Slant Curves in Contact Lorentzian Manifolds with CR Structures

Abstract

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the $\hat{\nabla }$-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when $c\le 0$, there does not exist a non-geodesic slant Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for the $\hat{\nabla }$-geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$.

1. Introduction

The notion of slant curves was introduced in [1] for a contact Riemannian three-manifold, that is, a curve in a contact three-manifold is said to be slant if its tangent vector field has a constant angle with the Reeb vector field. In [2], we showed that proper biharmonic curves are helices in three-dimensional Sasakian space forms of constant holomorphic sectional curvature $\tilde{H}(=2c-3)$. In particular, if $\tilde{H}\ne 1$, then it is a slant helix; that is, a helix such that $\eta ({\gamma }^{\prime })=\cos {\alpha }_0$ is a constant, with ${\kappa }^2+{\tau }^2=1+(\tilde{H}-1){\sin }^2{\alpha }_0$. In [3], we studied slant curves satisfying $\hat{\nabla }$-Jacobi equations for a $\hat{\nabla }$-geodesic vector field in Sasakian space forms with respect to the Tanaka–Webster connection $\hat{\nabla }$. In [4], we showed that proper biharmonic Frenet curves are pseudo-helices in three-dimensional Lorentzian Sasakian space forms of constant holomorphic sectional curvature $H(=2c+3)$. In particular, if $H\ne -1$, then it is a slant pseudo-helix; that is, a pseudo-helix such that $\eta ({\gamma }^{\prime })$ is a constant, with ${\kappa }^2-{\tau }^2=-1+(H+1)(1+{\epsilon }_1{a}^2)$ for $a=\eta ({\gamma }^{\prime }).$

In this paper, we study the slant curves in Lorentzian Sasakian space forms of constant holomorphic sectional curvature $\hat{H}=2c$ for the Tanaka–Webster connection $\hat{\nabla }.$

D. Perrone [5,6] showed that the notion of non-degenerate almost CR structures is equivalent to the notion of contact pseudo-metric structures. Thus, he defined the generalized Tanaka–Webster connection $\hat{\nabla }$ in a contact pseudo-metric manifold.

In Section 3, we find the properties of the Tanaka–Webster connection in a contact Lorentzian manifold. In Section 4.1, we find that a necessary and sufficient condition for a $\hat{\nabla }$-geodesic is a magnetic curve (for ∇) along slant curves.

Next, we investigate the $\hat{\nabla }$-Jacobi equation for a $\hat{\nabla }$-geodesic vector field in contact Lorentzian manifolds:

$$\left\{\begin{array}{c}{\hat{\nabla }}_{{\gamma }^{\prime }}{\gamma }^{\prime }=\hat{\sigma }(\gamma ),\hfill \\ {\hat{\nabla }}_{{\gamma }^{\prime }}^2\hat{\sigma }(\gamma )-{\hat{\nabla }}_{{\gamma }^{\prime }}\hat{T}(\hat{\sigma }(\gamma ),{\gamma }^{\prime })-\hat{R}(\hat{\sigma }(\gamma ),{\gamma }^{\prime }){\gamma }^{\prime }=0,\hfill \end{array}\right.$$

(1)

where the torsion $\hat{T}(X,Y)=[X,Y]-{\hat{\nabla }}_{X}Y+{\hat{\nabla }}_{Y}X$ and pseudo-Hermitian curvature $\hat{R}(X,Y)={\hat{\nabla }}_{[X,Y]}-[{\hat{\nabla }}_X,{\hat{\nabla }}_Y].$ Then, in Section 4.2, we prove that when $c\le 0$, there does not exist a non-geodesic slant Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for the $\hat{\nabla }$-geodesic vector fields in M. Thus, we obtain the explicit parametric equations satisfying (1) in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$.

2. Preliminaries

2.1. Contact Lorentzian Manifold

An almost contact structure $(\phi ,\xi ,\eta )$ on a $(2n+1)$-dimensional differentiable manifold M has a tensor field $\phi$ of $(1,1)$, a global vector field $\xi$, and a 1-form $\eta$ such that

$$\begin{array}{c}\hfill {\phi }^2=-I+\eta \otimes \xi ,\eta (\xi )=1,\end{array}$$

(2)

$$\begin{array}{c}\hfill \phi (\xi )=0,\eta \circ \phi =0.\end{array}$$

(3)

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 that 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 naturally define 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 $M\times \mathbb{R}$. If the almost complex structure J is integrable, then the contact Lorentzian manifold M is called normal or Sasakian. It is known that 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

([7,8]). 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 similar arguments and computations to those of [9], we obtain:

Proposition 2

([7,8]). Let $(M^{2n+1},\eta ,\xi ,\phi ,g)$ be a contact Lorentzian manifold. Then

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

(8)

where $h=\frac{1}{2}{L}_{\xi }\phi .$

If $\xi$ is a killing vector field with respect to the Lorentzian metric g, that is, $M^{2n+1}$ is a K-contact Lornetzian manifold. Then

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

(9)

Proposition 3.

Let $\{T,N,B\}$ be orthonormal Frame fields in a Lorentzian three-manifold. Then

$$T{\wedge }_{L}N={\epsilon }_{3}B,N{\wedge }_{L}B={\epsilon }_{1}T,B{\wedge }_{L}T={\epsilon }_{2}N.$$

2.2. Lorentzian Bianchi–Cartan–Vranceanu Model Space

The one-parameter family of Riemannian three-manifolds ${\left\{{\mathcal{M}}^3(\tilde{H})\right\}}_{\tilde{H}\in \mathbb{R}}$ is classically known by L. Bianchi [10], E. Cartan [11], and G. Vranceanu [12]. The model ${\mathcal{M}}^3(\tilde{H})$ of the Sasakian three-space form is called the Bianchi–Cartan–Vranceanu model of the three-dimensional Sasakian space form. Cartan classified all three-dimensional spaces with four-dimensional isometry groups in [11]. Thus, he proved that they are all homogeneous. Moreover, parallel surfaces in Bianchi–Cartan–Vranceanu spaces are classified in [13].

On the other hand, G. Calvaruso [7] proved that there is a one-to-one correspondence between homogeneous contact Riemannian three-manifolds and homogeneous contact Lorentzian three-manifolds.

Now, we construct a Lorentzian Bianchi–Cartan–Vranceanu model of three-dimensional Lorentzian Sasakian space forms.

Let c be a real number, and set

$$\mathcal{D}=\left\{(x,y,z)\in {\mathbb{R}}^3(x,y,z)|1+\frac{c}{2}(x^2+y^2)>0\right\}.$$

Note that $\mathcal{D}$ is the whole ${\mathbb{R}}^3(x,y,z)$ for $c\ge 0$. In the region $\mathcal{D}$, we take the contact form

$$\eta =dz+\frac{ydx-xdy}{1+\frac{c}{2}(x^2+y^2)}.$$

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

Next, we equip $\mathcal{D}$ with the Lorentzian metric $g_c$ as follows:

$$g_c=\frac{d{x}^2+d{y}^2}{{\{1+\frac{c}{2}(x^2+y^2)\}}^2}-{\left(dz+\frac{ydx-xdy}{1+\frac{c}{2}(x^2+y^2)}\right)}^2.$$

We take the following orthonormal frame field on $(\mathcal{D},g_c)$:

$$u_1=\{1+\frac{c}{2}(x^2+y^2)\}\frac{\partial }{\partial x}-y\frac{\partial }{\partial z},u_2=\{1+\frac{c}{2}(x^2+y^2)\}\frac{\partial }{\partial y}+x\frac{\partial }{\partial z},u_3=\frac{\partial }{\partial z}.$$

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

$$\phi u_1=u_2,\phi u_2=-u_1,\phi u_3=0.$$

The Levi–Civita connection ∇ of this Lorentzian three-manifold is described as

$${\nabla }_{u_1}{u}_1=cy{u}_2,{\nabla }_{u_1}{u}_2=-cy{u}_1+u_3,{\nabla }_{u_1}{u}_3=u_2,$$

$${\nabla }_{u_2}{u}_1=-cx{u}_2-u_3,{\nabla }_{u_2}{u}_2=cx{u}_1,{\nabla }_{u_2}{u}_3=-u_1,$$

(10)

$${\nabla }_{u_3}{u}_1=u_2,{\nabla }_{u_3}{u}_2=-u_1,{\nabla }_{u_3}{u}_3=0.$$

$$[u_1,u_2]=-cy{u}_1+cx{u}_2+2u_3,[u_2,u_3]=[u_3,u_1]=0.$$

The contact form $\eta$ on $\mathcal{D}$ satisfies

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

(11)

Moreover, the structure $(\phi ,\xi ,\eta ,g_c)$ is Sasakian. The curvature tensor $R(X,Y)={\nabla }_{[X,Y]}-[{\nabla }_X,{\nabla }_Y]$ on $(M^3,\eta ,\xi ,\phi ,g_c)$ is given by

$$R(u_1,u_2)u_2=-(2c+3)u_1,R(u_1,u_3)u_3=-u_1,$$

$$R(u_2,u_1)u_1=-(2c+3)u_2,R(u_2,u_3)u_3=-u_2,$$

$$R(u_3,u_1)u_1=u_3,R(u_3,u_2)u_2=u_3.$$

The sectional curvature ([7]) is given by

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

and

$$K(u_1,u_2)=R(u_1,u_2,u_1,u_2)=2c+3.$$

Hence, $(\mathcal{D},g_c)$ is of constant holomorphic sectional curvature $H=2c+3$.

Hereafter, we denote this model $(\mathcal{D},g_c)$ of a Lorentzian Sasakian space form by ${\mathcal{M}}_1^3(H)$.

The harmonic maps $\varphi :(M^m,g)\to (N^n,h)$ between two pseudo-Riemannian manifolds as critical points of the energy $E(\varphi )={\int }_M{\left|d\varphi \right|}^{2}dv.$ The tension field ${\tau }_{\varphi }$ is defined by

$${\tau }_{\varphi }=trace{\nabla }^{\varphi }d\varphi ={\sum }_{i=1}^m{\epsilon }_i({\nabla }_{e_i}^{\varphi }d\varphi (e_i)-d\varphi ({\nabla }_{e_i}{e}_i)),$$

where ${\nabla }^{\varphi }$ and $\left\{e_i\right\}$ denote the induced connection by $\varphi$ on the bundle ${\varphi }^{*}T{N}^n$. A smooth map $\varphi$ is called a harmonic map if its tension field vanishes.

Next, the bienergy $E_2(\varphi )$ of a map $\varphi$ is defined by $E_2(\varphi )={\int }_M{\left|{\tau }_{\varphi }\right|}^{2}dv,$; $\varphi$ is biharmonic if it is a critical point of the bienergy. Harmonic maps are clearly biharmonic. Non-harmonic biharmonic maps are called proper biharmonic maps. We define the bitension field ${\tau }_2(\varphi )$ by

$${\tau }_2(\varphi ):={\sum }_{i=1}^m{\epsilon }_i(({\nabla }_{e_i}^{\varphi }{\nabla }_{e_i}^{\varphi }-{\nabla }_{{\nabla }_{e_i}{e}_i}^{\varphi }){\tau }_{\varphi }-R^N({\tau }_{\varphi },d\varphi (e_i))d\varphi (e_i)),$$

where $R^N$ is the curvature tensor of $N^n$ and is defined by $R^N(X,Y)={\nabla }_{[X,Y]}-[{\nabla }_X,{\nabla }_Y]$ (see [14]).

We now restrict our attention to isometric immersions $\gamma :I\to (M,g)$ from an interval I to a pseudo-Riemannian manifold. The image $C=\gamma (I)$ is the trace of a curve in M, and $\gamma$ is a parametrization of C by arc length. In this case, the tension field becomes ${\tau }_{\gamma }={\epsilon }_1{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }$ and the biharmonic equation reduces to

$${\tau }_2(\gamma )={\epsilon }_1({\nabla }_{{\gamma }^{\prime }}^2{\tau }_{\gamma }-R({\tau }_{\gamma },{\gamma }^{\prime }){\gamma }^{\prime })=0.$$

Note that $C=\gamma (I)$ is part of a geodesic of M if and only if $\gamma$ is harmonic. Moreover, from the biharmonic equation, if $\gamma$ is harmonic, geodesics are a subclass of biharmonic curves.

In [4], we showed that proper biharmonic Frenet curves are pseudo-helices in three-dimensional Lorentzian Sasakian space forms of constant holomorphic sectional curvature $H(=2c+3)$. In particular, in [15], we studied proper biharmonic spacelike curves in Lorentzian Heisenberg space.

3. Almost CR Manifold

We recall the notions of CR structure and pseudo-Hermitian geometry.

Let $(M,\mathcal{H}(M),J,\theta )$ be a non-degenerate almost CR manifold. If we extend J to an endomorphism $\phi$ of the tangent bundle by $\phi {\mid }_{\mathcal{H}\left(M\right)}=J$ and $\phi \left(P\right)=0,$ where P is the Reeb vector field of $\theta ,$ then ${\phi }^2=-I+\theta \otimes P$. The Webster metric $g_{\theta }$ is given by

$$g_{\theta }(X,Y)=\left(d\theta \right)(X,JY),g_{\theta }(X,P)=0,g_{\theta }(P,P)=\epsilon (=\pm 1),$$

for any $X,Y\in \mathcal{H}\left(M\right).$ $g_{\theta }$ is a pseudo-Riemannian metric on $M.$ Hence,

$$\phi ,\xi =-P,\eta =-\theta ,g=g_{\theta }$$

is a contact pseudo-metric structure on $M.$ Conversely, a contact pseudo-metric structure $(\phi ,\xi ,\eta ,g)$ defines a non-degenerate almost CR structure on M given by $\left(\mathcal{H}\right(M),J,\theta )$, where $\mathcal{H}\left(M\right)=ker\eta$, $\theta =-\eta$ and $J=\phi {\mid }_{\mathcal{H}\left(M\right)}.$ Then, we have

Proposition 4

([5]). The notion of a non-degenerate almost CR structure is equivalent to the notion of a contact pseudo-metric structure.

Tanaka ([16]) defined the canonical affine connection, called the Tanaka–Webster connection, on a non-degenerate CR manifold. D. Perrone defined the generalized Tanaka–Webster connection [5] on a contact pseudo-metric manifold $M=(M^{2n+1},\eta ,\xi ,\phi ,g)$.

In this section, we consider the generalized Tanaka–Webster connection on a contact Lorentzian manifold $M.$

The generalized Tanaka–Webster connection $\hat{\nabla }$ is defined by (cf. [3,16] )

$${\hat{\nabla }}_{X}Y={\nabla }_{X}Y-\eta \left(X\right)\phi Y+\left({\nabla }_X\eta \right)\left(Y\right)\xi -\eta \left(Y\right){\nabla }_X\xi ,$$

for all vector fields $X,Y$ on M. $\hat{\nabla }$ may be rewritten as

$${\hat{\nabla }}_{X}Y={\nabla }_{X}Y+A(X,Y).$$

Then, using (5) and (8), we have

$$A(X,Y)=-\eta \left(X\right)\phi Y-\eta \left(Y\right)(\phi X-\phi hX)-g(\phi X-\phi hX,Y)\xi .$$

(12)

Next, if we define the torsion $\hat{T}(X,Y)=[X,Y]-{\hat{\nabla }}_{X}Y+{\hat{\nabla }}_{Y}X$ for the Tanaka–Webster connection $\hat{\nabla }$ in M ([17]), then we get

$$\hat{T}(X,Y)=2g(\phi X,Y)\xi +\eta \left(X\right)\phi hY-\eta \left(Y\right)\phi hX.$$

(13)

In particular, for a K-contact manifold, (12) and the above equation reduce as follows:

$$\begin{array}{ccc}\hfill A(X,Y)& =& -\eta \left(X\right)\phi Y-\eta \left(Y\right)\phi X-g(\phi X,Y)\xi ,\hfill \\ \hfill \hat{T}(X,Y)& =& 2g(\phi X,Y)\xi .\hfill \end{array}$$

Using (2)–(9), we have

Theorem 1.

The generalized Tanaka–Webster connection $\hat{\nabla }$ on a contact Lorentzian manifold $M=(M^{2n+1};\eta ,\xi ,\phi ,g)$ is the unique linear connection satisfying the following conditions:

(a)

$\hat{\nabla }\eta =0$, $\hat{\nabla }\xi =0$,

(b)

$\hat{\nabla }g=0$,

(c)

$\hat{T}(X,Y)=2g(\phi X,Y)\xi$, $X,Y\in \mathcal{D}$,

(d)

$\hat{T}(\xi ,\phi Y)=-\phi \hat{T}(\xi ,Y)$, $Y\in \mathcal{D}$,

(e)

$\left({\hat{\nabla }}_X\phi \right)Y=Q(X,Y)$, $X,Y\in TM$.

The Tanaka–Webster connection on a non-degenerate (integrable) CR manifold is defined as the unique linear connection satisfying (a), (b), (c), (d), and $Q=0$ (CR integrability), where Q is a $(1,2)$-tensor field on M defined by $Q(X,Y)=\left({\nabla }_X\phi \right)Y-g(X-hX,Y)\xi -\eta \left(Y\right)(X-hX).$

Thus, in [5] (page 217), we find:

Corollary 1.

Let $(M^{2n+1},\eta ,\xi ,\phi ,g)$ be a contact Lorentzian manifold. Then, the $(M^{2n+1},D)$ is a (strongly pseudoconvex) CR manifold if and only if

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

In particular, if $M^{2n+1}$ is a Lorentzian Sasakian manifold, then it satisfies (7). In fact, every three-dimensional contact Lorentzian manifold is a (strongly pseudoconvex) CR manifold. Thus, a three-dimensional K-contact manifold is Sasakian.

4. Slant Curves in Non-Degenerate CR 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 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 $\{T={\gamma }^{\prime },N,B\}$ along $\gamma$. Then, the Frenet–Serret equations, following [14,18], are:

$$\left\{\begin{array}{c}{\hat{\nabla }}_{{\gamma }^{\prime }}T={\epsilon }_2\hat{\kappa }N,\hfill \\ {\hat{\nabla }}_{{\gamma }^{\prime }}N=-{\epsilon }_1\hat{\kappa }T-{\epsilon }_3\hat{\tau }B,\hfill \\ {\hat{\nabla }}_{{\gamma }^{\prime }}B={\epsilon }_2\hat{\tau }N,\hfill \end{array}\right.$$

(14)

where $\hat{\kappa }=\left|{\hat{\nabla }}_{{\gamma }^{\prime }}{\gamma }^{\prime }\right|$ is the geodesic curvature of $\gamma$ and $\hat{\tau }$ is its geodesic torsion for the Tanaka–Webster connection $\hat{\nabla }$. The vector fields T, N, and B are called the tangent vector field, principal normal vector field, and binormal vector field of $\gamma$, respectively.

The constants ${\epsilon }_2$ and ${\epsilon }_3$ are defined by $g(N,N)={\epsilon }_2$ and $g(B,B)={\epsilon }_3,$ and are called the second causal character and third causal character of $\gamma$, respectively. Thus, this satisfies ${\epsilon }_1{\epsilon }_2=-{\epsilon }_3$.

A Frenet curve $\gamma$ is a pseudo-Hermitian geodesic if and only if $\hat{\kappa }=0$. A Frenet curve $\gamma$ with constant geodesic curvature and zero geodesic torsion is called a pseudo-Hermitian pseudo-circle. A pseudo-Hermitian pseudo-helix is a Frenet curve $\gamma$ whose geodesic curvature and torsion are constant.

4.1. Slant Curves

A one-dimensional integral submanifold of D in a three-dimensional contact manifold is called a Legendre curve, especially to avoid confusion with an integral curve of the vector field $\xi$. As a generalization of the Legendre curve, the notion of slant curves was introduced in [1] for a contact Riemannian three-manifold, that is, a curve in a contact three-manifold is said to be slant if its tangent vector field has a constant angle with the Reeb vector field.

Similarly to in the contact Riemannian three-manifolds, a curve in a contact Lorentzian three-manifold is said to be slant if its tangent vector field has a constant angle with the Reeb vector field (i.e., $g({\gamma }^{\prime },\xi )$ is a constant). In particular, if $g({\gamma }^{\prime },\xi )=0$ then $\gamma$ is a Legendre curve.

Let $\gamma$ be a Frenet curve in a Sasakian Lorentzian three-manifold $M^3$. Then, we get

$${\hat{\nabla }}_{{\gamma }^{\prime }}{\gamma }^{\prime }={\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }-2\eta \left({\gamma }^{\prime }\right)\phi {\gamma }^{\prime }.$$

If $\gamma$ is a slant curve, then since $\eta \left({\gamma }^{\prime }\right)=a$, a is a constant, ${\hat{\nabla }}_{{\gamma }^{\prime }}{\gamma }^{\prime }=0$ if and only if ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=2a\phi {\gamma }^{\prime }$. Hence, we have:

Proposition 5.

A Frenet curve γ in a Sasakian Lorentzian three-manifold $M^3$ is a slant curve. Then, γ is a geodesic for $\hat{\nabla }$ if and only if it is a magnetic curve (for ∇).

Recently, we studied slant curves and magnetic curves in Sasakian Lorentzian three-manifolds (see [19]). If a curve $\gamma$ satisfies ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=q\phi {\gamma }^{\prime }$, then we call it a contact magnetic curve in a contact Riemannian and Lorentzian manifold; we proved that $\gamma$ is a slant curve if and only if M is Sasakian.

Now, we assume that $\eta \left(T\right)=a$, where a is a function. Using (5) and differentiating $g(T,\xi )=-a$ along $\gamma$ for a Tanaka–Webster connection $\hat{\nabla }$, then

$$-a^{\prime }=g({\epsilon }_2\hat{\kappa }N,\xi )+g(T,{\hat{\nabla }}_T\xi )=-{\epsilon }_2\hat{\kappa }\eta \left(N\right).$$

This equation implies:

Proposition 6.

A non-geodesic Frenet curve γ for $\hat{\nabla }$ in a Sasakian Lorentzian three-manifold $M^3$ is a slant curve if and only if $\eta \left(N\right)=0$.

Moreover, we have:

Lemma 1.

Let γ be a non-geodesic slant curve in the three-dimensional almost contact Lorentzian manifold M. Then, we find an orthonormal frame field in M as follows:

$$T={\gamma }^{\prime },N=\frac{\phi T}{\sqrt{{\epsilon }_1+a^2}},B=\frac{\xi +{\epsilon }_{1}aT}{\sqrt{{\epsilon }_1+a^2}},$$

and $\xi =-{\epsilon }_{1}aT+\sqrt{{\epsilon }_1+a^2}B.$

Thus, γ is a spacelike curve with a spacelike normal vector field or timelike curve.

Differentiating $\xi =-{\epsilon }_{1}aT+\sqrt{{\epsilon }_1+a^2}B$ along $\gamma$ for $\hat{\nabla }$ and using (14), we have:

Proposition 7.

A non-geodesic Frenet curve γ in a Sasakian Lorentzian three-manifold $M^3$ is a slant curve. Then, the ratio of $\hat{\kappa }$ and $\hat{\tau }$ is constant.

4.2. $\hat{\nabla }$-Jacobi Equations

We find that the non-vanishing Tanaka–Webster connections $\hat{\nabla }$ of the Bianchi–Cartan–Vranceanu model space are

$${\hat{\nabla }}_{u_1}{u}_1=cy{u}_2,{\hat{\nabla }}_{u_1}{u}_2=-cy{u}_1,{\hat{\nabla }}_{u_2}{u}_1=-cx{u}_2,{\hat{\nabla }}_{u_2}{u}_2=cx{u}_1.$$

By using the above data, we calculate the Tanaka–Webster curvature tensor $\hat{R}(X,Y)Z={\hat{\nabla }}_{[X,Y]}Z-{\hat{\nabla }}_X\left({\hat{\nabla }}_{Y}Z\right)+{\hat{\nabla }}_Y\left({\hat{\nabla }}_{X}Z\right).$ Then, we find that

$$\hat{R}(u_1,u_2)u_2=-2c{u}_1,\hat{R}(u_1,u_2)u_1=2c{u}_2,$$

(15)

and all others are zero.

As $\hat{H}=H-3$, we find that constant holomorphic sectional curvature $\hat{H}=2c$ for the Tanaka–Webster connection $\hat{\nabla }.$ Hereafter, we denote the Lorentzian Bianchi–Cartan–Vranceanu model space for $\hat{\nabla }$ by ${\mathcal{M}}_1^3\left(\hat{H}\right)$.

Using (14), we get

$$\begin{array}{c}\hfill {\hat{\nabla }}_T^{3}T=3{\epsilon }_3\hat{\kappa }{\hat{\kappa }}^{\prime }T+{\epsilon }_2({\hat{\kappa }}^{″}-{\epsilon }_2\hat{\kappa }({\epsilon }_1{\hat{\kappa }}^2+{\epsilon }_3{\hat{\tau }}^2))N+{\epsilon }_1(2{\hat{\kappa }}^{\prime }\hat{\tau }+\hat{\kappa }{\hat{\tau }}^{\prime })B.\end{array}$$

From the curvature tensor (15) and Proposition 3, we have

$$\begin{array}{cc}\hfill & \hat{R}(\hat{\kappa }N,T)T\hfill \\ \hfill =& \hat{\kappa }\hat{R}(N_1{e}_1+N_2{e}_2+N_3{e}_3,T_1{e}_1+T_2{e}_2+T_3{e}_3)(T_1{e}_1+T_2{e}_2+T_3{e}_3)\hfill \\ \hfill =& -2c{\epsilon }_2\hat{\kappa }\left[B_3^{2}N-N_3{B}_{3}B\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\hat{\nabla }}_T^{3}T-{\epsilon }_2\hat{R}(\hat{\kappa }N,T)T& =& 3{\epsilon }_3\hat{\kappa }{\hat{\kappa }}^{\prime }T+\left[{\epsilon }_2{\hat{\kappa }}^{″}-\hat{\kappa }({\epsilon }_1{\hat{\kappa }}^2+{\epsilon }_3{\hat{\tau }}^2-2c{B}_3^2)\right]N\hfill \\ & & +\left[{\epsilon }_1(2{\hat{\kappa }}^{\prime }\hat{\tau }+\hat{\kappa }{\hat{\tau }}^{\prime })-2c\hat{\kappa }{N}_3{B}_3\right]B.\hfill \end{array}$$

Hence, we have:

Proposition 8.

Let $\gamma :I\to M$ be a non-geodesic slant Frenet curve in the Lorentzian Sasakian space forms ${\mathcal{M}}_1^3\left(\hat{H}\right)$ for the Tanaka–Webster connection $\hat{\nabla }$. Then, γ satisfies ${\hat{\nabla }}_T^{3}T-\hat{R}({\hat{\nabla }}_{T}T,T)T=0$ if and only if γ is a pseudo-Hermitian pseudo-helix with ${\hat{\kappa }}^2-{\hat{\tau }}^2=2c{\epsilon }_1\eta {\left(B\right)}^2.$

Using (14), we calculate

$${\hat{\nabla }}_T^{2}T={\epsilon }_3{\hat{\kappa }}^{2}T+{\epsilon }_2\hat{\kappa }N-{\epsilon }_1\hat{\kappa }\hat{\tau }B,$$

and we get

$$g({\hat{\nabla }}_T^{2}T,\phi {\gamma }^{\prime })={\epsilon }_2{\hat{\kappa }}^{\prime }\sqrt{{\epsilon }_1+a^2}.$$

Thus, we have:

Proposition 9.

A non-geodesic slant Frenet curve γ in a three-dimensional Sasakian Lorentzian manifold ${\mathcal{M}}_1^3\left(\hat{H}\right)$ satisfies $g({\hat{\nabla }}_T^{2}T,\phi {\gamma }^{\prime })=0$ if and only if $\hat{\kappa }$ is a non-zero constant.

Hence, we obtain:

Theorem 2.

Let $\gamma :I\to M$ be a non-geodesic slant Frenet curve in the Lorentzian Sasakian space forms ${\mathcal{M}}_1^3\left(\hat{H}\right)$ for the Tanaka–Webster connection $\hat{\nabla }$. Then, γ satisfies the $\hat{\nabla }$-Jacobi equation for a $\hat{\nabla }$-geodesic vector field if and only if it is a pseudo-Hermitian pseudo-helix with ${\hat{\kappa }}^2-{\hat{\tau }}^2=2c{\epsilon }_1\eta {\left(B\right)}^2.$

Let $\gamma$ be a slant Frenet curve in Lorentzian Sasakian space forms ${\mathcal{M}}_1^3(\hat{H})$ parametrized by arc-length. Then, the tangent vector field T has the form

$$T={\gamma }^{\prime }=\sqrt{{\epsilon }_1+a^2}\cos \beta u_1+\sqrt{{\epsilon }_1+a^2}\sin \beta u_2+a{u}_3,$$

(16)

where $a=constant,\beta =\beta \left(s\right).$ Using (10), since $\gamma$ is a non-geodesic, we may assume that $\hat{\kappa }=\sqrt{{\epsilon }_1+a^2}({\beta }^{\prime }+cy\sqrt{{\epsilon }_1+a^2}\cos \beta -cx\sqrt{{\epsilon }_1+a^2}\sin \beta )>0$ without loss of generality. Then, we get the normal vector field

$$N=-\sin \beta u_1+\cos \beta u_2.$$

The binormal vector field ${\epsilon }_{3}B=T{\wedge }_{L}N=-a\cos \beta u_1-a\sin \beta u_2-\sqrt{{\epsilon }_1+a^2}{u}_3.$ From the Lemma 1, we see that ${\epsilon }_2=1$, so we have ${\epsilon }_3=-{\epsilon }_1$. Hence, we have the binormal vector field

$$B={\epsilon }_1(a\cos \beta u_1+a\sin \beta u_2+\sqrt{{\epsilon }_1+a^2}{u}_3).$$

Using the Frenet–Serret Equation (14), we have:

Lemma 2.

Let γ be a slant Frenet curve in Lorentzian Sasakian space forms ${\mathcal{M}}_1^3\left(\hat{H}\right)$ parametrized by arc-length. Then, γ admits an orthonormal frame field $\{T,N,B\}$ along γ and

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

(17)

From this, we find that ${\hat{\kappa }}^2-{\hat{\tau }}^2=2c{\epsilon }_1\eta {(B)}^2$ if and only if ${\{{\beta }^{\prime }+c\sqrt{{\epsilon }_1+a^2}(y\cos \beta -x\sin \beta )\}}^2=2c({\epsilon }_1+a^2).$ Hence, we have:

Corollary 2.

Let ${\mathcal{M}}_1^3(\hat{H})$ be a Lorentzian Sasakian space form with $c\le 0$. Then, there does not exist a non-geodesic slant Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for $\hat{\nabla }$-geodesic vector fields.

Since for $c>0$, we get $\hat{H}=H-3=2c>0$, we now construct a non-geodesic slant Frenet curve $\gamma$ satisfying (1) in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$.

Let $\gamma (s)=(x(s),y(s),z(s))$ be a curve in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$. Then, the tangent vector field T of $\gamma$ is

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

using the relations:

$$\frac{\partial }{\partial x}=\frac{1}{\{1+\frac{c}{2}(x{(s)}^2+y{(s)}^2)\}}(u_1+y{u}_3),\frac{\partial }{\partial y}=\frac{1}{\{1+\frac{c}{2}(x{(s)}^2+y{(s)}^2)\}}(u_2-x{u}_3),\frac{\partial }{\partial z}=u_3.$$

If $\gamma$ is a slant Frenet curve in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$, then from (16), the system of differential equations for $\gamma$ is given by

$$\begin{array}{ccc}\hfill \frac{dx}{ds}(s)& =& \sqrt{{\epsilon }_1+a^2}\cos \beta (s)\{1+\frac{c}{2}(x{(s)}^2+y{(s)}^2)\},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill \frac{dy}{ds}(s)& =& \sqrt{{\epsilon }_1+a^2}\sin \beta (s)\{1+\frac{c}{2}(x{(s)}^2+y{(s)}^2)\},\hfill \end{array}$$

(19)

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

(20)

From the Theorem 2 and (17), we have:

Corollary 3.

Let $\gamma :I\to M_1^3(\hat{H})$ be a non-geodesic slant Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for the $\hat{\nabla }$-geodesic in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$. Then

$${\beta }^{\prime }+c\sqrt{{\epsilon }_1+a^2}(y\cos \beta -x\sin \beta )=\pm \sqrt{2c({\epsilon }_1+a^2)}.$$

(21)

Together with (21), we see that the Equation (20) becomes

$$\frac{dz}{ds}=\frac{1}{c}({\beta }^{\prime }\pm \sqrt{2c({\epsilon }_1+a^2)})+a.$$

Thus, we have

$$z\left(s\right)=\frac{1}{c}\beta \left(s\right)+\{a\pm \frac{1}{c}\sqrt{2c({\epsilon }_1+a^2)}\}s+z_0,$$

where $z_0$ is a constant. We now compute the x and y coordinates. We put $h(s):=1+\frac{c}{2}(x{(s)}^2+y{(s)}^2)$. Then, (18) and (19) become

$$\frac{dx}{ds}=\sqrt{{\epsilon }_1+a^2}\cos \beta \left(s\right)h\left(s\right),\frac{dy}{ds}=\sqrt{{\epsilon }_1+a^2}\sin \beta \left(s\right)h\left(s\right),$$

respectively. We note that the function $h\left(s\right)$ satisfies the following Ordinary Differential Equation:

$$\frac{d}{ds}\log \left|h\left(s\right)\right|=c\sqrt{{\epsilon }_1+a^2}(\cos \beta \left(s\right)x\left(s\right)+\sin \beta \left(s\right)y\left(s\right)).$$

Differentiating (21), we have

$$\frac{d^2}{d{s}^2}\beta \left(s\right)=\frac{d\beta }{ds}\left(s\right)\frac{d}{ds}\log \left|h\left(s\right)\right|.$$

First, if $d\beta /ds=0$ for all s, then $\left(x\right(s),y(s\left)\right)$ is a line in the orbit space. Hence, we have the following parametrization:

$$\left\{\begin{array}{c}x\left(s\right)=\sqrt{{\epsilon }_1+a^2}\cos {\beta }_0\int h\left(s\right)ds,\hfill \\ y\left(s\right)=\sqrt{{\epsilon }_1+a^2}\sin {\beta }_0\int h\left(s\right)ds,\hfill \\ z\left(s\right)=\{a\pm \frac{1}{c}\sqrt{2c({\epsilon }_1+a^2)}\}s+z_0,\hfill \end{array}\right.$$

where $\int h\left(s\right)ds=\sqrt{-\frac{2}{c({\epsilon }_1+a^2)}}+{\{p\exp \left(-\sqrt{-2c({\epsilon }_1+a^2)}s\right)-\sqrt{-\frac{c({\epsilon }_1+a^2)}{8}}\}}^{-1},p\in \mathbb{R}$, and $c<0.$ So, we conclude that $\beta$ is not constant along $\gamma .$

Next, we assume that $\frac{d\beta }{ds}{\mid }_{s=s_0}\ne 0$ for some $s=s_0$. Then, we get $h\left(s\right)=r\frac{d\beta }{ds}s,$ $r\in R.$ Thus, we have

$$\left\{\begin{array}{c}x\left(s\right)=r\sqrt{{\epsilon }_1+a^2}\sin \beta \left(s\right)+x_0,\hfill \\ y\left(s\right)=-r\sqrt{{\epsilon }_1+a^2}\cos \beta \left(s\right)+y_0.\hfill \end{array}\right.$$

Since $c>0$, the orbit space is the whole plane ${\mathbb{R}}^2(x,y)$. The projected curve $\overline{\gamma }\left(s\right)$ is a circle ${(x-x_0)}^2+{(y-y_0)}^2=r^2({\epsilon }_1+a^2)$. We may assume that $\overline{\gamma }$ is a circle centered at $(0,0)$. Then, the angle function $\beta$ is given by

$$\beta \left(s\right)=\frac{1}{r}\left(\frac{c}{2}{r}^2({\epsilon }_1+a^2)+1\right)s+{\beta }_0.$$

Therefore, we obtain:

Theorem 3.

Let $\gamma :I\to M_1^3(\hat{H})$ be a non-geodesic slant Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for the $\hat{\nabla }$-geodesic in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$. Then, its parametric equations are given by

$$\left\{\begin{array}{c}x(s)=r\sqrt{{\epsilon }_1+a^2}\sin (\frac{1}{r}\left(\frac{c}{2}{r}^2({\epsilon }_1+a^2)+1\right)s+{\beta }_0)+x_0,\hfill \\ y(s)=-r\sqrt{{\epsilon }_1+a^2}\cos (\frac{1}{r}\left(\frac{c}{2}{r}^2({\epsilon }_1+a^2)+1\right)s+{\beta }_0)+y_0,\hfill \\ z(s)=[a+\frac{1}{c}\{\frac{1}{r}\left(\frac{c}{2}{r}^2({\epsilon }_1+a^2)+1\right)\pm \sqrt{2c({\epsilon }_1+a^2)}\}]s+z_0,\hfill \end{array}\right.$$

where $r\in R$ and ${\beta }_0,x_0,y_0,z_0$ are constants.

If $\gamma$ is a timelike curve, then $\epsilon =-1$ and $a=\cosh {\alpha }_0$. If $\gamma$ is a spacelike curve, then $\epsilon =1$ and $a=\sinh {\alpha }_0$. In particular, if $\epsilon =1$ and $\eta ({\gamma }^{\prime })=a=0$, then we have:

Example 1

(Legendre curves). Let $\gamma :I\to M_1^3(\hat{H})$ be a non-geodesic Legendre Frenet curve satisfying the $\hat{\nabla }$-Jacobi equations for the $\hat{\nabla }$-geodesic in Lorentzian space forms $M_1^3(\hat{H})$ for $\hat{H}=2c>0$. Then, its parametric equations are given by

$$\left\{\begin{array}{c}x(s)=r\sin (\frac{1}{r}\left(\frac{c}{2}{r}^2+1\right)s+{\beta }_0)+x_0,\hfill \\ y(s)=-r\cos (\frac{1}{r}\left(\frac{c}{2}{r}^2+1\right)s+{\beta }_0)+y_0,\hfill \\ z(s)=\frac{1}{c}\{\frac{1}{r}\left(\frac{c}{2}{r}^2+1\right)\pm \sqrt{2c}\}s+z_0,\hfill \end{array}\right.$$

where $r\in R$ and ${\beta }_0,x_0,y_0,z_0$ are constants.