Frenet curves in 3-dimensional $\delta$-Lorentzian trans Sasakian manifolds

1.   Introduction

The differential geometry of curves especially in contact and para-contact manifolds studied by several authors. Olszak ^[16], derived certain necessary and sufficient conditions for an almost contact metric (a.c.m) structure on M to be normal and point out some of their consequences. Olszak completely characterized the local nature of normal a.c.m. structures on M by giving suitable examples. Moreover Olszak gave some restrictions on the scalar curvature in contact metric manifolds which are conformally flat or of constant $\phi$-sectional curvature in ^[15].

Welyczko ^[21], generalized some of results for Legendre curves to the case of 3-dimensional normal almost contact metric manifolds, especially, quasi-Sasakian manifolds. Welyczko ^[20], studied the curvature and torsion of slant Frenet curves in 3-dimensional normal almost paracontact metric manifolds.

Curvature and torsion of Legendre curves in 3-dimensional $\left(\varepsilon, \delta\right)$ trans-Sasakian manifolds was obtained in ^[1]. Lee defined Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, Lee proved that the ratio of $\kappa$ and $\tau$-1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, Lee proved 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 in^[11]. Lee, also gave the properties of the generalized Tanaka-Webster connection in a contact Lorentzian manifold in ^[12].

Yıldırım ^[22] obtained curvatures of non-geodesic Frenet curves on 3-dimensional normal almost contact manifolds without neglecting $\alpha$ and $\beta$, and provided the results of their characterization.

Trans-Sasakian structure on a manifold with Lorentzian metric and conformally flat Lorentzian trans-Sasakian manifolds was studied in ^[19].

Siddiki ^[17] studied $\delta$-Lorentzian trans-Sasakian manifolds with a semi-symmetric-metric connection and computed curvature tensors, Ricci curvature tensors and scalar curvature of the $\delta$-Lorentzian trans-Sasakian manifold with a semi-symmetric-metric connection.

In this framework, the paper is organized in the following way. In section 2, we give basic definitions and propositions of a $\delta$-Lorentzian trans-Sasakian manifold. We give the Frenet-Serret equations of a curve in Lorentzian 3-manifold. In section 3, we obtain an orthonormal basis $\left\{e_{1}, e_{2}, e_{3}\right\}$ by using the basis $\left(\zeta', \varphi \zeta', \xi\right)$ for the curve $\zeta$ in a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold. Also we calculate the Frenet elements of a non-geodesic Frenet curve, slant curve and Legendre curve in this manifold. Then, we give the curvatures of the curve $\zeta$ on some kinds of $\delta$-Lorentzian manifolds. In the last section, we give some examples for the spacelike curves on a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold.

2.   Materials and methods

2.1.   $\delta$-Lorentzian trans-Sasakian Manifolds

Let $\bar{N}$ be a $\delta$-almost contact metric manifold equipped with $\delta$-almost contact metric structure $(\varphi, \xi, \eta, \bar{g}, \delta)$ consisting of (1, 1) tensor field $\varphi$, a vector field $\xi$, a 1-form $\eta$ and an indefinite metric $\bar{g}$ such that

for all $U, V \in\bar{N}$, where $\delta^{2} = 1$ so that $\delta = \mp1$. The above structure $(\varphi, \xi, \eta, \bar{g}, \delta)$ is called the $\delta$-Lorentzian structure on $\bar{N}$. If $\delta = 1$, then the manifold becomes the usual Lorentzian structure^[2] on $\bar{N}$, the vector field $\xi$ is timelike ^[18].

In the classification of almost Hermitian manifolds, there appears a class $W_{4}$ of Hermitian manifolds which are closely related to the conformal Kaehler manifolds ^[17]. The class $C_{6}\oplus C_{5}$ coincides with the class of trans-Sasakian structue of type $(\alpha, \beta)$ ^[13]. In fact, the local nature of the two sub classes, namely $C_{6}$ and $C_{5}$ of trans-Sasakian structures are charactrized completely. An almost contact metric structure on $\bar{N}$ is called trans-Sasakian if $(\bar{N}\times \Re, J, G)$ belongs to the class $W_{4}$, where J is the almost complex structure on $\bar{N}\times \Re$ defined by

for all vector fields U on $\bar{N}$ and smooth functions f on $\bar{N}\times \Re$ and G is the product metric on $\bar{N}\times \Re$. This may be expressed by the condition

for any vector fields U and V on $\bar{N}$, $\nabla$ denotes the Levi-Civita connection with respect to $\bar{g}$, $\alpha$ and $\beta$ are smooth functions on $\bar{N}$ ^[17]. The existence of condition (2.3) is ensure by the above discussion.

With the above literature, the $\delta$-Lorentzian trans-Sasakian manifolds are defined as follows.

Definition 2.1. ^[2] A $\delta$-Lorentzian manifold with structure $(\varphi, \xi, \eta, \bar{g}, \delta)$ is said to be $\delta$-Lorentzian trans-Sasakian manifold of type $(\alpha, \beta)$ if it satisfies the condition

for any vector fields U and V on $\bar{N}$.

If $\delta$ = 1, then the $\delta$-Lorentzian trans-Sasakian manifold becomes the usual Lorentzian trans-Sasakian manifold of type $(\alpha, \beta)$ ^[17]. $\delta$-Lorentzian trans-Sasakian manifold of type $(0, 0)$, $(0, \beta)$, $(\alpha, 0)$ are the Lorentzian cosymplectic, Lorentzian $\beta$-Kenmotsu and Lorentzian $\alpha$-Sasakian manifolds respectively. In particular if $\alpha = 1$, $\beta = 0$ and $\alpha = 0$, $\beta = 1$, a $\delta$-Lorentzian trans-Sasakian manifold reduces to a $\delta$-Lorentzian Sasakian manifold and a $\delta$-Lorentzian Kenmotsu manifold respectively.

From (2.4), we have

and

Further for a $\delta$-Lorentzian trans-Sasakian manifold, we have

and

2.2.   Frenet curves

Let $\zeta:I\rightarrow \bar{N}$ be a unit speed curve in Lorentzian 3-manifold $\bar{N}$ such that $\bar{g}(\zeta^{'}, \zeta^{'}) = \varepsilon_{1} = \mp 1$. The constant $\varepsilon_{1}$ is called the casual character of $\zeta$. The constants $\varepsilon_{2}$ and $\varepsilon_{3}$ defined by $\bar{g}(n, n) = \varepsilon_{2}$ and $\bar{g}(b, b) = \varepsilon_{3}$ and called the second casual character and third casual character of $\zeta$, respectively. Thus we have $\varepsilon_{1}\varepsilon_{2} = -\varepsilon_{3}$.

A unit speed curve $\zeta$ is said to be spacelike or timelike if its casual character is 1 or -1, respectively. A unit speed curve $\zeta$ is said to be a Frenet curve if $\bar{g}(\zeta^{'}, \zeta^{'})\neq 0$. A Frenet curve $\zeta$ admits an orthonormal frame field $\{t = \zeta^{'}, n, b\}$ along $\zeta$. Then the Frenet-Serret equations are given as follows

where $\kappa = |\nabla_{\zeta^{'}}\zeta^{'}|$ is first curvature and $\tau$ is second curvature of $\zeta$ ^[11]. The vector fields t, n and b are called the tangent vector field, the principal normal vector field and the binormal vector field of $\zeta$, respectively.

A Frenet curve $\zeta$ is a geodesic if and only if $\kappa = 0$. A Frenet curve $\zeta$ with constant first curvature and zero second curvature is called a pseudo-circle. A pseudo-helix is a Frenet curve $\zeta$ whose curvatures are constant.

A curve in a Lorentzian three-manifold is said to be slant if its tangent vector field has constant angle with the Reeb vector field, i.e., $\eta(\zeta') = -\bar{g}(\zeta', \xi) = constant$. If $\eta(\zeta') = -\bar{g}(\zeta', \xi) = 0$, then the curve $\zeta$ is called a Legendre curve^[11].

3.   Main results

In this section, we consider a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold $\bar{N}$. Let $\zeta:I\rightarrow \bar{N}$ be a non-geodesic $(\kappa\neq 0)$ Frenet curve given with the arc-parameter s and $\bar{\nabla}$ be the Levi-Civita connection on $\bar{N}$. From the basis $(\zeta^{'}, \varphi\zeta^{'}, \xi)$ we obtain an orthonormal basis $\left\{e_{1}, e_{2}, e_{3}\right\}$ given by

where

Then if we write the covariant differentiation of $\zeta^{'}$ as

where $\nu$ is a certain function.

Moreover we obtain

where $\rho'(s) = \frac{d\rho(\zeta(s))}{ds}$. Then we find

and

The fundamental forms of the tangent vector $\zeta'$ on the basis of the Eq (3.1) is

and the Darboux vector connected to the vector $\zeta'$ is

So we can write

Furthermore, for any vector field $Z = \sum_{i = 1}^{3}\theta^{i}e_{i}\in \chi(\bar{N})$ strictly dependent on the curve $\zeta$ on $\bar{N}$, there exists the following equation

3.1.   Frenet elements of $\zeta$

Let $\zeta:I\rightarrow \bar{N}$ be a non-geodesic $(\kappa\neq 0)$ Frenet curve given with the arc parameter s and the elements $\{t, n, b, \kappa, \tau\}$. The Frenet elements of this curve are calculated as follows:

If we consider the Eq (3.3), then we get

From the Eqs (3.5) and (3.12) we find

On the other hand

By means of the Eqs (3.6) and (3.7) we obtain

By a direct computation we find

Taking the norm of the last equation and if we consider the Eqs (3.5) and (3.16) on (3.15) we obtain

Moreover we can write the Frenet vector fields of $\zeta$ as in the following theorem.

Theorem 3.1. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a Frenet curve on $\bar{N}$. The Frenet vector fields t, n and b are in the form of

Moreover we have

Let $\zeta$ be non-geodesic Frenet curve given with the arc-parameter s in 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold $\bar{N}$. So we can give the following theorem.

Theorem 3.2. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a Frenet curve on $\bar{N}$. $\zeta$ is a slant curve ($\rho = \eta(\zeta') = cos\theta = constant$) on $\bar{N}$ if and only if the Frenet elements $\{t, n, b, \kappa, \tau\}$ of $\zeta$ are as follows.

Proof. Let the curve $\zeta$ be a slant curve in 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold $\bar{N}$. If we take into account the condition $\rho = \eta(\zeta') = cos\theta = constant$ in the Eqs (3.1), (3.13) and (3.17) we find (3.21). If the equations in (3.21) hold, from the definition of slant curves it is obvious that the curve $\zeta$ is a slant curve.

If we consider the Theorem (3.1), we can give the following corollaries.

Corollary 3.1. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a slant curve on $\bar{N}$. If the first curvature $\kappa$ is non-zero constant, then $\zeta$ is a pseudo-helix with $\tau = \left|\delta\left(-\varepsilon_{1}\alpha+\frac{cos\theta\nu}{\sqrt{\varepsilon_{1}+\delta\cos^{2}\theta}}\right)\right|$.

Corollary 3.2. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a slant curve on this manifold $\bar{N}$. If $\kappa$ is not constant and $\tau = 0$, then, $\zeta$ is a plane curve and the following equation satisfies

Theorem 3.3. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a spacelike Frenet curve on $\bar{N}$. $\zeta$ is a Legendre curve($\rho = \eta(\zeta') = 0$) in this manifold if and only if the Frenet elements $\{t, n, b, \kappa, \tau\}$ of $\zeta$ satisfy the following equations:

Proof. Let the curve $\zeta$ be a Legendre curve 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold $\bar{N}$. If we take into account the condition $\rho = \eta(\zeta') = 0$ in the Eqs (3.1), (3.13) and (3.17) we find (3.23). If the equations in (3.23) hold, from the definition of Legendre curves it is obvious that the curve $\zeta$ is a Legendre curve on $\bar{N}$.

Corollary 3.3. Let the curve $\zeta$ be a Legendre curve in 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold $\bar{N}$. If $\kappa$ is non-zero constant and $\tau$ is equal to zero, then $\zeta$ is a plane curve and $\alpha = 0$.

If we consider the Eqs (3.13) and (3.17) and theorem (3.1) we can give the following corollaries.

Corollary 3.4. Let $\bar{N}$ be a 3-dimensional $\delta$-Lorentzian trans-Sasakian manifold and $\zeta$ be a Frenet curve on this manifold $\bar{N}$. The first curvature of the curve $\zeta$ is not dependent on $\alpha$ and $\beta$.

Corollary 3.5. From the Eqs (3.13) and (3.17) the first curvature and the second curvature of $\zeta$ on 3-dimensional $\delta$-Lorentzian cosymplectic manifold $\bar{N}$ are

and

i) If the curve $\zeta$ in 3-dimensional $\delta$-Lorentzian cosymplectic manifold $\bar{N}$ is a slant curve, then we have

ii) If the curve $\zeta$ in 3-dimensional $\delta$-Lorentzian cosymplectic manifold $\bar{N}$ is a Legendre curve, then we have

Corollary 3.6. Let $\zeta$ be a curve on 3-dimensional $\delta$-Lorentzian $\beta$-Kenmotsu manifold $\bar{N}$. Then, the first and second curvatures of $\zeta$ are

and

If the curve $\zeta$ is a slant curve on $\bar{N}$, then we have

If the curve $\zeta$ is a Legendre curve on $\bar{N}$, then we have

Corollary 3.7. Let $\zeta$ be a curve on 3-dimensional $\delta$-Lorentzian $\alpha$-Sasakian manifold $\bar{N}$. Then, the first curvature and the second curvature of $\zeta$ are

and

The curvatures of $\zeta$ are

where $\zeta$ is a slant curve in 3-dimensional $\delta$-Lorentzian $\alpha$-Sasakian manifold $\bar{N}$ and

where $\zeta$ is a Legendre curve in 3-dimensional $\delta$-Lorentzian $\alpha$-Sasakian manifold $\bar{N}$.

Corollary 3.8. From the Eqs (3.13) and (3.17) the first curvature and the second curvature of $\zeta$ on 3-dimensional $\delta$-Lorentzian Kenmotsu manifold $\bar{N}$ are

and

i) If the curve $\zeta$ in 3-dimensional $\delta$-Lorentzian Kenmotsu manifold $\bar{N}$ is a slant curve, then we obtain

ii) If the curve $\zeta$ in 3-dimensional $\delta$-Lorentzian Kenmotsu manifold $\bar{N}$ is a Legendre curve, then we have

Corollary 3.9. Let $\zeta$ be a curve on 3-dimensional $\delta$-Lorentzian Sasakian manifold $\bar{N}$. Then, the first and second curvatures of $\zeta$ are

and

If the curve $\zeta$ is a slant curve on $\bar{N}$, then we have

If the curve $\zeta$ is a Legendre curve on $\bar{N}$, then we obtain

4.   Examples

Let $\bar{N}$ be a 3-dimensional manifold given

where (x, y, z) denote the standart co-ordinates in $\Re^{3}$. Then

are linearly independent of each point of $\bar{N}$ ^[17]. Let $\bar{g}$ be the Lorentzian metric tensor defined by

for $i, j = 1, 2, 3$ and $\delta = \mp1$. Let $\eta$ be a 1-form defined by $\eta(Z) = \delta\bar{g}(Z, E_{3})$ for any vector field $Z\in \Gamma (T\bar{N})$. Let $\varphi$ be the (1, 1)-tensor field defined by

Then using the condition of the linearity of $\varphi$ and $\bar{g}$, we obtain $\eta(E_{3}) = 1$ and

for all $Z, W\in \Gamma (T\bar{N})$.

Now, let $\nabla$ be the Levi-Civita connection with respect to the Lorentzian metric $\bar{g}$. Then we obtain

The Riemannian connection $\nabla$ with respect to the metric $\bar{g}$ is given by

If we use this equation which is known as Koszul's formula for the Lorentzian metric tensor $\bar{g}$, we can easily calculate the covariant derivations as follows:

From the above relations, for any vector field $X$ on $\bar{N}$, we have

for $\xi = E_{3}$, $\alpha = 0$ and $\beta = 1$. Hence the manifold $\bar{N}$ under consideration is a $\delta$-Lorentzian trans-Sasakian of type $(0, 1)$ manifold of dimension three.

Example 4.1. Let $\gamma$ be a spacelike curve defined as

where the curve $\gamma$ parametrized by the arc length parameter t. If we differentiate $\gamma(t)$ and consider (3.1) we find

and

where $\rho = \eta(\gamma'(t))$. If we consider the Eqs (3.2), (3.3), (3.5), (3.13) and (3.17) we can write

Thus, the curve $\gamma$ is a spacelike helix in $\bar{N}$.

Example 4.2. Let $\omega$ be a spacelike Legendre curve defined as

where the curve $\omega$ parametrized by the arc length parameter t. If we differentiate $\omega(t)$ and using (3.1) we find

and

If we consider the Eqs (3.2), (3.3), (3.5), (3.13) and (3.17) we obtain

Conflict of interest

The author declares that there is no competing interest.