### Article

Kyungpook Mathematical Journal 2021; 61(4): 813-822

**Published online** December 31, 2021 https://doi.org/10.5666/KMJ.2021.61.4.813

Copyright © Kyungpook Mathematical Journal.

### Ricci-Yamabe Solitons and Gradient Ricci-Yamabe Solitons on Kenmotsu 3-manifolds

Arpan Sardar and Avijit Sarkar*

Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India

e-mail : arpansardar51@gmail.com and avjaj@yahoo.co.in

**Received**: February 14, 2021; **Revised**: April 24, 2021; **Accepted**: June 14, 2021

### Abstract

The aim of this paper is to characterize a Kenmotsu 3-manifold whose metric is either a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton. Finally, we verify the obtained results by an example.

**Keywords**: Ricci-Yamabe soliton, Gradient Yamabe soliton, Kenmotsu manifolds, η-paralle Ricci tensor, scalar curvature

### 1. Introduction

Hamilton introduced Yamabe flow in 1982 [8] at the same time he introduced Ricci flow. Ricci solitons and Yamabe solitons are the limit of the solutions of Ricci flow and Yamabe flow respectively. In dimension

Over the past twenty years the theory of geometric flows, such as Ricci flow and Yamabe flow has been the focus of attention of many geometers. Recently, in 2019, Guler and Crasmareanu [7] introduced the study of a new geometric flow which is a scalar combination of Ricci and Yamabe flows under the name Ricci-Yamabe map. This is also called the Ricci-Yamabe flow of type

A solution to the Ricci-Yamabe flow is called Ricci-Yamabe soliton if it depends only on one parameter group of diffeomorphism and scaling. A Ricci-Yamabe soliton on a Riemannian manifold

where

where

The Ricci-Yamabe soliton is said to be expanding, steady, or shrinking according to whether λ is negative, zero, or positive. It is called an almost Ricci-Yamabe soliton if

• Ricci soliton [8] if

α =1, β=0 .• Yamabe soliton[9] if

α =0, β = 1 .• Einstein soliton [2] if

α =1, β = -1 .• ρ- Einstein soliton [3] ifα =1, β = -2ρ .

When

The paper is organized as follows. After preliminaries in Section 2, we consider Ricci-Yamabe solitons on Kenmotsu 3-manifolds in Section 3. In Section 4 we study Ricci-Yamabe solitons on Kenmotsu 3-manifolds with η-parallel Ricci tensors. Section 5 is devoted to studying gradient Ricci-Yamabe solitons on Kenmotsu 3-manifolds. Finally, in Section 6 we construct an example of a 3-dimensional Kenmotsu manifold admitting a Ricci-Yamabe soliton.

### 2. Preliminaries

An almost contact structure [1] on a

where 'id' denotes the identity mapping and relation (2.1) implies that

or equivalently,

for all

which gives ^{*}^{*}

For a

for any

From [4], we know that for a 3-dimensional Kenmotsu manifold

where

An almost contact metric manifold is said to be η-Einstein if the Ricci tensor

for any vector field

Three-dimensional Kenmotsu manifold have been studied in the papers ([4], [13]).

**Lemma 2.1.**([13]) On any three-dimensional Kenmotsu manifold

**Lemma 2.2.**([4]) A 3-dimensional Riemannian manifold is a manifold of constant sectional curvature -1 if and only if the scalar curvature

**Lemma 2.3.**(Proposition 8 of ([10]) Let

### 3. Ricci-Yamabe Solitons on Kenmotsu 3-manifolds

Assume that the Kenmotsu 3-manifold admits a proper Ricci-Yamabe soliton

In Kenmotsu 3-manifolds

which implies

Hence we can state the following theorem using Lemma 2.3 :

**Theorem 3.1.**

### 4. Ricci-Yamabe Solitons on Kenmotsu 3-manifolds with η-parallel Ricci Tensor

In this section we study Ricci-Yamabe solitons on Kenmotsu 3-manifolds with η-parallel Ricci tensor. A Kenmotsu manifold is said to have η-parallel Ricci tensor if [6]

for all smooth vector fields

In Kenmotsu 3-manifolds

Putting

If

Case I: If

Case II: If

Hence we conclude the following:

**Theorem 4.1.**

### 5. Gradient Ricci-Yamabe Solitons on Kenmotsu 3-manifolds

Suppose a Kenmotsu 3-manifold admits the gradient Ricci-Yamabe soliton. Then from equation (1.3), we get

Differentiating (5.1) covariantly along any vector field

Interchanging

Hence from the above equations, we get

Now, in 3-dimension Kenmotsu manifolds

Contracting (5.6) and using Lemma 2.1, we get

Again, replacing

In view of (5.7) and (5.8) we infer

Putting

Taking inner product of (5.7) with the vector field ξ, we get

Putting

Using (5.10) and (5.12) in (5.9), we obtain

Hence above equation implies either,

Case I: If

Case II: If

which implies

Using the above equation in (5.1), we get

which implies

Thus, we state the following:

**Theorem 5.1.**

If we take

### 6. Example

We consider the 3-dimensional manifold

Let

Let η be the 1-form defined by

Then using the linearity of ϕ and

for any

Then for

Let ∇ be the Levi-Civita connection with respect to the metric

The Riemannian connection ∇ of the metric

For the above we see that

Now, we have

With the help of the above results and using (6.1), we obtain

From the above, we can easily calculate the non-vanishing components of the Ricci tensor

Hence from the above, we get

where

From (3.3) we obtain

Therefore

### Acknowledgement

The authors are thankful to the referee for his valuable suggestions towards the improvement of the paper.

### References

- D. E. Blair, Riemannian Geometry of contact and symplectic manifolds, Progress in Mathematics,
203 (2010), Birkhäuser, New work. - G. Catino and L. Mazzieri,
Gradient Einstein solitons , Nonlinear Anal.,132 (2016), 66-94. - G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri,
The Ricci-Bourguignon flow , Pacific J. Math.,28 (2017), 337-370. - U. C. De and G. Pathak,
On 3-dimensional Kenmotsu manifolds , Indian J. Pure Appl. Math.,35(2) (2004), 159-165. - D. Dey,
Almost Kenmotsu metric as Ricci-Yamabe soliton , arXiv: 2005.02322v1[math.DG], (2020 May 5). - A. Ghosh, R. Sharma and J. T. Cho,
Contact metric manifolds with η-parallel torsion tensor , Ann. Global Anal. Geom.,34(3) (2008), 287-299. - S. Guler and M. Crasmareanu,
Ricci-Yamabe maps for Riemannian flow and their volume variation and volume entropy , Turkish J. Math.,43(5) (2019), 2631-2641. - R. S. Hamilton,
The Ricci flow on surfaces, Mathematics and general relativity , Contemp. Math.,71 (1998), 237-262. - R. S. Hamilton, Lectures on geometric flows, 1989 (unpublished).
- K. Kenmotsu,
A class of almost contact Riemannian manifolds , Tohoku Math. J.,24 (1972), 93-103. - V. F. Kirichenko,
On the geometry of Kenmotsu manifolds , Dokl. Akad. Nauk,380(5) (2001), 585-587. - S. Sasaki and Y. Hatakeyama,
On differentiable manifolds with certain structures which are closely related to almost contact structure, II , Tohoku Math. J.,13 (1961), 281-294. - Y. Wang,
Yamabe soliton on 3-dimensional Kenmotsu manifolds , Bull. Belg. Math. Soc. Simon Stevin,23 (2016), 345-355. - K. Yano,
Concircular geometry. I. Concircular transformations , Proc. Imp. Acad. Tokyo,16 (1940), 195-200.