# Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field

^{1}

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*Axioms:*Geometry and Topology)

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Almost-Contact Complex Riemannian Manifolds

#### 2.2. Conformal Transformations of the accR Structure

## 3. Main Results

**Theorem 1.**

**Proof.**

- v is a vertical constant, i.e., constant on ${\mathcal{H}}^{\perp}$;
- w is a horizontal constant, i.e., constant on $\mathcal{H}$.

## 4. Example

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hamilton, R.S. The Ricci flow on surfaces. In Mathematics and General Relativity; Contemporary Mathematics: Santa Cruz, CA, USA; American Mathematical Society: Providence, RI, USA, 1988; pp. 237–262. Volume 71. [Google Scholar]
- Chow, B.; Lu, P.; Ni, L. Hamilton’s Ricci Flow. In Graduate Studies in Mathematics 77; American Mathematical Society: Providence, RI, USA; Science Press: Beijing, China, 2006. [Google Scholar]
- Barbosa, E.; Ribeiro, E., Jr. On conformal solutions of the Yamabe flow. Arch. Math.
**2013**, 101, 79–89. [Google Scholar] [CrossRef] - Cao, H.D.; Sun, X.; Zhang, Y. On the structure of gradient Yamabe solitons. Math. Res. Lett.
**2012**, 19, 767–774. [Google Scholar] [CrossRef] - Chen, B.-Y.; Deshmukh, S. A note on Yamabe solitons. Balk. J. Geom. Appl.
**2018**, 23, 37–43. [Google Scholar] - Daskalopoulos, P.; Sesum, N. The classification of locally conformally flat Yamabe solitons. Adv. Math.
**2013**, 240, 346–369. [Google Scholar] [CrossRef] - Ghosh, A. Yamabe soliton and Quasi Yamabe soliton on Kenmotsu manifold. Math. Slov.
**2020**, 70, 151–160. [Google Scholar] [CrossRef] - Hui, S.K.; Mandal, Y.C. Yamabe solitons on Kenmotsu manifolds. Commun. Korean Math. Soc.
**2019**, 34, 321–331. [Google Scholar] - Roy, S.; Dey, S.; Bhattacharyya, A. Yamabe Solitons on (LCS)
_{n}-manifolds. J. Dyn. Syst. Geom. Theor.**2020**, 18, 261–279. [Google Scholar] [CrossRef] - Manev, M. Yamabe solitons on conformal Sasaki-like almost contact B-metric manifolds. Math
**2022**, 10, 658. [Google Scholar] [CrossRef] - Manev, M. Contactly conformal transformations of general type of almost contact manifolds with B-metric. Appl. Math. Balkanica
**1997**, 11, 347–357. [Google Scholar] - Manev, M.; Gribachev, K. Contactly conformal transformations of almost contact manifolds with B-metric. Serdica Math. J.
**1993**, 19, 287–299. [Google Scholar] - Ganchev, G.; Mihova, V.; Gribachev, K. Almost contact manifolds with B-metric. Math. Balkanica
**1993**, 7, 261–276. [Google Scholar] - Ganchev, G.; Borisov, A. Note on almost contact manifolds with Norden metric. C. R. Acad. Bulg. Sci.
**1986**, 39, 31–34. [Google Scholar] - Manev, H. On the structure tensors of almost contact B-metric manifolds. Filomat
**2015**, 29, 427–436. [Google Scholar] [CrossRef] [Green Version] - Manev, H.; Mekerov, D. Lie groups as 3-dimensional almost contact B-metric manifolds. J. Geom.
**2015**, 106, 229–242. [Google Scholar] [CrossRef] [Green Version] - Manev, M.; Gribachev, K. Conformally invariant tensors on almost contact manifolds with B-metric. Serdica Math. J.
**1994**, 20, 133–147. [Google Scholar] - Manev, M.; Ivanova, M. Canonical type connections on almost contact manifold with B-matric. Ann. Glob. Anal. Geom.
**2013**, 43, 397–408. [Google Scholar] [CrossRef] [Green Version] - Ivanov, S.; Manev, H.; Manev, M. Sasaki-like almost contact complex Riemannian manifolds. J. Geom. Phys.
**2016**, 105, 136–148. [Google Scholar] [CrossRef] - Schouten, J.A. Ricci Calculus. An Introduction to Tensor Analysis and Its Geometrical Applications; Springer: Berlin/Heidelberg, Germany, 1954. [Google Scholar]
- Yano, K. On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo
**1944**, 20, 340–345. [Google Scholar] [CrossRef] - Mihai, A.; Mihai, I. Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications. J. Geom. Phys.
**2013**, 73, 200–208. [Google Scholar] [CrossRef] - Deshmukh, S.; Belova, O. On Killing vector fields on Riemannian manifolds. Math
**2021**, 9, 259. [Google Scholar] [CrossRef]

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Manev, M.
Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field. *Axioms* **2023**, *12*, 44.
https://doi.org/10.3390/axioms12010044

**AMA Style**

Manev M.
Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field. *Axioms*. 2023; 12(1):44.
https://doi.org/10.3390/axioms12010044

**Chicago/Turabian Style**

Manev, Mancho.
2023. "Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field" *Axioms* 12, no. 1: 44.
https://doi.org/10.3390/axioms12010044