s) are spacelike vectors, n
b (b
s) a timelike vector,
then α
b:x
b(b
s) is a dual Lorentzian Mannheim curve if and only if its curvature
b κ2 + τb2 ), where λ
b is a never pure
κ
b and torsion τb satisfy the formula κ
b = −λ(b
dual constant.
Proof. i) Let α
b : x
b(b
s) be a dual Lorentzian Mannheim curve in D31 with arc length
parameter b
s, and βb : x
b1 (b
s1 ) the dual Lorentzian Mannheim partner curve with arc length
parameter sb1 . Then by the definition we can assume that
→
b s )−
(3.1)
x
b1 (b
s) = x
b(b
s) + λ(b
n
b (b
s)
b s). By taking the derivative of (3.1) with respect to
for some never pure dual constant λ(b
s and applying the Frenet formulas we have
b
−
→
→ dλ
b−
→ b −
db
x1 (b
s)
bκ b
= 1 + λb
t +
n
b + λb
τb
b.
db
s
db
s
→
−
→
−
Since b
t1 is coincident with b
b1 in direction, we get
b s)
dλ(b
= 0.
db
s
b is a never pure dual constant. Thus we have
This means that λ
−
→
−
→
db
x1 (b
s)
bκ b
bτ b
= 1 + λb
t + λb
b.
db
s
On the other hand, we have
−
→ db
−
→
−
→
db
x1 db
s
s
bκ b
bτ b
b
t1 =
=
1 + λb
t + λb
b
.
db
s db
s1
db
s1
By taking the derivative of this equation with respect to b
s1 and applying the Frenet
formulas we obtain
→
−
→
→ db
→ db
t1 db
s
κ−
τ−
s
b db
bκ2 − λb
bτ 2 −
b db
b
b
= λ
t + κ
b + λb
n
b +λ
b
db
s db
s1
db
s
db
s
db
s1
2
−
→ db
−
→
s
bκ b
bτ b
+
1 + λb
t + λb
b
.
db
s1
From this equation we get
s
bκ2 − λb
bτ 2 db
κ + λb
b
= 0,
db
s1
b τ2 − κ
κ
b = λ(b
b2 ).
This completes the proof.
ii) Let α
b:x
b(b
s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter
b
s and β : x
b
b1 (b
s1 ) the dual Lorentzian Mannheim partner curve with arc length parameter
s1 . Then by the definition we can assume that
b
→
b s )−
(3.2)
x
b1 (b
s) = x
b(b
s) + λ(b
n
b (b
s)
654
S. Özkaldı, K. İlarslan, Y. Yaylı
b s). By taking the derivative of (3.2) with respect to
for some never pure dual constant λ(b
s and applying the Frenet formulas we have
b
−
→
→ dλ
b−
→ b −
db
x1 (b
s)
bκ b
= 1 − λb
t +
n
b + λb
τb
b.
db
s
db
s
→
−
→
−
Since b
t1 is coincident with b
b1 in direction, we get
b s)
dλ(b
= 0.
db
s
b is a never pure dual constant. Thus we have
This means that λ
−
→
−
→
db
x1 (b
s)
bκ b
bτ b
= 1 − λb
t + λb
b.
db
s
On the other hand, we have
−
→ db
−
→
−
→
db
x1 db
s
s
bκ b
bτ b
b
t1 =
=
1 − λb
t + λb
b
.
db
s db
s1
db
s1
By taking the derivative of this equation with respect to b
s1 and applying the Frenet
formulas we obtain
→
−
→
→ db
→ s
db
κ−
db
τ−
s
db
t1 db
2
2 −
b
b
b
b
b
b
= −λ
t + κ
b − λb
κ + λb
τ
n
b +λ
b
db
s db
s1
db
s
db
s
db
s1
2
−
→ db
−
→
s
bκ b
bτ b
+
1 − λb
t + λb
b
.
db
s1
From this equation we get
s
bκ2 + λb
bτ 2 db
κ − λb
b
= 0,
db
s1
b κ2 − τb2 ).
κ
b = λ(b
This completes the proof.
iii) Let α
b:x
b(b
s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter
s, and βb : x
b
b1 (b
s1 ) the dual Lorentzian Mannheim partner curve with arc length parameter
s1 . Then by the definition we can assume that
b
→
b s )−
(3.3)
x
b1 (b
s) = x
b(b
s) + λ(b
n
b (b
s)
b s). By taking the derivative of (3.3) with respect to
for some never pure dual constant λ(b
s and applying the Frenet formulas we have
b
−
→
→ dλ
b−
→ b −
db
x1 (b
s)
bκ b
= 1 + λb
t +
n
b + λb
τb
b.
db
s
db
s
→
−
→
−
Since b
t1 is coincident with b
b1 in direction, we get
b s)
dλ(b
= 0.
db
s
b is a never pure dual constant. Thus we have
This means that λ
−
→
−
→
db
x1 (b
s)
bκ b
bτ b
= 1 + λb
t + λb
b.
db
s
On the other hand, we have
−
→ db
−
→
−
→
db
x1 db
s
s
bκ b
bτ b
b
t1 =
=
1 + λb
t + λb
b
.
db
s db
s1
db
s1
On Mannheim Partner Curves in Dual Lorentzian Space
655
By taking the derivative of this equation with respect to b
s1 and applying the Frenet
formulas, we obtain
→
−
→
→ db
→ db
t1 db
s
κ−
τ−
s
b db
bκ2 + λb
bτ 2 −
b db
b
b
= λ
t + κ
b + λb
n
b +λ
b
db
s db
s1
db
s
db
s
db
s1
2
−
→ db
−
→
s
bκ b
bτ b
+
1 + λb
t + λb
b
.
db
s1
From this equation we get
s
bκ2 + λb
bτ 2 db
= 0,
κ + λb
b
db
s1
b τ2 + κ
κ
b = −λ(b
b2 ).
This completes the proof.
3.3. Theorem. Let α
b:x
b(b
s) be a dual Lorentzian Mannheim curve in D31 with arc length
parameter sb.
→
−
→
−
→
−
i) In the case where b
t (b
s) is a timelike vector, n
b (b
s) and b
b (b
s) are spacelike vectors,
then βb : x
b1 (b
s1 ) is the dual Mannheim partner curve of α
b if and only if the
curvature κ
b1 and the torsion τb1 of βb satisfy the following equation
db
τ1
κ
b1
=
1−µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b.
→
−
→
−
→
−
b
ii) In the case where t (b
s) and n
b (b
s) are spacelike vectors, b
b (b
s) a timelike vector,
then βb : x
b1 (b
s1 ) is the dual Mannheim partner curve of α
b if and only if the
curvature κ
b1 and the torsion τb1 of βb satisfy the following equation
db
τ1
κ
b1
=−
1+µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b.
→
−
→
−
→
−
b
b
iii) In the case where t (b
s) and b (b
s) are spacelike vectors, n
b (b
s) a timelike vector,
then βb : x
b1 (b
s1 ) is the dual Mannheim partner curve of α
b if and only if the
curvature κ
b1 and the torsion τb1 of βb satisfy the following equation
db
τ1
κ
b1
=−
1−µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b.
Proof. i) Suppose that α
b : x
b(b
s) is a dual Lorentzian Mannheim curve. Then by the
definition we can assume that
→
−
(3.4)
x
b(b
s1 ) = x
b1 (b
s1 ) + µ
b(b
s 1 )b
b1 (b
s1 )
for some function µ
b(b
s1 ). By taking the derivative of (3.4) with respect to b
s1 and applying
the Frenet formulas, we have
→
→ db
−
→
−
→
−
s
db
µ−
b
b
(3.5)
t
= b
t1+
b −µ
bτb1 n
b 1.
db
s1
db
s1
→
−
→
−
Since b
b is coincident with n
b in direction, we get
db
µ
= 0.
db
s1
656
S. Özkaldı, K. İlarslan, Y. Yaylı
This means that µ
b is a never pure dual constant. Thus we have
→ db
−
→
−
→
−
s
b
(3.6)
t
= b
t 1−µ
bτb1 n
b 1.
db
s1
On the other hand, we have
→ −
−
→
→
−
b
b
(3.7)
t = b
t 1 cosh θb + n
b 1 sinh θ,
→
−
→
−
where θb is the dual hyberbolic angle between b
t and b
t 1 at the corresponding points of
b By taking the derivative of this equation with respect to sb1 , we obtain
α
b and β.
!
!
→
−
→
−
→ db
−
→
−
s
dθb
dθb
κ
bn
b
= b
κ1 +
sinh θb b
t1+ κ
b1 +
cosh θb n
b 1 + τb1 sinh θb b
b 1.
db
s1
db
s1
db
s1
−
→
→
−
From this equation and the fact that the direction of n
b is coincident with that of b
b 1,
we get
!
dθb
κ
b1 +
cosh θb = 0.
db
s1
Therefore we have
dθb
= −b
κ1 .
(3.8)
db
s1
−
→
→
−
From (3.6) and (3.7) and noting that b
t 1 is orthogonal to b
b 1 , we find that
1
µ
bτb1
db
s
=
=−
.
db
s1
cosh θb
sinh θb
Then we have
b
µ
bτb1 = − tanh θ.
By taking the derivative of this equation and applying (3.8), we get
db
τ1
=κ
b1 1 − µ
b2 τb12 ,
µ
b
db
s1
that is
db
τ1
κ
b1
=
1−µ
b2 τb12 .
db
s1
µ
b
Conversely, if the curvature κ
b1 and torsion τb1 of the dual Lorentzian curve βb satisfy
db
τ1
κ
b1
=
1−µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b(b
s), then we define a dual curve by
→
−
(3.9)
x
b(b
s1 ) = x
b1 (b
s1 ) + µ
bb
b1 (b
s1 ) ,
and we will prove that α
b is a dual Lorentzian Mannheim curve and that βb is the dual
Lorentzian partner curve of α
b.
By taking the derivative of (3.9) with respect to sb1 twice, we get
→ db
−
→
−
→
−
s
b
(3.10)
t
= b
t 1−µ
bτb1 n
b 1,
db
s1
2
2
→
−
→ d sb
−
→
−
→ db
−
→
s
db
τ1 −
(3.11) κ
bn
b
+ b
t 2 = −b
µκ
b1 τb1 b
t1+ κ
b1 − µ
b
n
b1−µ
bτb12 b
b 1,
db
s1
db
s1
db
s1
On Mannheim Partner Curves in Dual Lorentzian Space
657
respectively. Taking the cross product of (3.10) with (3.11) and noticing that
we have
(3.12)
κ
b1 − µ
b
db
τ1
−µ
b2 κ
b1 τb12 = 0,
db
s1
3
→ db
−
−
→
→
−
s
= −b
µ2 τb13 b
t 1+µ
bτb12 n
b1
κ
bb
b
db
s1
By taking the cross product of (3.12) with (3.10), we obtain also
4
→
−
→ db
−
s
κ
bn
b
= −b
µτb12 1 − µ
b2 τb12 b
b 1.
db
s1
→
−
This means that the principal normal direction n
b of α
b:x
b(b
s) coincides with the binormal
→
−
b
b
direction b 1 of β : x
b1 (b
s1 ). Hence α
b : x
b(b
s) is a dual Lorentzian Mannheim curve and
βb : x
b1 (b
s1 ) is its dual Lorentzian Mannheim partner curve.
ii) Suppose that α
b:x
b(b
s) is a dual Lorentzian Mannheim curve. Then by the definition
we can assume that
→
−
(3.13) x
b(b
s1 ) = x
b1 (b
s1 ) + µ
b(b
s 1 )b
b1 (b
s1 )
for some function µ
b(b
s1 ). By taking the derivative of (3.13) with respect to sb1 and applying
the Frenet formulas, we have
→
→ db
−
→
−
→
−
db
µ−
s
b
b
= b
t1+
b +µ
bτb1 n
b 1.
(3.14)
t
db
s1
db
s1
→
−
→
−
Since b
b is coincident with n
b in direction, we get
db
µ
= 0.
db
s1
This means that µ
b is a never pure dual constant. Thus we have
→ db
−
→
−
→
−
s
b
(3.15)
t
= b
t 1+µ
bτb1 n
b 1.
db
s1
On the other hand, we have
→ −
−
→
→
−
b
b
(3.16)
t = b
t 1 cos θb + n
b 1 sin θ,
→
−
→
−
b
where θb is the dual angle between b
t and b
t 1 at the corresponding points of α
b and β.
By taking the derivative of this equation with respect to sb1 , we obtain
!
!
→
−
→
−
→ db
−
→
−
s
dθb
dθb
b
b
κ
bn
b
=− κ
b1 +
sin θ t 1 + κ
b1 +
cos θb n
b 1 + τb1 sin θb b
b 1.
db
s1
db
s1
db
s1
−
→
→
−
From this equation and the fact that the direction of n
b is coincident with that of b
b 1,
we get

b
dθ
 κ
b1 + db
sin θb = 0
s1
b
d
θ
 κ
b1 + dbs1 cos θb = 0.
Therefore we have
dθb
(3.17)
= −b
κ1 .
db
s1
658
S. Özkaldı, K. İlarslan, Y. Yaylı
−
→
→
−
From (3.15), (3.16) and noticing that b
t 1 is orthogonal to b
b 1 , we find that
db
s
1
µ
bτb1
=
=
.
b
db
s1
cos θ
sin θb
Then we have
b
µ
bτb1 = tan θ.
By taking the derivative of this equation and applying (3.17), we get
db
τ1
µ
b
= −b
κ1 1 + µ
b2 τb12 ,
db
s1
that is
db
τ1
κ
b1
=−
1+µ
b2 τb12 .
db
s1
µ
b
Conversely, if the curvature κ
b1 and torsion τb1 of the dual Lorentzian curve βb satisfy
db
τ1
κ
b1
=−
1+µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b(b
s), then we define a dual curve by
→
−
(3.18) x
b(b
s1 ) = x
b1 (b
s1 ) + µ
bb
b1 (b
s1 ) ,
and we will prove that α
b is a dual Lorentzian Mannheim curve and that βb is the dual
Lorentzian partner curve of α
b.
By taking the derivative of (3.18) with respect to sb1 twice, we get
→
−
→ db
−
→
−
s
b
(3.19)
= b
t 1+µ
bτb1 n
b 1,
t
db
s1
2
2
→
−
→ d sb
−
→
−
→
→ db
−
s
db
τ1 −
(3.20) κ
bn
b
+ b
t 2 = −b
µκ
b1 τb1 b
t1+ κ
b1 + µ
b
n
b1+µ
bτb12 b
b 1,
db
s1
db
s1
db
s1
respectively. Taking the cross product of (3.19) with (3.20) and noticing that
we have
(3.21)
−b
κ1 − µ
b
db
τ1
−µ
b2 κ
b1 τb12 = 0,
db
s1
3
→ db
−
−
→
→
−
s
κ
bb
b
= −b
µ2 τb13 b
t 1+µ
bτb12 n
b 1.
db
s1
By taking the cross product of (3.21) with (3.19), we obtain also
4
→
−
→ db
−
s
κ
bn
b
=µ
bτb12 1 + µ
b2 τb12 b
b 1.
db
s1
→
−
This means that the principal normal direction n
b of α
b:x
b(b
s) coincides with the binormal
→
−
direction b
b 1 of βb : x
b1 (b
s1 ). Hence, α
b :x
b(b
s) is a dual Lorentzian Mannheim curve and
βb : x
b1 (b
s1 ) is its dual Lorentzian Mannheim partner curve.
iii) Suppose that α
b:x
b(b
s) is a dual Lorentzian Mannheim curve. Then by the definition
we can assume that
−
→
(3.22) x
b(b
s1 ) = x
b1 (b
s1 ) + µ
b(b
s 1 )b
b1 (b
s1 )
for some function µ
b(b
s1 ). By taking the derivative of (3.22) with respect to sb1 and applying
the Frenet formulas, we have
→
→ db
−
→
−
→
−
s
db
µ−
b
b
(3.23)
t
= b
t1+
b +µ
bτb1 n
b 1.
db
s1
db
s1
On Mannheim Partner Curves in Dual Lorentzian Space
659
→
−
−
→
Since b
b is coincident with n
b in direction, we get
db
µ
= 0.
db
s1
This means that µ
b is a never pure dual constant. Thus we have
→ db
−
→
−
→
−
s
b
(3.24)
t
= b
t 1+µ
bτb1 n
b 1.
db
s1
On the other hand, we have
→ −
−
→
→
−
b
b
(3.25)
t = b
t 1 cosh θb + n
b 1 sinh θ,
where
α
b and
→
−
→
−
θb is the dual hyberbolic angle between b
t and b
t 1 at the corresponding points of
b By taking the derivative of this equation with respect to sb1 , we obtain
β.
!
!
→
−
→
−
→ db
−
→
−
s
dθb
dθb
b
b
κ
bn
b
= b
κ1 +
sinh θ t 1 + κ
b1 +
cosh θb n
b 1 + τb1 sinh θb b
b 1.
db
s1
db
s1
db
s1
−
→
→
−
From this equation and the fact that the direction of n
b is coincident with that of b
b 1,
we get
!
dθb
κ
b1 +
cosh θb = 0.
db
s1
Therefore we have
dθb
(3.26)
= −b
κ1 .
db
s1
−
→
→
−
From (3.24), (3.25) and noticing that b
t 1 is orthogonal to b
b 1 , we find that
db
s
1
µ
bτb1
=
=
.
db
s1
cosh θb
sinh θb
Then we have
b
µ
bτb1 = tanh θ.
By taking the derivative of this equation and applying (3.26), we get
that is
µ
b
db
τ1
= −b
κ1 1 − µ
b2 τb12 ,
db
s1
db
τ1
κ
b1
=−
1−µ
b2 τb12 .
db
s1
µ
b
Conversely, if the curvature κ
b1 and torsion τb1 of the dual curve βb satisfy
db
τ1
κ
b1
=−
1−µ
b2 τb12
db
s1
µ
b
for some never pure dual constant µ
b(b
s), then we define a dual Lorentzian curve by
→
−
(3.27) x
b(b
s1 ) = x
b1 (b
s1 ) + µ
bb
b1 (b
s1 ) ,
and we will prove that α
b is a dual Lorentzian Mannheim curve and that βb is the dual
Lorentzian partner curve of α
b.
660
S. Özkaldı, K. İlarslan, Y. Yaylı
By taking the derivative of (3.27) with respect to sb1 twice, we get
→ db
−
→
−
→
−
s
b
(3.28)
t
= b
t 1+µ
bτb1 n
b 1,
db
s1
2
2
→
−
→ d sb
−
→
−
→ db
−
→
s
db
τ1 −
(3.29) κ
bn
b
+ b
t 2 =µ
bκ
b1 τb1 b
t1+ κ
b1 + µ
b
n
b1 +µ
bτb12 b
b 1,
db
s1
db
s1
db
s1
respectively. Taking the cross product of (3.28) with (3.29) and noticing that
we have
(3.30)
κ
b1 + µ
b
db
τ1
−µ
b2 κ
b1 τb12 = 0,
db
s1
3
− db
→
→
−
→
−
s
b
κ
bb
=µ
b2 τb13 b
t 1 +µ
bτb12 n
b1
db
s1
By taking the cross product of (3.30) with (3.28), we obtain also
4
→
−
→ db
−
s
κ
bn
b
= −b
µτb12 1 − µ
b2 τb12 b
b 1.
db
s1
→
−
This means that the principal normal direction n
b of α
b:x
b(b
s) coincides with the binormal
→
−
direction b
b 1 of βb : x
b1 (b
s1 ). Hence, α
b :x
b(b
s) is a dual Lorentzian Mannheim curve and
βb : x
b1 (b
s1 ) is its dual Lorentzian Mannheim partner curve.
3.4. Remark. Let α
b = α + εα∗ and βb = β + εβ ∗ be non-null dual curves in the dual
3
Lorentzian space D1 . As a special case, if we consider the dual part of the given curves
as α∗ = 0 and β ∗ = 0, then α
b and βb are non-null Lorentzian curves in R31 . In this case
all the results of this paper are similar to the results given in [9].
Acknowledgement
We wish to thank the referee for the careful reading of the manuscript and for the
constructive comments that have substantially improved the presentation of the paper.
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