# Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

## 3. Proof of the Main Results

#### 3.1. Type I

**Theorem**

**1.**

- ${M}^{2}$ has zero mean curvature;
- ${M}^{2}$ is an open piece of the pseudo-sphere ${\mathbb{S}}_{1}^{2}(0,c)$ centered at the origin with radius c;
- ${M}^{2}$ is an open piece of the hyperbolic space ${\mathbb{H}}_{1}^{2}(0,c)$ centered at the origin with radius c.

#### 3.2. Type II

**Theorem**

**2.**

- ${M}^{2}$ has zero mean curvature;
- ${M}^{2}$ is an open piece of the pseudo-sphere ${\mathbb{S}}_{1}^{2}(0,c)$ centered at the origin with real radius c;
- ${M}^{2}$ is an open piece of the hyperbolic space ${\mathbb{H}}_{1}^{2}(0,c)$ centered at the origin with real radius c.

#### 3.3. Type III

**Theorem**

**3.**

- ${M}^{2}$ has zero mean curvature;
- ${M}^{2}$ is an open piece of the pseudo sphere ${\mathbb{S}}_{1}^{2}(0,c)$ of real radius c;
- ${M}^{2}$ is an open piece of the hyperbolic space ${\mathbb{H}}_{1}^{2}(0,c)$ of real radius c.

**Theorem**

**4.**

- ${M}^{2}$ is an open part of catenoid of the 1st kind, the 2nd kind, the 3rd kind, the 4th kind, or the 5th kind.
- ${M}^{2}$ is an open part of the surface of Enneper of the 2nd kind or the 3rd kind,
- ${M}^{2}$ is an open part of the pseudo sphere ${\mathbb{S}}_{1}^{2}(0,c)$ centered at the origin with radius c,
- ${M}^{2}$ is an open part of the hyperbolic space ${\mathbb{H}}_{1}^{2}(0,c)$ centered at the origin with radius c.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Al-Zoubi, H.; Akbay, A.K.; Hamadneh, T.; Al-Sabbagh, M.
Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space. *Axioms* **2022**, *11*, 326.
https://doi.org/10.3390/axioms11070326

**AMA Style**

Al-Zoubi H, Akbay AK, Hamadneh T, Al-Sabbagh M.
Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space. *Axioms*. 2022; 11(7):326.
https://doi.org/10.3390/axioms11070326

**Chicago/Turabian Style**

Al-Zoubi, Hassan, Alev Kelleci Akbay, Tareq Hamadneh, and Mutaz Al-Sabbagh.
2022. "Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space" *Axioms* 11, no. 7: 326.
https://doi.org/10.3390/axioms11070326