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

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### Contact Lorentzian Manifold

**Proposition**

**1**

## 3. Slant Curves in Contact Lorentzian Three-Manifolds

#### 3.1. Lorentzian Cross Product

**Proposition**

**3.**

**Definition**

**1.**

**Proposition**

**4.**

- (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,$$det(X,Y,Z)=-g(X,\phi Y)\eta (Z)-g(Y,\phi Z)\eta (X)-g(Z,\phi X)\eta (Y)$$
- (6)
- $g(X,\phi Y)Z+g(Y,\phi Z)X+g(Z,\phi X)Y=-(X,Y,Z)\xi .$

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 3.2. Frenet Slant Curves

**Proposition**

**6.**

**Proposition**

**7.**

**Theorem**

**1.**

**Corollary**

**1.**

#### 3.3. Null Slant Curves

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Contact Magnetic Curves

**Theorem**

**3.**

**Theorem**

**4.**

**Corollary**

**2.**

**Theorem**

**5.**

- (i)
- a spacelike curve with spacelike normal vector field; or
- (ii)
- a timelike curve.

**Corollary**

**3.**

#### Example

**Lemma**

**1.**

**Lemma**

**2.**

**Proposition**

**8.**

**Theorem**

**6.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

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

**AMA Style**

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

**Chicago/Turabian Style**

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