# New Type Direction Curves in 3-Dimensional Compact Lie Group

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Proposition**

**1**

**.**Suppose that the curve $\alpha (s)$ is a curve in Lie group G such that the parameter s is the arc length parameter of $\alpha (s)$, and the Frenet apparatus of $\alpha (s)$ is $(T,N,B,\kappa ,\tau )$. Then,

**Definition**

**1.**

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

## 3. Main Results

**Theorem**

**3.**

**Remark**

**1.**

**Definition**

**2.**

#### 3.1. Osculating-Direction Curves

**Definition**

**3.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**2.**

- (i)
- α is a general helix in $G.$
- (ii)
- α is an osculating-donor curve of a slant helix.
- (iii)
- γ is a slant helix.

#### 3.2. Normal-Direction Curves

**Definition**

**4.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Corollary**

**3.**

- (i)
- α is a general helix in $G.$
- (ii)
- α is a normal-donor curve of a slant helix
- (iii)
- γ is a slant helix.

#### 3.3. Rectifying-Direction Curves

**Definition**

**5.**

**Theorem**

**12.**

**Proof.**

**Corollary**

**4.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Proof.**

**Corollary**

**5.**

**Corollary**

**6.**

**Theorem**

**15.**

**Proof.**

**Corollary**

**7.**

- (i)
- α is not a general helix in $G.$
- (ii)
- α is not a rectifying-donor curve of a general helix.
- (iii)
- A rectifying-direction curve γ of α is not a general helix.

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Çakmak, A.
New Type Direction Curves in 3-Dimensional Compact Lie Group. *Symmetry* **2019**, *11*, 387.
https://doi.org/10.3390/sym11030387

**AMA Style**

Çakmak A.
New Type Direction Curves in 3-Dimensional Compact Lie Group. *Symmetry*. 2019; 11(3):387.
https://doi.org/10.3390/sym11030387

**Chicago/Turabian Style**

Çakmak, Ali.
2019. "New Type Direction Curves in 3-Dimensional Compact Lie Group" *Symmetry* 11, no. 3: 387.
https://doi.org/10.3390/sym11030387