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Article

Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds

Institute of Basic Science, Chonnam National University, Gwangju 61186, Korea
Symmetry 2019, 11(6), 784; https://doi.org/10.3390/sym11060784
Submission received: 23 May 2019 / Revised: 5 June 2019 / Accepted: 10 June 2019 / Published: 12 June 2019
(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

Abstract

:
In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of κ and τ 1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that γ is a slant curve if and only if M is Sasakian for a contact magnetic curve γ in contact Lorentzian three-manifold M. As an example, we find contact magnetic curves in Lorentzian Heisenberg three-space.

1. Introduction

As a generalization of Legendre curve, we defined the notion of slant curves in [1,2]. A curve in a contact three-manifold is said to be slant if its tangent vector field has constant angle with the Reeb vector field. For a contact Riemannian manifold, we proved that a slant curve in a Sasakian three-manifold is that its ratio of κ and τ 1 is constant. Baikoussis and Blair proved that, on a three-dimensional Sasakian manifold, the torsion of the Legendre curve is + 1 ([3]).
A magnetic curve represents a trajectory of a charged particle moving on the manifold under the action of a magnetic field in [4]. A magnetic field on a semi-Riemannian manifold ( M , g ) is a closed two-form F. The Lorentz force of the magnetic field F is a ( 1 , 1 ) -type tensor field Φ given by
g ( Φ ( X ) , Y ) = F ( X , Y ) , X , Y Γ ( T M ) .
The magnetic trajectories of F are curves γ on M that satisfy the Lorentz equation
γ γ = Φ ( γ ) ,
where ∇ is the Levi–Civita connection of g. The Lorentz equation generalizes the equation satisfied by the geodesics of M, namely γ γ = 0 . Since the Lorentz force Φ is skew-symmetric, we have
d d t g ( γ , γ ) = 2 g ( Φ ( γ ) , γ ) = 0 ,
that is, magnetic curve have constant speed γ = v 0 . When the magnetic curve γ ( t ) is arc-length parameterized, it is called a normal magnetic curve. Cabreizo et al. studied a contact magnetic field in three-dimensional Sasakian manifold ([5]).
In this article, we define the magnetic curve γ with contact magnetic field F ξ , q of the length q in three-dimensional Sasakian Lorentzian manifold M 3 . We call it the contact magnetic curve or trajectories of F ξ , q .
In Section 3, we define a Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using the Lorentzian cross product, we prove that the ratio of κ and τ 1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold.
In Section 4, we prove that γ is a slant curve if and only if M is Sasakian for a contact magnetic curve γ in contact Lorentzian three-manifolds M. For example, we find contact magnetic curves in Lorentzian Heisenberg three-space.

2. Preliminaries

Contact Lorentzian Manifold

Let M be a ( 2 n + 1 ) -dimensional differentiable manifold. M has an almost contact structure ( φ , ξ , η ) if it admits a tensor field φ of ( 1 , 1 ) , a vector field ξ and a 1-form η satisfying
φ 2 = I + η ξ , η ( ξ ) = 1 .
Suppose M has an almost contact structure ( φ , ξ , η ) . Then, φ ξ = 0 and η φ = 0 . Moreover, the endomorphism φ has rank 2 n .
If a ( 2 n + 1 ) -dimensional smooth manifold M with almost contact structure ( φ , ξ , η ) admits a compatible Lorentzian metric such that
g ( φ X , φ Y ) = g ( X , Y ) + η ( X ) η ( Y ) ,
then we say M has an almost contact Lorentzian structure ( η , ξ , φ , g ) . Setting Y = ξ , we have
η ( X ) = g ( X , ξ ) .
Next, if the compatible Lorentzian metric g satisfies
d η ( X , Y ) = g ( X , φ Y ) ,
then η is a contact form on M, ξ is the associated Reeb vector field, g is an associated metric and ( M , φ , ξ , η , g ) is called a contact Lorentzian manifold.
For a contact Lorentzian manifold M, one may define naturally an almost complex structure J on M × R by
J ( X , f d d t ) = ( φ X f ξ , η ( X ) d d t ) ,
where X is a vector field tangent to M, t is the coordinate of R and f is a function on M × R . When the almost complex structure J is integrable, the contact Lorentzian manifold M is said to be normal or Sasakian. A contact Lorentzian manifold M is normal if and only if M satisfies
[ φ , φ ] + 2 d η ξ = 0 ,
where [ φ , φ ] is the Nijenhuis torsion of φ .
Proposition 1
([6,7]). An almost contact Lorentzian manifold ( M 2 n + 1 , η , ξ , φ , g ) is Sasakian if and only if
( X φ ) Y = g ( X , Y ) ξ + η ( Y ) X .
Using the similar arguments and computations in [8], we obtain
Proposition 2
([6,7]). Let ( M 2 n + 1 , η , ξ , φ , g ) be a contact Lorentzian manifold. Then,
X ξ = φ X φ h X .
If ξ is a killing vector field with respect to the Lorentzian metric g. Then, we have
X ξ = φ X .

3. Slant Curves in Contact Lorentzian Three-Manifolds

Let γ : I M 3 be a unit speed curve in Lorentzian three-manifolds M 3 such that γ satisfies g ( γ , γ ) = ε 1 = ± 1 . The constant ε 1 is called the causal character of γ . A unit speed curve γ is said to be a spacelike or timelike if its causal character is 1 or 1 , respectively.
A unit speed curve γ is said to be a Frenet curve if g ( γ , γ ) 0 . A Frenet curve γ admits an orthonormal frame field { E 1 = γ ˙ , E 2 , E 3 } along γ . The constants ε 2 and ε 3 are defined by
g ( E i , E i ) = ε i , i = 2 , 3
and called second causal character and third causal character of γ , respectively. Thus, ε 1 ε 2 = ε 3 is satisfied. Then, the Frenet–Serret equations are the following ([9,10]):
γ ˙ E 1 = ε 2 κ E 2 , γ ˙ E 2 = ε 1 κ E 1 ε 3 τ E 3 , γ ˙ E 3 = ε 2 τ E 2 ,
where κ = | γ ˙ γ ˙ | is the geodesic curvature of γ and τ its geodesic torsion. The vector fields E 1 , E 2 and E 3 are called tangent vector field, principal normal vector field, and binormal vector field of γ , respectively.
A Frenet curve γ is a geodesic if and only if κ = 0 . A Frenet curve γ with constant geodesic curvature and zero geodesic torsion is called a pseudo-circle. A pseudo-helix is a Frenet curve γ whose geodesic curvature and torsion are constant.

3.1. Lorentzian Cross Product

C. Camci ([11]) defined a cross product in three-dimensional almost contact Riemannian manifolds ( M ˜ , η , ξ , φ , g ˜ ) as following:
X Y = g ˜ ( X , φ Y ) ξ η ( Y ) φ X + η ( X ) φ Y .
If we define the cross product ∧ as Equation (11) in three-dimensional almost contact Lorentzian manifold ( M , η , ξ , φ , g ) , then
g ( X Y , X ) = 2 η ( X ) g ( X , φ Y ) 0 .
In fact, we see already the cross product for a Lorentzian three-manifold as following:
Proposition 3.
Let { E 1 , E 2 , E 3 } be an orthonomal frame field in a Lorentzian three-manifold. Then,
E 1 L E 2 = ε 3 E 3 , E 2 L E 3 = ε 1 E 1 , E 3 L E 1 = ε 2 E 2 .
Now, in three-dimensional almost contact Lorentzian manifold M 3 , we define Lorentzian cross product as the following:
Definition 1.
Let ( M 3 , φ , ξ , η , g ) be a three-dimensional almost contact Lorentzian manifold. We define a Lorentzian cross product L by
X L Y = g ( X , φ Y ) ξ η ( Y ) φ X + η ( X ) φ Y ,
where X , Y T M .
The Lorentzian cross product L has the following properties:
Proposition 4.
Let ( M 3 , φ , ξ , η , g ) be a three-dimensional almost contact Lorentzian manifold. Then, for all X , Y , Z T M the Lorentzian cross product has the following properties:
(1) 
The Lorentzian cross product is bilinear and anti-symmetric.
(2) 
X L Y is perpendicular both of X and Y.
(3) 
X L φ Y = g ( X , Y ) ξ η ( X ) Y .
(4) 
φ X = ξ L X .
(5) 
Define a mixed product by d e t ( X , Y , Z ) = g ( X L Y , Z ) Then,
d e t ( X , Y , Z ) = g ( X , φ Y ) η ( Z ) g ( Y , φ Z ) η ( X ) g ( Z , φ X ) η ( Y )
and d e t ( X , Y , Z ) = d e t ( Y , Z , X ) = d e t ( Z , X , Y ) .
(6) 
g ( X , φ Y ) Z + g ( Y , φ Z ) X + g ( Z , φ X ) Y = ( X , Y , Z ) ξ .
Proof. 
(We can prove by a similar way as in [11])
( 1 ) and ( 2 ) are trivial.
( 3 ) using Equations (3), (5) and (13),
X L φ Y = g ( X , Y + η ( Y ) ξ ) ξ + η ( X ) ( Y + η ( Y ) ξ ) = g ( X , Y ) ξ η ( X ) Y .
( 4 ) by Equation (13),
ξ L X = g ( ξ , φ X ) ξ η ( X ) φ ξ + η ( ξ ) φ X = φ X .
( 5 ) from Equations (5) and (13),
g ( X L Y , Z ) = g ( g ( X , φ Y ) ξ η ( Y ) φ X + η ( X ) φ Y , Z ) = g ( X , φ Y ) η ( Z ) g ( Y , φ Z ) η ( X ) g ( Z , φ X ) η ( Y ) .
( 6 ) is easily obtained by ( 5 ) . □
From Equations (7) and (9), we have:
Proposition 5.
Let ( M 3 , φ , ξ , η , g ) be a three-dimensional Sasakian Lorentzian manifold. Then, we have
Z ( X L Y ) = ( Z X ) L Y + X L ( Z Y ) ,
for all X , Y , Z T M .
Proof. 
From Equation (13), we get
Z ( X L Y ) = Z ( g ( X , φ Y ) ξ + η ( Y ) φ X η ( X ) φ Y ) = g ( Z X , φ Y ) ξ + g ( X , ( Z φ ) Y ) ξ + g ( X , φ Z Y ) ξ + g ( X , φ Y ) Z ξ η ( Z Y ) φ X + g ( Y , Z ξ ) φ X + η ( Y ) ( Z φ ) X + η ( Y ) φ Z X + η ( Z X ) φ Y g ( X , Z ξ ) φ Y η ( X ) ( Z φ ) Y η ( X ) φ Z Y = ( Z X ) L Y + X L ( Z Y ) + P ( X , Y , Z ) ,
where
P ( X , Y , Z ) = g ( X , ( Z φ ) Y ) ξ + g ( X , φ Y ) Z ξ + g ( Y , Z ξ ) φ X η ( Y ) ( Z φ ) X g ( X , Z ξ ) φ Y + η ( X ) ( Z φ ) Y .
Since M is a three-dimensional Sasakian Lorentzian manifold, it satisfies Equations (7) and (9). Hence, we have
P ( X , Y , Z ) = g ( X , φ Y ) φ Z + g ( Y , φ Z ) φ X + g ( Z , φ X ) φ Y .
Using Equation (6) of Proposition 4, we obtain P ( X , Y , Z ) = 0 and Equation (14). □

3.2. Frenet Slant Curves

In this subsection, we study a Frenet slant curve in contact Lorentzian three-manifolds.
A curve in a contact Lorentzian three-manifold is said to be slant if its tangent vector field has constant angle with the Reeb vector field (i.e., η ( γ ) = g ( γ , ξ ) is a constant).
Since the Reeb vector field ξ is denoted by
ξ = i = 1 3 ε i g ( ξ , E i ) E i = i = 1 3 ε i η ( E i ) E i ,
using Equation (4) of Proposition 4 and Proposition 3, we have:
Proposition 6.
Let ( M 3 , φ , ξ , η , g ) be a three-dimensional almost contact Lorentzian manifold. Then, for a Frenet curve γ in M 3 , we have
φ E 1 = ε 2 ε 3 ( η ( E 2 ) E 3 η ( E 3 ) E 2 ) , φ E 2 = ε 3 ε 1 ( η ( E 3 ) E 1 η ( E 1 ) E 3 ) , φ E 3 = ε 1 ε 2 ( η ( E 1 ) E 2 η ( E 2 ) E 1 ) .
By using Proposition 6, we find that differentiating η ( E i ) (for i = 1 , 2 , 3 ) along a Frenet curve γ
η ( E 1 ) = ε 2 κ η ( E 2 ) + g ( E 1 , φ h E 1 ) , η ( E 2 ) = ε 1 κ η ( E 1 ) ε 3 ( τ 1 ) η ( E 3 ) + g ( E 2 , φ h E 1 ) , η ( E 3 ) = ε 2 ( τ 1 ) η ( E 2 ) + g ( E 3 , φ h E 1 ) .
Now, we assume that M 3 is a Sasakian Lorentzian manifold; then,
η ( E 1 ) = ε 2 κ η ( E 2 ) ,
η ( E 2 ) = ε 1 κ η ( E 1 ) ε 3 ( τ 1 ) η ( E 3 ) ,
η ( E 3 ) = ε 2 ( τ 1 ) η ( E 2 ) .
From Equation (15), if γ is a geodesic curve, that is κ = 0 , in a Sasakian Lorentzian three-manifold M 3 , then γ is naturally a slant curve. Now, let us consider a non-geodesic curve γ ; then, we have:
Proposition 7.
A non-geodesic Frenet curve γ in a Sasakian Lorentzian three-manifold M 3 is slant curve if and only if η ( E 2 ) = 0 .
From Equations (15) and (17) and Proposition 7, we get that η ( E 1 ) and η ( E 3 ) are constants. Hence, using Equation (16), we obtain:
Theorem 1.
The ratio of κ and τ 1 is a constant along a non-geodesic Frenet slant curve in a Sasakian Lorentzian three-manifold M 3 .
Next, let us consider a Legendre curve γ as a spacelike curve with spacelike normal vector. For a Legendre curve γ , η ( γ ) = η ( E 1 ) = 0 , η ( E 2 ) = 0 and η ( E 3 ) is a constant. Hence, using Equation (16), we have:
Corollary 1.
Let M be a three-dimensional Sasakian Lorentzian manifold ( M 3 , η , ξ , φ , g ) . Then, the torsion of a Legendre curve is 1.
From this, we see that the ratio of κ and τ 1 is a constant along non-geodesic Frenet slant curve containing Legendre curve.

3.3. Null Slant Curves

In this section, let us consider a null curve γ that has a null tangent vector field g ( γ , γ ) = 0 and γ is not a geodesic (i.e., g ( γ γ , γ γ ) 0 ). We take a parameterization of γ such that g ( γ γ , γ γ ) = 1 . Then, Duggal, K.L. and Jin, D.H ([12]) proved that there exists only one Cartan frame { T , N , W } and the function τ along γ whose Cartan equations are
T T = N , T W = τ N , T N = τ T W ,
where
T = γ , N = T T , τ = 1 2 g ( T N , T N ) , W = T N τ T .
Hence,
g ( T , W ) = g ( N , N ) = 1 , g ( T , T ) = g ( T , N ) = g ( W , W ) = g ( W , N ) = 0 .
For a null Legendre curve γ , we easily prove that γ is geodesic. Hence, we suppose that γ is non-geodesic; then, we have:
Theorem 2.
Let γ be a non-geodesic null slant curve in a Sasakian Lorentzian three-manifold. We assume that κ = 1 , then we have
N = ± 1 a φ γ , τ = 1 2 a 2 1 , W = 1 2 a 2 γ 1 a ξ ,
where a = η ( γ ) is non-zero constant.
Proof. 
Let φ T = l T + m N + n W for some l , m , n . We find l = g ( φ T , T ) = 0 , then φ T = m N + n W . From this, we get
g ( φ T , φ T ) = m 2 = a 2 a n d 0 = g ( φ T , ξ ) = n ( a τ + m ) .
Hence, m = ± a and n = 0 or m = a τ .
If n = 0 , then N = 1 m φ T = ± 1 a φ T . Using the Cartan equation, we find that τ = 1 2 a 2 1 and W = 1 2 a 2 γ 1 a ξ .
Next, if n 0 and m = a τ then since γ is a slant curve, differentiating g ( φ T , N ) = m = ± a , we have n = g ( φ T , W ) = 0 , which gives a contradiction. □
From the second equation of Equation (19), we have:
Remark 1.
Let γ be a non-geodesic null slant curve in a Sasakian Lorentzian three-manifold. We assume that κ = 1 then τ is constant such that τ = 1 2 a 2 1 .

4. Contact Magnetic Curves

In a three-dimensional Sasakian Lorentzian manifold M 3 , the Reeb vector field ξ is Killing. By Equation (6), the 2-form Φ is d η , that is d η ( X , Y ) = g ( X , φ Y ) , for all X , Y Γ ( T M ) .
Let γ : I M be a smooth curve on a contact Lorentzian manifold ( M , φ , ξ , η , g ) . Then, we define a magnetic field on M by
F ξ , q ( X , Y ) = q d η ( X , Y ) ,
where X , Y X ( M ) and q is a non-zero constant. We call F ξ , q the contact magnetic field with strength q.
Using Equations (1), (4) and (6) we get Φ ( X ) = q φ X . Hence, from Equation (2) the Lorentz equation is
γ γ = q φ γ .
This is the generalized equation of geodesics under arc length parameterization, that is γ γ = 0 . For q = 0 , we find that the contact magnetic field vanishes identically and the magnetic curves are geodesics of M. The solutions of Equation (20) are called contact magnetic curve or trajectories of F ξ , q .
By using Equations (8) and (20), differentiating g ( ξ , γ ) along a contact magnetic curve γ in contact Lorentzian three-manifold
d d t g ( ξ , γ ) = g ( γ ξ , γ ) + g ( ξ , γ γ ) = g ( φ γ φ h γ , γ ) + g ( ξ , q φ γ ) = g ( φ h γ , γ ) .
Hence, we have:
Theorem 3.
Let γ be a contact magnetic curve in a contact Lorentzian three-manifold M. γ is a slant curve if and only if M is Sasakian.
Next, we find the curvature κ and torsion τ along non-geodesic Frenet contact magnetic curves γ .
We suppose that η ( E 1 ) = a , for a constant a. Then, using Equations (4), (10) and (20), we get
ε 2 κ 2 = q 2 g ( φ γ , φ γ ) = q 2 ( ε 1 + a 2 ) .
Hence, we find that γ has a constant curvature
κ = q ε 2 ( ε 1 + a 2 ) ,
and, from Equations (10), (20) and (21), the binormal vector field
E 2 = q ε 2 κ φ γ = δ ε 2 ε 2 ( ε 1 + a 2 ) φ γ ,
where δ = q / q .
Using Proposition 3 and Equation (22), the binormal E 3 is computed as
ε 3 E 3 = E 1 L E 2 = γ L ( δ ε 2 ε 2 ( ε 1 + a 2 ) φ γ ) = δ ε 2 ε 2 ( ε 1 + a 2 ) ( ε 1 ξ + a γ ) .
Differentiating binormal vector field E 3 , we have
γ E 3 = δ ε 2 ε 3 ε 2 ( ε 1 + a 2 ) γ ( ε 1 ξ + a γ ) = δ ε 2 ε 3 ε 2 ( ε 1 + a 2 ) ( ε 1 + q a ) φ γ .
On the other hand, by Equation (10), we have
γ E 3 = ε 2 τ E 2 = τ δ φ γ ε 2 ( ε 1 + a 2 ) .
From Equations (23) and (24), since ε 1 ε 2 ε 3 = 1 , we obtain
τ = 1 + ε 1 q a .
Moreover, if γ is a non-geodesic curve, then
τ 1 κ = δ ε 1 a ε 2 ( ε 1 + a 2 ) .
Therefore, we obtain:
Theorem 4.
Let γ be a non-geodesic Frenet curve in a Sasakian Lorentzian three-manifold M. If γ is a contact magnetic curve, then it is slant pseudo-helix with curvature κ = q ε 2 ( ε 1 + a 2 ) and torsion τ = 1 + ε 1 q a . Moreover, the ratio of κ and τ 1 is a constant.
Since a Legendre curve is a spacelike curve with spacelike normal vector field and η ( γ ) = a = 0 , we assume that γ is a Legendre curve and we have:
Corollary 2.
Let γ be a non-geodesic Legendre curve in a Sasakian Lorentzian three-manifold M. If γ is a contact magnetic curve, then it is Legendre pseudo-helix with curvature κ = | q | and torsion τ = 1 .
Now, from the geodesic curvature in Equation (21), if ε 1 = 1 , then η ( γ ) = a and 1 1 + a 2 , and we have ε 2 = 1 . Moreover, using ε 3 = ε 1 · ε 2 , we obtain ε 3 = 1 . Next, if ε 1 = 1 , then η ( γ ) = a = cosh α 0 . Since γ is a geodesic for a = cosh α 0 = 1 , we assume that γ is non-geodesic, and we get a 2 > 1 . Hence, 1 + a 2 > 0 and we get ε 2 = ε 3 = 1 . Therefore, we obtain:
Theorem 5.
Let γ be a non-geodesic Frenet curve in a Sasakian Lorentzian three-manifold M. If γ is a contact magnetic curve. then γ is one of the following:
(i) 
a spacelike curve with spacelike normal vector field; or
(ii) 
a timelike curve.
Moreover, we have:
Corollary 3.
Let γ be a non-geodesic Frenet curve in a Sasakian Lorentzian three-manifold M. If γ is a contact magnetic curve, then there does not exist a spacelike curve with timelike normal vector field.
In a similar with a Frenet curve, we study null contact magnetic curves in a Sasakian Lorentzian three-manifold M. Hence, we find that there exist a null contact magnetic curve with q = ± a and same the result with Theorem 2.

Example

The Heisenberg group H 3 is a Lie group which is diffeomorphic to R 3 and the group operation is defined by
( x , y , z ) ( x ¯ , y ¯ , z ¯ ) = ( x + x ¯ , y + y ¯ , z + z ¯ + x y ¯ 2 x ¯ y 2 ) .
The mapping
H 3 1 a b 0 1 c 0 0 1 | a , b , c R : ( x , y , z ) 1 x z + x y 2 0 1 y 0 0 1
is an isomorphism between H 3 and a subgroup of G L ( 3 , R ) .
Now, we take the contact form
η = d z + ( y d x x d y ) .
Then, the characteristic vector field of η is ξ = z .
Now, we equip the Lorentzian metric as following:
g = d x 2 + d y 2 d z + ( y d x x d y ) 2 .
We take a left-invariant Lorentzian orthonormal frame field ( e 1 , e 2 , e 3 ) on ( H 3 , g ) :
e 1 = x y z , e 2 = y + x z , e 3 = z ,
and the commutative relations are derived as follows:
[ e 1 , e 2 ] = 2 e 3 , [ e 2 , e 3 ] = [ e 3 , e 1 ] = 0 .
Then, the endomorphism field φ is defined by
φ e 1 = e 2 , φ e 2 = e 1 , φ e 3 = 0 .
The Levi–Civita connection ∇ of ( H 3 , g ) is described as
e 1 e 1 = e 2 e 2 = e 3 e 3 = 0 , e 1 e 2 = e 3 = e 2 e 1 , e 2 e 3 = e 1 = e 3 e 2 , e 3 e 1 = e 2 = e 1 e 3 .
The contact form η satisfies d η ( X , Y ) = g ( X , φ Y ) . Moreover, the structure ( η , ξ , φ , g ) is Sasakian.
The Riemannian curvature tensor R of ( H 3 , g ) is given by
R ( e 1 , e 2 ) e 1 = 3 e 2 , R ( e 1 , e 2 ) e 2 = 3 e 1 , R ( e 2 , e 3 ) e 2 = e 3 , R ( e 2 , e 3 ) e 3 = e 2 , R ( e 3 , e 1 ) e 3 = e 1 , R ( e 3 , e 1 ) e 1 = e 3 ,
and the other components are zero.
The sectional curvature is given by [6]
K ( ξ , e i ) = R ( ξ , e i , ξ , e i ) = 1 , f o r i = 1 , 2 ,
and
K ( e 1 , e 2 ) = R ( e 1 , e 2 , e 1 , e 2 ) = 3 .
Thus, we see that the Lorentzian Heisenberg space ( H 3 , g ) is the Lorentzian Sasakian space forms with constant holomorphic sectional curvature μ = 3 .
Let γ be a Frenet slant curve in Lorentzian Heisenberg space ( H 3 , g ) parameterized by arc-length. Then, the tangent vector field has the form
T = γ = ε 1 + a 2 cos β e 1 + ε 1 + a 2 sin β e 2 + a e 3 ,
where a = c o n s t a n t , β = β ( s ) . Using Equation (26), we get
γ γ = ε 1 + a 2 ( β + 2 a ) ( sin β e 1 + cos β e 2 ) .
Since γ is a non-geodesic, we may assume that κ = ε 1 + a 2 ( β + 2 a ) > 0 without loss of generality.
Then, the normal vector field
N = sin β e 1 + cos β e 2 .
The binormal vector field ε 3 B = T L N = a cos β e 1 a sin β e 2 ε 1 + a 2 e 3 . From Theorem 5, we see that ε 2 = 1 , thus we have ε 3 = ε 1 . Hence,
B = ε 1 ( a cos β e 1 + a sin β e 2 + ε 1 + a 2 e 3 ) .
Using the Frenet–Serret Equation (10), we have
Lemma 1.
Let γ be a Frenet slant curve in Lorentzian Heisenberg space ( H 3 , g ) parameterized by arc-length. Then, γ admits an orthonormal frame field { T , N , B } along γ and
κ = ε 1 + a 2 ( β + 2 a ) , τ = 1 + ε 1 a ( β + 2 a ) .
Next, if γ is a null slant curve in the Lorentzian Heisenberg space ( H 3 , g ) , then the tangent vector field has the form
T = γ = a cos β e 1 + a sin β e 2 + a e 3 ,
where a = c o n s t a n t , β = β ( s ) . Using Equation (26), we get
γ γ = a ( β + 2 a ) ( sin β e 1 + cos β e 2 ) .
Since γ is non-geodesic, using Equation (18) we have a ( β + 2 a ) = 1 and
N = sin β e 1 + cos β e 2 .
Differentiating N, we get
γ N = ( β + a ) cos β e 1 ( β + a ) sin β e 2 + a e 3 .
From Equation (18), τ = 1 2 g ( γ N , γ N ) = 1 2 ( β ) 2 + a β . Since W = γ N τ T , we have
W = { 1 2 ( β ) 2 + ( 1 a a ) β + 1 } T ( β + 2 a ) ξ = 1 2 a ( cos β e 1 + sin β e 2 e 3 ) .
Therefore, we have
Lemma 2.
Let γ be a non-geodesic null slant curve in the Lorentzian Heisenberg space ( H 3 , g ) . We assume that κ = a ( β + 2 a ) = 1 . Then, its torsion is constant such that τ = 1 2 a 2 1 .
Let γ ( s ) = ( x ( s ) , y ( s ) , z ( s ) ) be a curve in Lorentzian Heisenberg space ( H 3 , g ) . Then, the tangent vector field γ of γ is
γ = d x d s , d y d s , d z d s = d x d s x + d y d s y + d z d s z .
Using the relations:
x = e 1 + y e 3 , y = e 2 x e 3 , z = e 3 ,
if γ is a slant curve in ( H 3 , g ) , then from Equation (27) the system of differential equations for γ is given by
d x d s ( s ) = ε 1 + a 2 cos β ( s ) ,
d y d s ( s ) = ε 1 + a 2 sin β ( s ) , d z d s ( s ) = a + ε 1 + a 2 ( x ( s ) sin β ( s ) y ( s ) cos β ( s ) ) .
Now, we construct a magnetic curve γ (containing Frenet and null curve) in the Lorentzian Heisenberg space ( H 3 , g ) . From Equations (20) and (28), we have:
Proposition 8.
Let γ : I ( H 3 , g ) be a magnetic curve parameterized by arc-length in the Lorentzian Heisenberg space ( H 3 , g ) . Then,
β = q 2 a , f o r a = η ( γ ) .
Namely, β is a constant, e.g., A, hence β ( s ) = A s + b , b R . If γ is a null curve, then q = ± 1 a . Finally, from Equations (32) and (33), we have the following result:
Theorem 6.
Let γ : I ( H 3 , g ) be a non-geodesic curve parameterized by arc-length s in the Lorentzian Heisenberg group ( H 3 , g ) . If γ is a contact magnetic curve, then the parametric equations of γ are given by
x ( s ) = 1 A ε 1 + a 2 sin ( A s + b ) + x 0 , y ( s ) = 1 A ε 1 + a 2 cos ( A s + b ) + y 0 , z ( s ) = { a + ε 1 + a 2 A } s ε 1 + a 2 A x 0 cos ( A s + b ) + y 0 sin ( A s + b ) + z 0 ,
where b , x 0 , y 0 , z 0 are constants. If ε 1 = 0 then γ is a null curve.
In particular, for a Frenet Legendre curve γ , we get β = q = A .

Funding

The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2019R1l1A1A01043457).

Acknowledgments

The author would like to thank the reviewers for their valuable comments on this paper to improve the quality.

Conflicts of Interest

The author declares no conflict of interest.

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Lee, J.-E. Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds. Symmetry 2019, 11, 784. https://doi.org/10.3390/sym11060784

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Lee J-E. Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds. Symmetry. 2019; 11(6):784. https://doi.org/10.3390/sym11060784

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Lee, Ji-Eun. 2019. "Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds" Symmetry 11, no. 6: 784. https://doi.org/10.3390/sym11060784

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