#
Geodesics and Translation Curves in ${\mathrm{Sol}}_{0}^{4}$

## Abstract

**:**

## 1. Introduction

Complex space forms | Direct Product Spaces | Direct Product Spaces | Warped Product Spaces |

${\mathbb{E}}^{4}$, ${\mathbb{H}}^{4}$, ${\mathbb{S}}^{4}$, | ${\mathbb{S}}^{2}\times {\mathbb{S}}^{2}$, ${\mathbb{S}}^{2}\times {\mathbb{E}}^{2}$, ${\mathbb{S}}^{2}\times {\mathbb{H}}^{2}$, | ${\mathbb{S}}^{3}\times {\mathbb{E}}^{1}$, ${\mathbb{H}}^{3}\times {\mathbb{E}}^{1}$ | ${\mathrm{Sol}}_{0}^{4}$, ${\mathrm{Sol}}_{1}^{4}$, ${\mathrm{F}}^{4}$, |

$\mathbb{C}{P}^{2}$, $\mathbb{C}{H}^{2}$ | ${\mathbb{E}}^{2}\times {\mathbb{H}}^{2}$, ${\mathbb{H}}^{2}\times {\mathbb{H}}^{2}$ | ${\mathrm{Nil}}_{3}\times {\mathbb{E}}^{1}$, ${\tilde{\mathrm{SL}}}_{2}\mathbb{R}\times {\mathbb{E}}^{1}$ | ${\mathrm{Nil}}^{4}$, ${\mathrm{Sol}}_{m,n}^{4}$ |

## 2. The Model Space ${\mathrm{Sol}}_{\mathbf{0}}^{\mathbf{4}}$

#### 2.1. Lie Group and Lie Algebra

**Remark 1.**

#### 2.2. Metric and Basis

#### 2.3. Levi–Civita Connection

#### 2.4. Riemannian and Sectional Curvatures

## 3. Geodesics in ${\mathrm{Sol}}_{\mathbf{0}}^{\mathbf{4}}$

**Theorem 1.**

#### 3.1. Geodesics in Hypersurface $\mathit{t}=\mathit{const}$

#### 3.2. Geodesics in Hypersurface $\mathit{z}=\mathit{const}$

#### 3.3. Geodesics in Hypersurface $\mathit{y}=\mathit{const}$

**Remark 2.**

## 4. Translation Curves in ${\mathrm{Sol}}_{\mathbf{0}}^{\mathbf{4}}$

#### 4.1. Translation Curves in $So{l}_{0}^{4}$

**Theorem 2.**

**Remark 3.**

#### 4.2. Curvature Properties of Translation Curves

**Definition 1.**

**Theorem 3.**

#### 4.3. Translation Spheres in $So{l}_{0}^{4}$

**Proposition 1.**

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Erjavec, Z.
Geodesics and Translation Curves in *Mathematics* **2023**, *11*, 1533.
https://doi.org/10.3390/math11061533

**AMA Style**

Erjavec Z.
Geodesics and Translation Curves in *Mathematics*. 2023; 11(6):1533.
https://doi.org/10.3390/math11061533

**Chicago/Turabian Style**

Erjavec, Zlatko.
2023. "Geodesics and Translation Curves in *Mathematics* 11, no. 6: 1533.
https://doi.org/10.3390/math11061533