### Article

Kyungpook Mathematical Journal 2022; 62(2): 333-345

**Published online** June 30, 2022

Copyright © Kyungpook Mathematical Journal.

∗ -Ricci Soliton on ( κ < 0 , μ ) -almost Cosymplectic Manifolds

Savita Rani and Ram Shankar Gupta*

University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector-16C, Dwarka, New Delhi-110078, India

e-mail : mansavi.14@gmail.com and ramshankar.gupta@gmail.com

**Received**: June 22, 2021; **Revised**: October 10, 2021; **Accepted**: November 15, 2021

### Abstract

We study *-Ricci solitons on non-cosymplectic

**Keywords**: *-Ricci soliton, (κ, μ)-almost cosymplectic manifolds, Nullity distribution, Torse forming vector field

### 1. Introduction

In the framework of Riemannian geometry, Blair et al. [2] introduced a

for any

If

for smooth functions _{5}

for any

In the last few decades there has been extensive study about Ricci solitons and *-Ricci solitons on manifolds. The notion of Ricci solitons was introduced by Hamilton as a natural generalization of Einstein metrics.

A Ricci soliton on a Riemannian manifold satisfies the following equation [8]

where

In 1959, Tachibana [21] introduced the notion of *-Ricci tensors on almost Hermitian manifolds. Later, Hamada [13] defined *-Ricci tensors of real hypersurfaces in non-flat complex space forms, and then Kaimakamis et al. [15] introduced the notion of *-Ricci solitons in non-flat complex space forms.

The *-Ricci tensor on an almost contact metric (a.c.m) manifold

where ψ is a

A *-Ricci soliton on a Riemannian manifold

Many authors have studied solitons on a.c.m manifolds: Sharma initiated the study of Ricci solitons in contact geometry as a

In view of the above, we study the existence of *-Ricci solitons on a non-cosymplectic

This paper is organised as follows: in Section 2, we give some background which is necessary to understand the subsequent sections. In Section 3, we study *-Ricci solitons on non-cosymplectic

### 2. Preliminaries

A smooth Riemannian manifold

for any

On an acs manifold, we have [4]

where

Also, on an acs manifold, we have [11]

for any

The

Endo [11] introduced

On

Also, on a Riemannian manifold

### 3. *-Ricci Solitons on ( κ < 0 , μ ) -acs Manifolds

Let

Let

From (3.3), we find that

On an acs manifold

where

**Lemma 3.1.** Let

for any

Differentiating (3.7) with respect to

Using (3.8) in (2.9), we obtain

Similarly, we get

Adding (3.9) and (3.10) and subtracting (3.11), we get

which gives

Differentiating (3.12) along

Using (3.13) in (3.10), we obtain

Using (2.8) in (3.14), we get (3.6). Hence, the proof of Lemma is complete.

**Theorem 3.2.** Let

for

Contracting (3.15) over

which gives

Using (3.7) in (3.16), we obtain

Using (3.2) and (3.4) in (3.17), we find

From (3.4), we get

Lie-derivative of (3.19) along

Comparing (3.18) and (3.20), we obtain

Putting

Taking inner product of (3.22) with ξ, we get

which implies

Using (1.5), (2.4), (3.4), (3.5), and Theorem 3.2, we get

**Corollary 3.3.** Let

**Corollary 3.4.** Let

Taking trace of (3.23) and using (2.4), we obtain

**Remark 3.5.** In [9] the author proved that there do not exist *-Ricci soliton on non-cosymplectic

In page 5 of [9], the equation (3.12) should be corrected as

Thereafter, the argument given in [9] is not useful to obtain correct result.

### 4. *-Ricci Solitons on ( κ < 0 , μ ) -acs Manifolds with Some Particular Potential Vector Fields

In 1944, Yano [22] introduced a torse-forming vector field as a generalization of concircular, concurrent and parallel vector fields.

**Definition 4.1.** A vector field

where

The vector field

If

then

Recently, in 2017, Chen [6] introduced a torqued vector field. If a non-vanishing

Now, we have

**Theorem 4.2.** Let

Contracting (4.3) over

From (4.4), we find that

Using

**Theorem 4.3.** Let

Contracting (4.6) over

**Theorem 4.4.** Let

where

Putting

**Corollary 4.5.** Let

**Theorem 4.6.** Let

Using (3.5), (4.10) and Theorem 3.2 in (1.4), we get

Contracting (4.11) over

**Theorem 4.7.** Let

Taking trace of (4.13) over

**Theorem 4.8.** Let

Contracting (4.15) over

**Corollary 4.9.** Let

### 5. Examples of *-Ricci Soliton on ( κ < 0 , μ ) -acs Manifolds

In the following examples, the (1,1)-tensor ψ is defined as

**Example 5.1.** Consider

From (5.1), we have

The Koszul's formula with Riemannian connection ∇ is given by

Computing Riemann curvature tensors using (5.4), we get

Further, from (1.1), we obtain

From (2.4) and (3.3), we find that

Using (5.1), (5.5), (5.6) and (5.7), we find that

Further, from (3.5), we obtain

Also, we have

Now, we can see that

for

Hence

**Example 5.2.** Consider

where _{1}

**Example 5.3.** Consider

Then,

Moreover,

### Acknowledgements.

The authors are thankful to the referees for their valuable suggestions for improvement of the article. The first author is thankful to GGSIP University for research fellowship F.No. GGSIPU/DRC/2021/685.

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