# Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Proposition**

**1.**

**Remark**

**1.**

**Proposition**

**2**

**Definition**

**2**

**Definition**

**3.**

**Proposition**

**3**

**Proposition**

**4**

**Definition**

**4**

- If u and v span a timelike vector subspace. Then we have $|\langle u,v\rangle |>\parallel u\parallel \parallel v\parallel $ and hence, there is a unique positive real number θ such that$$|\langle u,v\rangle |=\parallel u\parallel \parallel v\parallel cosh\theta .$$The real number θ is called the Lorentz timelike angle between u and v.
- If u and v span a spacelike vector subspace. Then we have $|\langle u,v\rangle |\le \parallel u\parallel \parallel v\parallel $ and hence, there is a unique real number $\theta \in [0,\frac{\pi}{2}]$ such that$$|\langle u,v\rangle |=\parallel u\parallel \parallel v\parallel cos\theta .$$The real number θ is called the Lorentz spacelike angle between u and v.

**Definition**

**5**

**Definition**

**6.**

**Remark**

**2**

**Remark**

**3.**

## 3. Main Conclusions

#### 3.1. Associate Normal Curves of a Pseudo Null Curve on de-Sitter Space

**Theorem**

**1.**

- If ${\beta}_{1}={\beta}_{1}^{+}$ is spacelike, then the Frenet frame of $r(s)$ and the pseudo spherical Frenet frame of ${b}_{1}(s)$ can be related by $f=\sqrt{2}({\lambda}^{\prime}+\lambda \kappa )$ as$$\left\{\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{1}^{+}{\alpha}_{1}^{+}=-\frac{f}{\sqrt{1-{f}^{2}}}T+\frac{1}{\sqrt{2(1-{f}^{2})}}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\u03f5}_{0}{\beta}_{1}^{+}=-\frac{1}{\sqrt{1-{f}^{2}}}T+\frac{f}{\sqrt{2(1-{f}^{2})}}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\gamma}_{1}^{+}=\frac{1}{\sqrt{2}}(\lambda N+\frac{1}{\lambda}B)\hfill \end{array}\right.$$or the Lorentz timelike angle ${\theta}_{1}^{+}$ between T and ${\beta}_{1}^{+}$ as$$\left\{\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{1}^{+}{\alpha}_{1}^{+}=-\u03f5sinh{\theta}_{1}^{+}T+\frac{1}{\sqrt{2}}cosh{\theta}_{1}^{+}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{0}{\beta}_{1}^{+}=-\u03f5cosh{\theta}_{1}^{+}T+\frac{1}{\sqrt{2}}sinh{\theta}_{1}^{+}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\gamma}_{1}^{+}=\frac{1}{\sqrt{2}}(\lambda N+\frac{1}{\lambda}B),\hfill \end{array}\right.$$where ${\u03f5}_{0}=\pm 1$, ${\u03f5}_{1}^{+}=\pm 1$, $f=\u03f5tanh{\theta}_{1}^{+}$ and when $0<f<1$, $\u03f5=1$; when $-1<f<0$, $\u03f5=-1$.
- If ${\beta}_{1}={\beta}_{1}^{-}$ is timelike, then the Frenet frame of $r(s)$ and the pseudo spherical Frenet frame of ${b}_{1}(s)$ can be related by $f=\sqrt{2}({\lambda}^{\prime}+\lambda \kappa )$ as$$\left\{\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{1}^{-}{\alpha}_{1}^{-}=-\frac{f}{\sqrt{{f}^{2}-1}}T+\frac{1}{\sqrt{2({f}^{2}-1)}}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\u03f5}_{0}{\beta}_{1}^{-}=-\frac{1}{\sqrt{{f}^{2}-1}}T+\frac{f}{\sqrt{2({f}^{2}-1)}}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\gamma}_{1}^{-}=\frac{1}{\sqrt{2}}(\lambda N+\frac{1}{\lambda}B)\hfill \end{array}\right.$$or the Lorentz timelike angle ${\theta}_{1}^{-}$ between T and ${\beta}_{1}^{-}$ as$$\left\{\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{1}^{-}{\alpha}_{1}^{-}=-\u03f5cosh{\theta}_{1}^{-}T+\frac{1}{\sqrt{2}}sinh{\theta}_{1}^{-}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \u03f5{\u03f5}_{0}{\beta}_{1}^{-}=-\u03f5sinh{\theta}_{1}^{-}T+\frac{1}{\sqrt{2}}cosh{\theta}_{1}^{-}(\lambda N-\frac{1}{\lambda}B),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\gamma}_{1}^{-}=\frac{1}{\sqrt{2}}(\lambda N+\frac{1}{\lambda}B),\hfill \end{array}\right.$$where ${\u03f5}_{0}=\pm 1$, ${\u03f5}_{1}^{-}=\pm 1$, $f=\u03f5coth{\theta}_{1}^{-}$ and when $f>1$, $\u03f5=1$; when $f<-1$, $\u03f5=-1$.

**Theorem**

**2.**

- If ${\beta}_{1}={\beta}_{1}^{+}$ is spacelike, the pseudo spherical curvature ${\kappa}_{1}^{+}$ of ${b}_{1}(s)$ can be expressed by$${\u03f5}_{1}^{+}{\kappa}_{1}^{+}=\u03f5\frac{{f}^{2}+\sqrt{2}\lambda {f}^{\prime}-1}{{(1-{f}^{2})}^{\frac{3}{2}}}=(\sqrt{2}\lambda {\theta}_{1}^{{+}^{\prime}}-\u03f5)cosh{\theta}_{1}^{+}.$$
- If ${\beta}_{1}={\beta}_{1}^{-}$ is timelike, the pseudo spherical curvature ${\kappa}_{1}^{-}$ of ${b}_{1}(s)$ can be expressed by$${\u03f5}_{1}^{-}{\kappa}_{1}^{-}=\u03f5\frac{{f}^{2}+\sqrt{2}\lambda {f}^{\prime}-1}{{({f}^{2}-1)}^{\frac{3}{2}}}=(\sqrt{2}\lambda {\theta}_{1}^{{-}^{\prime}}-\u03f5)sinh{\theta}_{1}^{-},$$

**Corollary**

**1.**

- the arc-length $\overline{s}$ of ${b}_{1}(s)$ can be expressed by $\overline{s}={c}_{0}g(s),\phantom{\rule{4pt}{0ex}}(0\ne {c}_{0}\in \mathbb{R})$;
- the pseudo spherical curvature of ${b}_{1}(s)$ is ${\kappa}_{1}^{+}=\pm 1$;
- the Frenet frame of $r(s)$ and the pseudo spherical Frenet frame of ${b}_{1}(s)$ can be related by$$\begin{array}{c}\left(\begin{array}{c}{\u03f5}_{1}^{+}{\alpha}_{1}^{+}\\ {\u03f5}_{0}{\beta}_{1}^{+}\\ {\gamma}_{1}^{+}\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{\lambda}{\sqrt{2}}& \frac{1}{\sqrt{2}\lambda}\\ -1& 0& 0\\ 0& \frac{\lambda}{\sqrt{2}}& \frac{1}{\sqrt{2}\lambda}\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right),\phantom{\rule{1.em}{0ex}}({\u03f5}_{0}=\pm 1,{\u03f5}_{1}^{+}=\pm 1).\end{array}$$

**Proof**

**of**

**Corollary**

**1.**

**Remark**

**4.**

#### 3.2. Associate Normal Curves of a Pseudo Null Curve on Hyperbolic Space

**Theorem**

**3.**

**Theorem**

**4.**

**Corollary**

**2.**

- the arc-length $\overline{s}$ of ${b}_{2}(s)$ can be expressed by $\overline{s}={c}_{0}g(s),\phantom{\rule{4pt}{0ex}}(0\ne {c}_{0}\in \mathbb{R})$;
- the hyperbolic curvature of ${b}_{2}(s)$ is ${\kappa}_{2}=\pm 1$;
- the Frenet frame of $r(s)$ and the hyperbolic Frenet frame of ${b}_{2}(s)$ can be related by$$\begin{array}{c}\left(\begin{array}{c}{\u03f5}_{2}{\alpha}_{2}\\ {\u03f5}_{0}{\beta}_{2}\\ {\gamma}_{2}\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{\lambda}{\sqrt{2}}& -\frac{1}{\sqrt{2}\lambda}\\ 1& 0& 0\\ 0& \frac{\lambda}{\sqrt{2}}& -\frac{1}{\sqrt{2}\lambda}\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right),\phantom{\rule{1.em}{0ex}}({\u03f5}_{0}=\pm 1,{\u03f5}_{2}=\pm 1).\end{array}$$

**Remark**

**5.**

#### 3.3. The Relationships of the Normal Partner Curves

**Theorem**

**5.**

- If ${\beta}_{1}={\beta}_{1}^{+}$ is spacelike, then the pseudo spherical Frenet frame of ${b}_{1}(s)$ and the hyperbolic Frenet frame of ${b}_{2}(s)$ can be related by $f=\sqrt{2}({\lambda}^{\prime}+\lambda \kappa )$ as$$\begin{array}{c}\left(\begin{array}{c}\u03f5{\u03f5}_{1}^{+}{\alpha}_{1}^{+}\\ {\u03f5}_{0}{\beta}_{1}^{+}\\ {\gamma}_{1}^{+}\end{array}\right)=-\left(\begin{array}{ccc}\frac{{f}^{2}}{\sqrt{1-{f}^{4}}}& \frac{f}{\sqrt{1-{f}^{4}}}& -\frac{1}{\sqrt{1-{f}^{2}}}\\ \frac{f}{\sqrt{1-{f}^{4}}}& \frac{1}{\sqrt{1-{f}^{4}}}& -\frac{f}{\sqrt{1-{f}^{2}}}\\ \frac{1}{\sqrt{1+{f}^{2}}}& -\frac{f}{\sqrt{1+{f}^{2}}}& 0\end{array}\right)\left(\begin{array}{c}\u03f5{\u03f5}_{2}{\alpha}_{2}\\ {\u03f5}_{0}{\beta}_{2}\\ {\gamma}_{2}\end{array}\right)\end{array}$$or the Lorentz timelike angle ${\theta}_{1}^{+}$ between T and ${\beta}_{1}^{+}$, the Lorentz spacelike angle ${\theta}_{2}$ between T and ${\beta}_{2}$ as$$\begin{array}{c}\left(\begin{array}{c}\u03f5{\u03f5}_{1}^{+}{\alpha}_{1}^{+}\\ \u03f5{\u03f5}_{0}{\beta}_{1}^{+}\\ {\gamma}_{1}^{+}\end{array}\right)=-\left(\begin{array}{ccc}sinh{\theta}_{1}^{+}sin{\theta}_{2}& sinh{\theta}_{1}^{+}cos{\theta}_{2}& -cosh{\theta}_{1}^{+}\\ cosh{\theta}_{1}^{+}sin{\theta}_{2}& cosh{\theta}_{1}^{+}cos{\theta}_{2}& -sinh{\theta}_{1}^{+}\\ cos{\theta}_{2}& -sin{\theta}_{2}& 0\end{array}\right)\left(\begin{array}{c}\u03f5{\u03f5}_{2}{\alpha}_{2}\\ \u03f5{\u03f5}_{0}{\beta}_{2}\\ {\gamma}_{2}\end{array}\right),\end{array}$$where ${\u03f5}_{0}$, ${\u03f5}_{1}^{+}$, ${\theta}_{1}^{+}$ and ${\u03f5}_{2}$, ${\theta}_{2}$ as stated in Theorems 1 and 3, respectively. $f=\u03f5tanh{\theta}_{1}^{+}=\u03f5tan{\theta}_{2}$ and when $0<f<1$, $\u03f5=1$; when $-1<f<0$, $\u03f5=-1$.
- If ${\beta}_{1}={\beta}_{1}^{-}$ is timelike, then the pseudo spherical Frenet frame of ${b}_{1}(s)$ and the hyperbolic Frenet frame of ${b}_{2}(s)$ can be related by $f=\sqrt{2}({\lambda}^{\prime}+\lambda \kappa )$ as$$\begin{array}{c}\left(\begin{array}{c}\u03f5{\u03f5}_{1}^{-}{\alpha}_{1}^{-}\\ {\u03f5}_{0}{\beta}_{1}^{-}\\ {\gamma}_{1}^{-}\end{array}\right)=-\left(\begin{array}{ccc}\frac{{f}^{2}}{\sqrt{{f}^{4}-1}}& \frac{f}{\sqrt{{f}^{4}-1}}& -\frac{1}{\sqrt{{f}^{2}-1}}\\ \frac{f}{\sqrt{{f}^{4}-1}}& \frac{1}{\sqrt{{f}^{4}-1}}& -\frac{f}{\sqrt{{f}^{2}-1}}\\ \frac{1}{\sqrt{1+{f}^{2}}}& -\frac{f}{\sqrt{1+{f}^{2}}}& 0\end{array}\right)\left(\begin{array}{c}\u03f5{\u03f5}_{2}{\alpha}_{2}\\ {\u03f5}_{0}{\beta}_{2}\\ {\gamma}_{2}\end{array}\right)\end{array}$$or the Lorentz timelike angle ${\theta}_{1}^{-}$ between T and ${\beta}_{1}^{-}$, the Lorentz spacelike angle ${\theta}_{2}$ between T and ${\beta}_{2}$ as$$\begin{array}{c}\left(\begin{array}{c}\u03f5{\u03f5}_{1}^{-}{\alpha}_{1}^{-}\\ \u03f5{\u03f5}_{0}{\beta}_{1}^{-}\\ {\gamma}_{1}^{-}\end{array}\right)=-\left(\begin{array}{ccc}cosh{\theta}_{1}^{-}sin{\theta}_{2}& cosh{\theta}_{1}^{-}cos{\theta}_{2}& -sinh{\theta}_{1}^{-}\\ sinh{\theta}_{1}^{-}sin{\theta}_{2}& sinh{\theta}_{1}^{-}cos{\theta}_{2}& -cosh{\theta}_{1}^{-}\\ cos{\theta}_{2}& -sin{\theta}_{2}& 0\end{array}\right)\left(\begin{array}{c}\u03f5{\u03f5}_{2}{\alpha}_{2}\\ \u03f5{\u03f5}_{0}{\beta}_{2}\\ {\gamma}_{2}\end{array}\right),\end{array}$$

**Proof**

**of**

**Theorem**

**5.**

**Theorem**

**6.**

- if ${\beta}_{1}={\beta}_{1}^{+}$ is spacelike, ${\kappa}_{1}={\kappa}_{1}^{+}$, then we have$${\kappa}_{1}^{{+}^{2}}=\frac{{({f}^{2}+\sqrt{2}\lambda {f}^{\prime}-1)}^{2}}{{(1-{f}^{2})}^{3}}={(\sqrt{2}\lambda {\theta}_{1}^{{+}^{\prime}}-\u03f5)}^{2}{cosh}^{2}{\theta}_{1}^{+},$$$${\kappa}_{2}^{2}=\frac{{({f}^{2}+\sqrt{2}\lambda {f}^{\prime}+1)}^{2}}{{(1+{f}^{2})}^{3}}={(\sqrt{2}\lambda {\theta}_{2}^{\prime}+\u03f5)}^{2}{cos}^{2}{\theta}_{2}$$and they are related by $f=\u03f5tanh{\theta}_{1}^{+}=\u03f5tan{\theta}_{2}$, when $0<f<1$, $\u03f5=1$; when $-1<f<0$, $\u03f5=-1$. ${\theta}_{1}^{+}$ and ${\theta}_{2}$ as stated in Theorems 1 and 3, respectively;
- if ${\beta}_{1}={\beta}_{1}^{-}$ is timelike, ${\kappa}_{1}={\kappa}_{1}^{-}$, then we have$${\kappa}_{1}^{{-}^{2}}=\frac{{({f}^{2}+\sqrt{2}\lambda {f}^{\prime}-1)}^{2}}{{({f}^{2}-1)}^{3}}={(\sqrt{2}\lambda {\theta}_{1}^{{-}^{\prime}}-\u03f5)}^{2}{sinh}^{2}{\theta}_{1}^{-},$$$${\kappa}_{2}^{2}=\frac{{({f}^{2}+\sqrt{2}\lambda {f}^{\prime}+1)}^{2}}{{({f}^{2}+1)}^{3}}={(\sqrt{2}\lambda {\theta}_{2}^{\prime}+\u03f5)}^{2}{cos}^{2}{\theta}_{2}$$and they are related by $f=\u03f5coth{\theta}_{1}^{-}=\u03f5tan{\theta}_{2},$ when $f>1$, $\u03f5=1$; when $f<-1$, $\u03f5=-1$. ${\theta}_{1}^{-}$ and ${\theta}_{2}$ as stated in Theorems 1 and 3, respectively.

**Corollary**

**3.**

- the arc-length of ${b}_{1}(s)$ and ${b}_{2}(s)$ all can be expressed by $\overline{s}={c}_{0}g(s),\phantom{\rule{4pt}{0ex}}(0\ne {c}_{0}\in \mathbb{R})$;
- the pseudo spherical curvature ${\kappa}_{1}^{+}$ of ${b}_{1}(s)$ and the hyperbolic curvature ${\kappa}_{2}$ of ${b}_{2}(s)$ satisfy$${\kappa}_{1}^{{+}^{2}}={\kappa}_{2}^{2}=1;$$
- the pseudo spherical Frenet frame of ${b}_{1}(s)$ and the hyperbolic Frenet frame of ${b}_{2}(s)$ can be related by$$\begin{array}{c}\left(\begin{array}{c}{\u03f5}_{2}{\alpha}_{2}\\ {\beta}_{2}\\ {\gamma}_{2}\end{array}\right)=-\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{c}{\u03f5}_{1}^{+}{\alpha}_{1}^{+}\\ {\beta}_{1}^{+}\\ {\gamma}_{1}^{+}\end{array}\right),\phantom{\rule{1.em}{0ex}}({\u03f5}_{1}^{+}=\pm 1,{\u03f5}_{2}=\pm 1).\end{array}$$

**Remark**

**6.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Qian, J.; Tian, X.; Kim, Y.H.
Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms. *Mathematics* **2020**, *8*, 919.
https://doi.org/10.3390/math8060919

**AMA Style**

Qian J, Tian X, Kim YH.
Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms. *Mathematics*. 2020; 8(6):919.
https://doi.org/10.3390/math8060919

**Chicago/Turabian Style**

Qian, Jinhua, Xueqian Tian, and Young Ho Kim.
2020. "Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms" *Mathematics* 8, no. 6: 919.
https://doi.org/10.3390/math8060919