# Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction and Motivations

## 2. Background and Notations

- (i)
- The mean curvature $\mathbf{H}$ of ${M}^{n}$ in ${\tilde{M}}^{m}$ is expressed as $\mathbf{H}=\frac{1}{n}trace\left(h\right)$. If $\mathbf{H}=0$, then ${M}^{n}$ is minimal in ${\tilde{M}}^{m}$ [19].
- (ii)
- If for h, holds the following relation $h({X}_{1},{X}_{2})=g({X}_{1},{X}_{2})\mathbf{H},$ then ${M}^{n}$ is totally umbilical. It is referred to be totally geodesic when $h=0.$
- (iii)
- Let the shape operator be endowed with the eigenvalue of multiplicity $\delta $. If this condition holds, a hypersurface of $(n+1)$-dimensional ${\mathbb{R}}^{n+1}$ is said to be a quasiumbilical hypersurface. On the subset $\mu $ of ${M}^{n}$ such that $mult\left(\delta \right)=n-1$, a characterized direction of an quasiumbilical hypersurface has an eigenvector with the eigenvalue of multiplicity one.
- (iv)
- Let M be a smooth n-dimensional manifold. A smooth map $\psi :M\u27f6{\mathbb{R}}^{n+1}$ is a hypersurface (an immersion) if its differential is injective. It is an embedding if it is also a homeomorphism onto its image $\psi \left(M\right)$. In this case, it is called orientation hypersuface of M [20].

**Theorem**

**1**

## 3. Main Results

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 4. Gradient $\mathbf{k}$-Yamabe Soliton

**Theorem**

**5.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Example**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ali, A.; Mofarreh, F.; Laurian-Ioan, P.; Alluhaibi, N.
Geometry of *k*-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields. *Symmetry* **2021**, *13*, 222.
https://doi.org/10.3390/sym13020222

**AMA Style**

Ali A, Mofarreh F, Laurian-Ioan P, Alluhaibi N.
Geometry of *k*-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields. *Symmetry*. 2021; 13(2):222.
https://doi.org/10.3390/sym13020222

**Chicago/Turabian Style**

Ali, Akram, Fatemah Mofarreh, Pişcoran Laurian-Ioan, and Nadia Alluhaibi.
2021. "Geometry of *k*-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields" *Symmetry* 13, no. 2: 222.
https://doi.org/10.3390/sym13020222