#
Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E_{1}^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Hypersurfaces of Minkowski Space

#### 2.2. Helicoidal Hypersurfaces with Timelike Axis in Minkowskian Spaces

## 3. Dini-Type Helicoidal Hypersurface with a Timelike Axis

- ${\alpha}_{1}=-{sin}^{8}ucosu{cos}^{5}w,$
- ${\alpha}_{2}=b{sin}^{7}u{cos}^{2}usinw{cos}^{4}w,$
- ${\alpha}_{3}={a}^{2}{sin}^{8}u{cos}^{3}u{sin}^{2}wcosw,$
- ${\alpha}_{4}=-{sin}^{9}u{cos}^{5}w,$
- ${\alpha}_{5}=b{sin}^{8}ucosusinw{cos}^{4}w,$
- ${\alpha}_{6}=\left({a}^{2}\left({cos}^{2}u{cos}^{2}w+{sin}^{2}u{sin}^{2}w\right)-{b}^{2}{cos}^{4}u\right){sin}^{5}u{cos}^{2}ucosw,$:
- ${\alpha}_{7}=-b(2{a}^{2}+{b}^{2}{cos}^{2}w){sin}^{4}u{cos}^{5}usinw{cos}^{2}w.$

- ${\beta}_{1}=-\left(\left({b}^{2}-{sin}^{2}u\right){cos}^{2}w+{a}^{2}\right){sin}^{4}ucosucosw,$
- ${\beta}_{2}=-2{sin}^{5}u{cos}^{3}w,$
- ${\beta}_{3}=-b{sin}^{4}ucosusinw{cos}^{2}w,$
- ${\beta}_{4}=-\left(\left({\left({b}^{2}+{cos}^{2}u\right)}^{2}-1\right){cos}^{2}w+{a}^{2}\left(2{cos}^{2}u+1\right)\right){sin}^{3}ucosw,$
- ${\beta}_{5}=b(\left({b}^{2}-{sin}^{2}u\right){cos}^{2}w+2{a}^{2}){sin}^{2}u{cos}^{3}usinw.$

**Theorem**

**1.**

**Theorem**

**2.**

**Proposition**

**1.**

- ${A}_{8}={\alpha}_{1},$
- ${A}_{7}=-4{\alpha}_{1}sinu+2{\alpha}_{2}+2{\alpha}_{4},$
- ${A}_{6}=\left(2+4{sin}^{2}u-4cosu\right){\alpha}_{1}-4{\alpha}_{2}sinu+4{\alpha}_{3}-12{\alpha}_{4}sinu+4{\alpha}_{5}u,$
- ${A}_{5}=\left(16cosu-4\right){\alpha}_{1}sinu+\left(-8cosu+2\right){\alpha}_{2}+\left(24{sin}^{2}u+6\right){\alpha}_{4}-16{\alpha}_{5}sinu+8{\alpha}_{6},$
- $\begin{array}{cc}\hfill {A}_{4}=& \left(-16{sin}^{2}u-8\right){\alpha}_{1}cosu+16{\alpha}_{2}cosusinu-16{\alpha}_{3}cosu\hfill \\ & +\left(-16{sin}^{2}u-24\right){\alpha}_{4}sinu+\left(16{sin}^{2}u+8\right){\alpha}_{5}-16{\alpha}_{6}sinu+16{\alpha}_{7}\hfill \end{array}$
- ${A}_{3}=\left(16cosu+4\right){\alpha}_{1}sinu+\left(-8cosu-2\right){\alpha}_{2}+\left(24{sin}^{2}u+6\right){\alpha}_{4}-16{\alpha}_{5}sinu+8{\alpha}_{6},$
- ${A}_{2}=\left(-2-4{sin}^{2}u-4cosu\right){\alpha}_{1}+4{\alpha}_{2}sinu-4{\alpha}_{3}-12{\alpha}_{4}sinu+4{\alpha}_{5},$
- ${A}_{1}=4{\alpha}_{1}sinu-2{\alpha}_{2}+2{\alpha}_{4},$
- ${A}_{0}=-{\alpha}_{1}.$

**Proposition**

**2.**

- ${B}_{6}={\beta}_{2},$
- ${B}_{5}=2{\beta}_{1}-6{\beta}_{2}sinu+2{\beta}_{3},$
- ${B}_{4}=\left(3+12{sin}^{2}u\right){\beta}_{2}-8{\beta}_{3}sinu+4{\beta}_{4},$
- ${B}_{3}=-8{\beta}_{1}cosu-\left(12sinu+8{sin}^{3}u\right){\beta}_{2}+\left(4+8{sin}^{2}u\right){\beta}_{3}-8{\beta}_{4}sinu+8{\beta}_{5},$
- ${B}_{2}=\left(3+12{sin}^{2}u\right){\beta}_{2}-8{\beta}_{3}sinu+4{\beta}_{4},$
- ${B}_{1}=-2{\beta}_{1}-6{\beta}_{2}sinu+2{\beta}_{3},$
- ${B}_{0}={\beta}_{2}.$

**Corollary**

**1.**

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Güler, E.; Kişi, Ö.
Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E_{1}^{4}

. *Mathematics* **2019**, *7*, 205.
https://doi.org/10.3390/math7020205

**AMA Style**

Güler E, Kişi Ö.
Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E_{1}^{4}

. *Mathematics*. 2019; 7(2):205.
https://doi.org/10.3390/math7020205

**Chicago/Turabian Style**

Güler, Erhan, and Ömer Kişi.
2019. "Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E_{1}^{4}

" *Mathematics* 7, no. 2: 205.
https://doi.org/10.3390/math7020205