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

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 $\kappa$ and $\tau -1$ is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that $\gamma$ is a slant curve if and only if M is Sasakian for a contact magnetic curve $\gamma$ 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 $\kappa$ and $\tau -1$ 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 $\mathsf{\Phi }$ given by

$$g(\mathsf{\Phi }(X),Y)=F(X,Y),\forall X,Y\in \mathsf{\Gamma }(TM).$$

(1)

The magnetic trajectories of F are curves $\gamma$ on M that satisfy the Lorentz equation

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=\mathsf{\Phi }({\gamma }^{\prime }),$$

(2)

where ∇ is the Levi–Civita connection of g. The Lorentz equation generalizes the equation satisfied by the geodesics of M, namely ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=0$. Since the Lorentz force $\mathsf{\Phi }$ is skew-symmetric, we have

$$\frac{d}{dt}g({\gamma }^{\prime },{\gamma }^{\prime })=2g(\mathsf{\Phi }({\gamma }^{\prime }),{\gamma }^{\prime })=0,$$

that is, magnetic curve have constant speed $\mid {\gamma }^{\prime }\mid =v_0$. When the magnetic curve $\gamma (t)$ 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 $\gamma$ with contact magnetic field $F_{\xi ,q}$ of the length q in three-dimensional Sasakian Lorentzian manifold $M^3$. We call it the contact magnetic curve or trajectories of $F_{\xi ,q}$.

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 $\kappa$ and $\tau -1$ is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold.

In Section 4, we prove that $\gamma$ is a slant curve if and only if M is Sasakian for a contact magnetic curve $\gamma$ 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 $(2n+1)$-dimensional differentiable manifold. M has an almost contact structure $(\phi ,\xi ,\eta )$ if it admits a tensor field $\phi$ of $(1,1)$, a vector field $\xi$ and a 1-form $\eta$ satisfying

$${\phi }^2=-I+\eta \otimes \xi ,\eta (\xi )=1.$$

(3)

Suppose M has an almost contact structure $(\phi ,\xi ,\eta )$. Then, $\phi \xi =0$ and $\eta \circ \phi =0$. Moreover, the endomorphism $\phi$ has rank $2n$.

If a $(2n+1)$-dimensional smooth manifold M with almost contact structure $(\phi ,\xi ,\eta )$ admits a compatible Lorentzian metric such that

$$g(\phi X,\phi Y)=g(X,Y)+\eta (X)\eta (Y),$$

(4)

then we say M has an almost contact Lorentzian structure $(\eta ,\xi ,\phi ,g)$. Setting $Y=\xi$, we have

$$\eta (X)=-g(X,\xi ).$$

(5)

Next, if the compatible Lorentzian metric g satisfies

$$d\eta (X,Y)=g(X,\phi Y),$$

(6)

then $\eta$ is a contact form on M, $\xi$ is the associated Reeb vector field, g is an associated metric and $(M,\phi ,\xi ,\eta ,g)$ is called a contact Lorentzian manifold.

For a contact Lorentzian manifold M, one may define naturally an almost complex structure J on $M\times \mathbb{R}$ by

$$J(X,f\frac{\mathrm{d}}{\mathrm{d}t})=(\phi X-f\xi ,\eta (X)\frac{\mathrm{d}}{\mathrm{d}t}),$$

where X is a vector field tangent to M, t is the coordinate of $\mathbb{R}$ and f is a function on $M\times \mathbb{R}$. 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

$$[\phi ,\phi ]+2d\eta \otimes \xi =0,$$

where $[\phi ,\phi ]$ is the Nijenhuis torsion of $\phi$.

Proposition 1

([6,7]). An almost contact Lorentzian manifold $(M^{2n+1},\eta ,\xi ,\phi ,g)$ is Sasakian if and only if

$$({\nabla }_X\phi )Y=g(X,Y)\xi +\eta (Y)X.$$

(7)

Using the similar arguments and computations in [8], we obtain

Proposition 2

([6,7]). Let $(M^{2n+1},\eta ,\xi ,\phi ,g)$ be a contact Lorentzian manifold. Then,

$${\nabla }_X\xi =\phi X-\phi hX.$$

(8)

If $\xi$ is a killing vector field with respect to the Lorentzian metric g. Then, we have

$${\nabla }_X\xi =\phi X.$$

(9)

3. Slant Curves in Contact Lorentzian Three-Manifolds

Let $\gamma :I\to M^3$ be a unit speed curve in Lorentzian three-manifolds $M^3$ such that ${\gamma }^{\prime }$ satisfies $g({\gamma }^{\prime },{\gamma }^{\prime })={\epsilon }_1=\pm 1.$ The constant ${\epsilon }_1$ is called the causal character of $\gamma$. A unit speed curve $\gamma$ is said to be a spacelike or timelike if its causal character is 1 or $-1$, respectively.

A unit speed curve $\gamma$ is said to be a Frenet curve if $g({\gamma }^{″},{\gamma }^{″})\ne 0$. A Frenet curve $\gamma$ admits an orthonormal frame field $\{E_1=\dot{\gamma },E_2,E_3\}$ along $\gamma$. The constants ${\epsilon }_2$ and ${\epsilon }_3$ are defined by

$$g(E_i,E_i)={\epsilon }_i,i=2,3$$

and called second causal character and third causal character of $\gamma$, respectively. Thus, ${\epsilon }_1{\epsilon }_2=-{\epsilon }_3$ is satisfied. Then, the Frenet–Serret equations are the following ([9,10]):

$$\left\{\begin{array}{c}{\nabla }_{\dot{\gamma }}{E}_1={\epsilon }_2\kappa E_2,\hfill \\ {\nabla }_{\dot{\gamma }}{E}_2=-{\epsilon }_1\kappa E_1-{\epsilon }_3\tau E_3,\hfill \\ {\nabla }_{\dot{\gamma }}{E}_3={\epsilon }_2\tau E_2,\hfill \end{array}\right.$$

(10)

where $\kappa =|{\nabla }_{\dot{\gamma }}\dot{\gamma }|$ is the geodesic curvature of $\gamma$ and $\tau$ 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 $\gamma$, respectively.

A Frenet curve $\gamma$ is a geodesic if and only if $\kappa =0$. A Frenet curve $\gamma$ with constant geodesic curvature and zero geodesic torsion is called a pseudo-circle. A pseudo-helix is a Frenet curve $\gamma$ 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 $(\tilde{M},\eta ,\xi ,\phi ,\tilde{g})$ as following:

$$\begin{array}{c}\hfill X\wedge Y=-\tilde{g}(X,\phi Y)\xi -\eta (Y)\phi X+\eta (X)\phi Y.\end{array}$$

(11)

If we define the cross product ∧ as Equation (11) in three-dimensional almost contact Lorentzian manifold $(M,\eta ,\xi ,\phi ,g)$, then

$$g(X\wedge Y,X)=2\eta (X)g(X,\phi Y)\ne 0.$$

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,

$$E_1{\wedge }_L{E}_2={\epsilon }_3{E}_3,E_2{\wedge }_L{E}_3={\epsilon }_1{E}_1,E_3{\wedge }_L{E}_1={\epsilon }_2{E}_2.$$

(12)

Now, in three-dimensional almost contact Lorentzian manifold $M^3$, we define Lorentzian cross product as the following:

Definition 1.

Let $(M^3,\phi ,\xi ,\eta ,g)$ be a three-dimensional almost contact Lorentzian manifold. We define a Lorentzian cross product ${\wedge }_L$ by

$$X{\wedge }_{L}Y=g(X,\phi Y)\xi -\eta (Y)\phi X+\eta (X)\phi Y,$$

(13)

where $X,Y\in TM.$

The Lorentzian cross product ${\wedge }_L$ has the following properties:

Proposition 4.

Let $(M^3,\phi ,\xi ,\eta ,g)$ be a three-dimensional almost contact Lorentzian manifold. Then, for all $X,Y,Z\in TM$ the Lorentzian cross product has the following properties:

(1) 

The Lorentzian cross product is bilinear and anti-symmetric.

(2) 

$X{\wedge }_{L}Y$ is perpendicular both of X and Y.

(3) 

$X{\wedge }_L\phi Y=-g(X,Y)\xi -\eta (X)Y$.

(4) 

$\phi X=\xi {\wedge }_{L}X.$

(5) 

Define a mixed product by $det(X,Y,Z)=g(X{\wedge }_{L}Y,Z)$ Then,

$$det(X,Y,Z)=-g(X,\phi Y)\eta (Z)-g(Y,\phi Z)\eta (X)-g(Z,\phi X)\eta (Y)$$

and $det(X,Y,Z)=det(Y,Z,X)=det(Z,X,Y).$

(6) 

$g(X,\phi Y)Z+g(Y,\phi Z)X+g(Z,\phi X)Y=-(X,Y,Z)\xi .$

Proof. 

(We can prove by a similar way as in [11])

$(1)$ and $(2)$ are trivial.

$(3)$ using Equations (3), (5) and (13),

$$\begin{array}{ccc}\hfill X{\wedge }_L\phi Y& =& g(X,-Y+\eta (Y)\xi )\xi +\eta (X)(-Y+\eta (Y)\xi )\hfill \\ & =& -g(X,Y)\xi -\eta (X)Y.\hfill \end{array}$$

$(4)$ by Equation (13),

$$\xi {\wedge }_{L}X=g(\xi ,\phi X)\xi -\eta (X)\phi \xi +\eta (\xi )\phi X=\phi X.$$

$(5)$ from Equations (5) and (13),

$$\begin{array}{ccc}\hfill g(X{\wedge }_{L}Y,Z)& =& g(g(X,\phi Y)\xi -\eta (Y)\phi X+\eta (X)\phi Y,Z)\hfill \\ & =& -g(X,\phi Y)\eta (Z)-g(Y,\phi Z)\eta (X)-g(Z,\phi X)\eta (Y).\hfill \end{array}$$

$(6)$ is easily obtained by $(5)$. □

From Equations (7) and (9), we have:

Proposition 5.

Let $(M^3,\phi ,\xi ,\eta ,g)$ be a three-dimensional Sasakian Lorentzian manifold. Then, we have

$${\nabla }_Z(X{\wedge }_{L}Y)=({\nabla }_{Z}X){\wedge }_{L}Y+X{\wedge }_L({\nabla }_{Z}Y),$$

(14)

for all $X,Y,Z\in TM$.

Proof. 

From Equation (13), we get

$$\begin{array}{ccc}\hfill {\nabla }_Z(X{\wedge }_{L}Y)& =& {\nabla }_Z(-g(X,\phi Y)\xi +\eta (Y)\phi X-\eta (X)\phi Y)\hfill \\ & =& g({\nabla }_{Z}X,\phi Y)\xi +g(X,({\nabla }_Z\phi )Y)\xi +g(X,\phi {\nabla }_{Z}Y)\xi +g(X,\phi Y){\nabla }_Z\xi \hfill \\ & & -\eta ({\nabla }_{Z}Y)\phi X+g(Y,{\nabla }_Z\xi )\phi X+\eta (Y)({\nabla }_Z\phi )X+\eta (Y)\phi {\nabla }_{Z}X\hfill \\ & & +\eta ({\nabla }_{Z}X)\phi Y-g(X,{\nabla }_Z\xi )\phi Y-\eta (X)({\nabla }_Z\phi )Y-\eta (X)\phi {\nabla }_{Z}Y\hfill \\ & =& ({\nabla }_{Z}X){\wedge }_{L}Y+X{\wedge }_L({\nabla }_{Z}Y)+P(X,Y,Z),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill P(X,Y,Z)& =& g(X,({\nabla }_Z\phi )Y)\xi +g(X,\phi Y){\nabla }_Z\xi +g(Y,{\nabla }_Z\xi )\phi X-\eta (Y)({\nabla }_Z\phi )X\hfill \\ & & -g(X,{\nabla }_Z\xi )\phi Y+\eta (X)({\nabla }_Z\phi )Y.\hfill \end{array}$$

Since M is a three-dimensional Sasakian Lorentzian manifold, it satisfies Equations (7) and (9). Hence, we have

$$\begin{array}{ccc}\hfill P(X,Y,Z)& =& g(X,\phi Y)\phi Z+g(Y,\phi Z)\phi X+g(Z,\phi X)\phi Y.\hfill \end{array}$$

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., $\eta ({\gamma }^{\prime })=-g({\gamma }^{\prime },\xi )$ is a constant).

Since the Reeb vector field $\xi$ is denoted by

$$\xi =\sum _{i=1}^3{\epsilon }_{i}g(\xi ,E_i)E_i=-\sum _{i=1}^3{\epsilon }_i\eta (E_i)E_i,$$

using Equation (4) of Proposition 4 and Proposition 3, we have:

Proposition 6.

Let $(M^3,\phi ,\xi ,\eta ,g)$ be a three-dimensional almost contact Lorentzian manifold. Then, for a Frenet curve γ in $M^3$, we have

$$\begin{array}{c}\phi E_1={\epsilon }_2{\epsilon }_3(\eta (E_2)E_3-\eta (E_3)E_2),\hfill \\ \phi E_2={\epsilon }_3{\epsilon }_1(\eta (E_3)E_1-\eta (E_1)E_3),\hfill \\ \phi E_3={\epsilon }_1{\epsilon }_2(\eta (E_1)E_2-\eta (E_2)E_1).\hfill \end{array}$$

By using Proposition 6, we find that differentiating $\eta (E_i)$ (for $i=1,2,3$) along a Frenet curve $\gamma$

$$\begin{array}{c}\eta {(E_1)}^{\prime }={\epsilon }_2\kappa \eta (E_2)+g(E_1,\phi h{E}_1),\hfill \\ \eta {(E_2)}^{\prime }=-{\epsilon }_1\kappa \eta (E_1)-{\epsilon }_3(\tau -1)\eta (E_3)+g(E_2,\phi h{E}_1),\hfill \\ \eta {(E_3)}^{\prime }={\epsilon }_2(\tau -1)\eta (E_2)+g(E_3,\phi h{E}_1).\hfill \end{array}$$

Now, we assume that $M^3$ is a Sasakian Lorentzian manifold; then,

$$\begin{array}{c}\eta {(E_1)}^{\prime }={\epsilon }_2\kappa \eta (E_2),\hfill \end{array}$$

(15)

$$\begin{array}{c}\eta {(E_2)}^{\prime }=-{\epsilon }_1\kappa \eta (E_1)-{\epsilon }_3(\tau -1)\eta (E_3),\hfill \end{array}$$

(16)

$$\begin{array}{c}\eta {(E_3)}^{\prime }={\epsilon }_2(\tau -1)\eta (E_2).\hfill \end{array}$$

(17)

From Equation (15), if $\gamma$ is a geodesic curve, that is $\kappa =0$, in a Sasakian Lorentzian three-manifold $M^3$, then $\gamma$ is naturally a slant curve. Now, let us consider a non-geodesic curve $\gamma$; 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 $\eta (E_2)=0$.

From Equations (15) and (17) and Proposition 7, we get that $\eta (E_1)$ and $\eta (E_3)$ are constants. Hence, using Equation (16), we obtain:

Theorem 1.

The ratio of κ and $\tau -1$ 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 $\gamma$ as a spacelike curve with spacelike normal vector. For a Legendre curve $\gamma$, $\eta ({\gamma }^{\prime })=\eta (E_1)=0$, $\eta (E_2)=0$ and $\eta (E_3)$ is a constant. Hence, using Equation (16), we have:

Corollary 1.

Let M be a three-dimensional Sasakian Lorentzian manifold $(M^3,\eta ,\xi ,\phi ,g)$. Then, the torsion of a Legendre curve is 1.

From this, we see that the ratio of $\kappa$ and $\tau -1$ 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 $\gamma$ that has a null tangent vector field $g({\gamma }^{\prime },{\gamma }^{\prime })=0$ and $\gamma$ is not a geodesic (i.e., $g({\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime })\ne 0$). We take a parameterization of $\gamma$ such that $g({\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime })=1$. Then, Duggal, K.L. and Jin, D.H ([12]) proved that there exists only one Cartan frame $\{T,N,W\}$ and the function $\tau$ along $\gamma$ whose Cartan equations are

$$\begin{array}{c}\hfill {\nabla }_{T}T=N,{\nabla }_{T}W=\tau N,{\nabla }_{T}N=-\tau T-W,\end{array}$$

where

$$T={\gamma }^{\prime },N={\nabla }_{T}T,\tau =\frac{1}{2}g({\nabla }_{T}N,{\nabla }_{T}N),W=-{\nabla }_{T}N-\tau T.$$

(18)

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 $\gamma$, we easily prove that $\gamma$ is geodesic. Hence, we suppose that $\gamma$ 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 $\kappa =1$, then we have

$$\begin{array}{c}\hfill N=\pm \frac{1}{a}\phi {\gamma }^{\prime },\tau =\frac{1}{2a^2}\mp 1,W=\frac{1}{2a^2}{\gamma }^{\prime }-\frac{1}{a}\xi ,\end{array}$$

(19)

where $a=\eta ({\gamma }^{\prime })$ is non-zero constant.

Proof. 

Let $\phi T=lT+mN+nW$ for some $l,m,n.$ We find $l=g(\phi T,T)=0$, then $\phi T=mN+nW.$ From this, we get

$$g(\phi T,\phi T)=m^2=a^{2}and0=g(\phi T,\xi )=n(a\tau +m).$$

Hence, $m=\pm a$ and $n=0$ or $m=-a\tau$.

If $n=0$, then $N=\frac{1}{m}\phi T=\pm \frac{1}{a}\phi T$. Using the Cartan equation, we find that $\tau =\frac{1}{2a^2}\mp 1$ and $W=\frac{1}{2a^2}{\gamma }^{\prime }-\frac{1}{a}\xi$.

Next, if $n\ne 0$ and $m=-a\tau$ then since $\gamma$ is a slant curve, differentiating $g(\phi T,N)=m=\pm a$, we have $n=g(\phi T,W)=0$, 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 $\kappa =1$ then τ is constant such that $\tau =\frac{1}{2a^2}\mp 1$.

4. Contact Magnetic Curves

In a three-dimensional Sasakian Lorentzian manifold $M^3$, the Reeb vector field $\xi$ is Killing. By Equation (6), the 2-form $\mathsf{\Phi }$ is $d\eta$, that is $d\eta (X,Y)=g(X,\phi Y)$, for all $X,Y\in \mathsf{\Gamma }(TM)$.

Let $\gamma :I\to M$ be a smooth curve on a contact Lorentzian manifold $(M,\phi ,\xi ,\eta ,g)$. Then, we define a magnetic field on M by

$$F_{\xi ,q}(X,Y)=-qd\eta (X,Y),$$

where $X,Y\in \mathbb{X}(M)$ and q is a non-zero constant. We call $F_{\xi ,q}$ the contact magnetic field with strength q.

Using Equations (1), (4) and (6) we get $\mathsf{\Phi }(X)=q\phi X$. Hence, from Equation (2) the Lorentz equation is

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=q\phi {\gamma }^{\prime }.$$

(20)

This is the generalized equation of geodesics under arc length parameterization, that is ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=0.$ 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 $F_{\xi ,q}$.

By using Equations (8) and (20), differentiating $g(\xi ,{\gamma }^{\prime })$ along a contact magnetic curve $\gamma$ in contact Lorentzian three-manifold

$$\begin{array}{ccc}\hfill \frac{d}{dt}g(\xi ,{\gamma }^{\prime })& =& g({\nabla }_{{\gamma }^{\prime }}\xi ,{\gamma }^{\prime })+g(\xi ,{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime })\hfill \\ & =& g(\phi {\gamma }^{\prime }-\phi h{\gamma }^{\prime },{\gamma }^{\prime })+g(\xi ,q\phi {\gamma }^{\prime })\hfill \\ & =& -g(\phi h{\gamma }^{\prime },{\gamma }^{\prime }).\hfill \end{array}$$

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 $\kappa$ and torsion $\tau$ along non-geodesic Frenet contact magnetic curves $\gamma$.

We suppose that $\eta (E_1)=a$, for a constant a. Then, using Equations (4), (10) and (20), we get

$${\epsilon }_2{\kappa }^2=q^{2}g(\phi {\gamma }^{\prime },\phi {\gamma }^{\prime })=q^2({\epsilon }_1+a^2).$$

Hence, we find that $\gamma$ has a constant curvature

$$\kappa =\mid q\mid \sqrt{{\epsilon }_2({\epsilon }_1+a^2)},$$

(21)

and, from Equations (10), (20) and (21), the binormal vector field

$$E_2=\frac{q}{{\epsilon }_2\kappa }\phi {\gamma }^{\prime }=\frac{\delta {\epsilon }_2}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}\phi {\gamma }^{\prime },$$

(22)

where $\delta =q/\mid q\mid$.

Using Proposition 3 and Equation (22), the binormal $E_3$ is computed as

$$\begin{array}{ccc}\hfill {\epsilon }_3{E}_3& =& E_1{\wedge }_L{E}_2\hfill \\ & =& {\gamma }^{\prime }{\wedge }_L(\frac{\delta {\epsilon }_2}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}\phi {\gamma }^{\prime })\hfill \\ & =& -\frac{\delta {\epsilon }_2}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}({\epsilon }_1\xi +a{\gamma }^{\prime }).\hfill \end{array}$$

Differentiating binormal vector field $E_3$, we have

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{E}_3& =& -\frac{\delta {\epsilon }_2{\epsilon }_3}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}{\nabla }_{{\gamma }^{\prime }}({\epsilon }_1\xi +a{\gamma }^{\prime })\hfill \\ & =& -\frac{\delta {\epsilon }_2{\epsilon }_3}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}({\epsilon }_1+qa)\phi {\gamma }^{\prime }.\hfill \end{array}$$

(23)

On the other hand, by Equation (10), we have

$${\nabla }_{{\gamma }^{\prime }}{E}_3={\epsilon }_2\tau E_2=\tau \frac{\delta \phi {\gamma }^{\prime }}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}.$$

(24)

From Equations (23) and (24), since ${\epsilon }_1{\epsilon }_2{\epsilon }_3=-1$, we obtain

$$\tau =1+{\epsilon }_{1}qa.$$

(25)

Moreover, if $\gamma$ is a non-geodesic curve, then

$$\frac{\tau -1}{\kappa }=\frac{\delta {\epsilon }_{1}a}{\sqrt{{\epsilon }_2({\epsilon }_1+a^2)}}.$$

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 $\kappa =\mid q\mid \sqrt{{\epsilon }_2({\epsilon }_1+a^2)}$ and torsion $\tau =1+{\epsilon }_{1}qa$. Moreover, the ratio of κ and $\tau -1$ is a constant.

Since a Legendre curve is a spacelike curve with spacelike normal vector field and $\eta ({\gamma }^{\prime })=a=0$, we assume that $\gamma$ 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 $\kappa =\left|q\right|$ and torsion $\tau =1$.

Now, from the geodesic curvature in Equation (21), if ${\epsilon }_1=1$, then $\eta ({\gamma }^{\prime })=a$ and $1\le 1+a^2$, and we have ${\epsilon }_2=1.$ Moreover, using ${\epsilon }_3=-{\epsilon }_1·{\epsilon }_2$, we obtain ${\epsilon }_3=-1.$ Next, if ${\epsilon }_1=-1$, then $\eta ({\gamma }^{\prime })=a=cosh{\alpha }_0$. Since $\gamma$ is a geodesic for $a=cosh{\alpha }_0=1$, we assume that $\gamma$ is non-geodesic, and we get $a^2>1$. Hence, $-1+a^2>0$ and we get ${\epsilon }_2={\epsilon }_3=1$. 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 $q=\pm a$ and same the result with Theorem 2.

Example

The Heisenberg group ${\mathbb{H}}_3$ is a Lie group which is diffeomorphic to ${\mathbb{R}}^3$ and the group operation is defined by

$$(x,y,z)\ast (\overline{x},\overline{y},\overline{z})=(x+\overline{x},y+\overline{y},z+\overline{z}+\frac{x\overline{y}}{2}-\frac{\overline{x}y}{2}).$$

The mapping

$${\mathbb{H}}_3\to \left\{\left(\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}\right)|a,b,c\in \mathbb{R}\right\}:(x,y,z)\mapsto \left(\begin{array}{ccc}1& x& z+\frac{xy}{2}\\ 0& 1& y\\ 0& 0& 1\end{array}\right)$$

is an isomorphism between ${\mathbb{H}}_3$ and a subgroup of $GL(3,\mathbb{R})$.

Now, we take the contact form

$$\eta =dz+(ydx-xdy).$$

Then, the characteristic vector field of $\eta$ is $\xi =\frac{\partial }{\partial z}.$

Now, we equip the Lorentzian metric as following:

$$g=d{x}^2+d{y}^2-{\left(dz+(ydx-xdy)\right)}^2.$$

We take a left-invariant Lorentzian orthonormal frame field $(e_1,e_2,e_3)$ on $({\mathbb{H}}_3,g)$:

$$e_1=\frac{\partial }{\partial x}-y\frac{\partial }{\partial z},e_2=\frac{\partial }{\partial y}+x\frac{\partial }{\partial z},e_3=\frac{\partial }{\partial z},$$

and the commutative relations are derived as follows:

$$[e_1,e_2]=2e_3,[e_2,e_3]=[e_3,e_1]=0.$$

Then, the endomorphism field $\phi$ is defined by

$$\phi e_1=e_2,\phi e_2=-e_1,\phi e_3=0.$$

The Levi–Civita connection ∇ of $({\mathbb{H}}_3,g)$ is described as

$$\begin{array}{c}{\nabla }_{e_1}{e}_1={\nabla }_{e_2}{e}_2={\nabla }_{e_3}{e}_3=0,{\nabla }_{e_1}{e}_2=e_3=-{\nabla }_{e_2}{e}_1,\\ {\nabla }_{e_2}{e}_3=-e_1={\nabla }_{e_3}{e}_2,{\nabla }_{e_3}{e}_1=e_2={\nabla }_{e_1}{e}_3.\end{array}$$

(26)

The contact form $\eta$ satisfies $d\eta (X,Y)=g(X,\phi Y).$ Moreover, the structure $(\eta ,\xi ,\phi ,g)$ is Sasakian.

The Riemannian curvature tensor R of $({\mathbb{H}}_3,g)$ is given by

$$\begin{array}{c}R(e_1,e_2)e_1=3e_2,R(e_1,e_2)e_2=-3e_1,\\ R(e_2,e_3)e_2=-e_3,R(e_2,e_3)e_3=-e_2,\\ R(e_3,e_1)e_3=e_1,R(e_3,e_1)e_1=e_3,\end{array}$$

and the other components are zero.

The sectional curvature is given by [6]

$$K(\xi ,e_i)=-R(\xi ,e_i,\xi ,e_i)=-1,fori=1,2,$$

and

$$K(e_1,e_2)=R(e_1,e_2,e_1,e_2)=3.$$

Thus, we see that the Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$ is the Lorentzian Sasakian space forms with constant holomorphic sectional curvature $\mu =3$.

Let $\gamma$ be a Frenet slant curve in Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$ parameterized by arc-length. Then, the tangent vector field has the form

$$T={\gamma }^{\prime }=\sqrt{{\epsilon }_1+a^2}cos\beta e_1+\sqrt{{\epsilon }_1+a^2}sin\beta e_2+a{e}_3,$$

(27)

where $a=constant,\beta =\beta (s).$ Using Equation (26), we get

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=\sqrt{{\epsilon }_1+a^2}({\beta }^{\prime }+2a)(-sin\beta e_1+cos\beta e_2).$$

(28)

Since $\gamma$ is a non-geodesic, we may assume that $\kappa =\sqrt{{\epsilon }_1+a^2}({\beta }^{\prime }+2a)>0$ without loss of generality.

Then, the normal vector field

$$N=-sin\beta e_1+cos\beta e_2.$$

The binormal vector field ${\epsilon }_{3}B=T{\wedge }_{L}N=-acos\beta e_1-asin\beta e_2-\sqrt{{\epsilon }_1+a^2}{e}_3.$ From Theorem 5, we see that ${\epsilon }_2=1$, thus we have ${\epsilon }_3=-{\epsilon }_1$. Hence,

$$B={\epsilon }_1(acos\beta e_1+asin\beta e_2+\sqrt{{\epsilon }_1+a^2}{e}_3).$$

Using the Frenet–Serret Equation (10), we have

Lemma 1.

Let γ be a Frenet slant curve in Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$ parameterized by arc-length. Then, γ admits an orthonormal frame field $\{T,N,B\}$ along γ and

$$\begin{array}{c}\kappa =\sqrt{{\epsilon }_1+a^2}({\beta }^{\prime }+2a),\\ \tau =1+{\epsilon }_{1}a({\beta }^{\prime }+2a).\end{array}$$

(29)

Next, if $\gamma$ is a null slant curve in the Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$, then the tangent vector field has the form

$$T={\gamma }^{\prime }=acos\beta e_1+asin\beta e_2+a{e}_3,$$

(30)

where $a=constant,\beta =\beta (s).$ Using Equation (26), we get

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=a({\beta }^{\prime }+2a)(-sin\beta e_1+cos\beta e_2).$$

(31)

Since $\gamma$ is non-geodesic, using Equation (18) we have $\mid a({\beta }^{\prime }+2a)\mid =1$ and

$$N=-sin\beta e_1+cos\beta e_2.$$

Differentiating N, we get

$${\nabla }_{{\gamma }^{\prime }}N=-({\beta }^{\prime }+a)cos\beta e_1-({\beta }^{\prime }+a)sin\beta e_2+a{e}_3.$$

From Equation (18), $\tau =\frac{1}{2}g({\nabla }_{{\gamma }^{\prime }}N,{\nabla }_{{\gamma }^{\prime }}N)=\frac{1}{2}{({\beta }^{\prime })}^2+a{\beta }^{\prime }.$ Since $W=-{\nabla }_{{\gamma }^{\prime }}N-\tau T$, we have

$$W=\{-\frac{1}{2}{({\beta }^{\prime })}^2+(\frac{1}{a}-a){\beta }^{\prime }+1\}T-({\beta }^{\prime }+2a)\xi =\frac{1}{2a}(cos\beta e_1+sin\beta e_2-e_3).$$

Therefore, we have

Lemma 2.

Let γ be a non-geodesic null slant curve in the Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$. We assume that $\kappa =\mid a({\beta }^{\prime }+2a)\mid =1$. Then, its torsion is constant such that $\tau =\frac{1}{2a^2}\mp 1.$

Let $\gamma (s)=(x(s),y(s),z(s))$ be a curve in Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$. Then, the tangent vector field ${\gamma }^{\prime }$ of $\gamma$ is

$${\gamma }^{\prime }=\left(\frac{dx}{ds},\frac{dy}{ds},\frac{dz}{ds}\right)=\frac{dx}{ds}\frac{\partial }{\partial x}+\frac{dy}{ds}\frac{\partial }{\partial y}+\frac{dz}{ds}\frac{\partial }{\partial z}.$$

Using the relations:

$$\frac{\partial }{\partial x}=e_1+y{e}_3,\frac{\partial }{\partial y}=e_2-x{e}_3,\frac{\partial }{\partial z}=e_3,$$

if $\gamma$ is a slant curve in $({\mathbb{H}}_3,g)$, then from Equation (27) the system of differential equations for $\gamma$ is given by

$$\begin{array}{ccc}\hfill \frac{dx}{ds}(s)& =& \sqrt{{\epsilon }_1+a^2}cos\beta (s),\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \frac{dy}{ds}(s)& =& \sqrt{{\epsilon }_1+a^2}sin\beta (s),\hfill \\ \hfill \frac{dz}{ds}(s)& =& a+\sqrt{{\epsilon }_1+a^2}(x(s)sin\beta (s)-y(s)cos\beta (s)).\hfill \end{array}$$

(33)

Now, we construct a magnetic curve $\gamma$ (containing Frenet and null curve) in the Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$. From Equations (20) and (28), we have:

Proposition 8.

Let $\gamma :I\to ({\mathbb{H}}_3,g)$ be a magnetic curve parameterized by arc-length in the Lorentzian Heisenberg space $({\mathbb{H}}_3,g)$. Then,

$${\beta }^{\prime }=q-2a,fora=\eta ({\gamma }^{\prime }).$$

Namely, ${\beta }^{\prime }$ is a constant, e.g., A, hence $\beta (s)=As+b,b\in \mathbb{R}$. If $\gamma$ is a null curve, then $q=\pm \frac{1}{a}$. Finally, from Equations (32) and (33), we have the following result:

Theorem 6.

Let $\gamma :I\to ({\mathbb{H}}_3,g)$ be a non-geodesic curve parameterized by arc-length s in the Lorentzian Heisenberg group $({\mathbb{H}}_3,g)$. If γ is a contact magnetic curve, then the parametric equations of γ are given by

$$\left\{\begin{array}{c}x(s)=\frac{1}{A}\sqrt{{\epsilon }_1+a^2}sin(As+b)+x_0,\hfill \\ y(s)=-\frac{1}{A}\sqrt{{\epsilon }_1+a^2}cos(As+b)+y_0,\hfill \\ z(s)=\{a+\frac{{\epsilon }_1+a^2}{A}\}s-\frac{\sqrt{{\epsilon }_1+a^2}}{A}\left\{x_{0}cos(As+b)+y_{0}sin(As+b)\right\}+z_0,\hfill \end{array}\right.$$

where $b,x_0,y_0,z_0$ are constants. If ${\epsilon }_1=0$ then γ is a null curve.

In particular, for a Frenet Legendre curve $\gamma$, we get ${\beta }^{\prime }=q=A$.