### Article

Kyungpook Mathematical Journal 2022; 62(4): 715-728

**Published online** December 31, 2022

Copyright © Kyungpook Mathematical Journal.

h -almost Ricci Solitons on Generalized Sasakian-space-forms

Doddabhadrappla Gowda Prakasha and Amruthalakshmi Malleshrao Ravindranatha, Sudhakar Kumar Chaubey, Pundikala Veeresha, Young Jin Suh^{*}

Department of Mathematics, Davangere University, Shivagangothri Campus, Davangere - 577 007, India

e-mail : prakashadg@gmail.com and amruthamirajkar@gmail.com

Section of Mathematics, Department of Information Technology, University of Technology and Applied Sciences, Shinas, P. O. Box 77, Postal Code 324, Oman

e-mail : sk22_math@yahoo.co.in and sudhakar.chaubey@shct.edu.om

Center for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru - 560 029, India

e-mail : viru0913@gmail.com and pundikala.veeresha@christuniversity.in

Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Korea

e-mail : yjsuh@knu.ac.kr

**Received**: August 30, 2022; **Revised**: September 30, 2022; **Accepted**: October 7, 2022

### Abstract

The aim of this article is to study the

**Keywords**: Generalized Sasakian-space-form, h-almost Ricci soliton, h-almost gradient Ricci soliton, Three-dimensional quasi-Sasakian manifold, Scalar curvature

### 1. Introduction

Nowadays, the Ricci solitons and their generalizations are enjoying rapid growth by providing new techniques in understanding the geometry and topology of arbitrary Riemannian manifolds. Ricci soliton is a natural generalization of Einstein metric, and is also a self-similar solution to Hamilton's Ricci flow [20, 21]. It plays a specific role in the study of singularities of the Ricci flow. A solution

on

The generalized version of Ricci soliton, so called

that

where λ and

(where, ^{2n+1}

The paper is organized as follows: Section 2 is concerned with the preliminaries on generalized Sasakian-space-forms. In section 3,

### 2. Preliminaries

A

for all vector field

for any vector fields

for any vector fields

An almost contact metric manifold

for some smooth functions _{1}_{2}_{3}_{1}_{2}_{3}

for any vector fields

In addition to the relation (2.4), for a

for any vector fields

Also, for a generalized Sasakian-space-form

and

respectively, for any vector fields

**Definition 2.1.** ([7]) A vector field

for some smooth function ρ on

**Definition 2.2.** ([15]) An infinitesimal automorphism

### 3. h -almost Ricci Solitons on Generalized Sasakian-space-forms

Let

Replacing ξ instead of

Plugging

Observing (3.2) in (3.3) we have

Making use of (3.4) in (3.3) we get

On the other hand, from (2.3) we deduce that

Multiplying both sides of (3.6) by

Let us suppose that the potential vector field

This together with (3.8) provides

Inserting ξ in place of

With the help of

Applying (3.12) in (3.11) we have

Taking the inner product of (3.13) with

or equivalently,

Taking exterior derivative of (3.15) we get

which implies

Taking wedge product of (3.16) with η we gave

from which it follows that

Further, with the help of (2.13) and noting that

As a volume form, ω is closed and by thus the Cartan's formula provides

Next, taking the Lie differentiation to volume form

Integrating (3.20) over

(and so

**Theorem 3.1.** Let

The trace of (3.1) and with the fact that

Now it is easy to check from (2.10), (3.22) and (3.23) that

By virtue of (3.24) and (2.7) we have

That is,

Therefore, the scalar curvature

Next, with the help of (3.22) and (3.25), from (3.1) we get

Since ρ is constant, it follows from (3.10) that

We employ (3.22) and (3.24) in the above equation to achieve

**Theorem 3.2.** Let

**Remark 3.3.** In [14], De and Sarkar studied projective curvature tensor on a generalized Sasakian-space-forms and proved that a

Now, at this junction, we recall the following theorem due to Kim [27]:

**Theorem 3.4.** Let

(i) If _{2} =0

(ii) If

Also, it is known that projectively flat and conformally flat conditions for a generalized Sasakian-space-form of dimension greater than three are equivalent. By taking account of this fact along with previous discussion, we are able to conclude the following:

**Theorem 3.5.** Let

### 4.h -almost Gradient Ricci Solitons on Three-dimensional Quasi-Sasakian Generalized Sasakian-space-forms

In [28], authors have studied the notion of quasi-Sasakian generalized Sasakian-space-forms. This notion is an analogous version of the trans-Sasakian generalized Sasakian-space-forms studied in [2]. An almost contact metric manifold ^{3}

manifold if and only if [30]

for any vector field ^{3}^{3}

From (4.2) it follows that

for any vector field ^{3}

Therefore (4.3) and (4.4) give us

In this section, before entering into the main part we prove the following:

**Lemma 4.1.** On a three-dimensional quasi-Sasakian generalized Sasakian-space-form

for any vector field

Taking covariant differentiation of (4.1) along an arbitrary vector field

Since ξ is Killing on a three-dimensional quasi-Sasakian generalized Sasakian-space-form

for any vector field

Next, suppose that in a three-dimensional quasi-Sasakian generalized Sasakian-space-form

for any vector field ^{3}

and the repeated use of equation (4.9) gives

Replacing ξ instead of

for any vector field

for any vector fields

for all vector fields

Anti-symmetrizing the foregoing equation provides

for all vector fields

for all vector fields

for any vector field

For the scalar curvature

for any vector fields

**Theorem 4.2.** Let _{1} - f_{3}

It is noted that the Weyl tensor vanishes on any three dimensional Riemannian manifold. Therefore we may consider Cotton tensor which is another conformal invariant of a three-dimensional Riemannian manifold. The Cotton tensor

for any vector fields ^{3}

Since the manifold under consideration is of constant curvature, that is, the scalar curvature

**Corollory 4.3.** Let

### Footnote

This work was supported by Department of Science and Technology (DST), Ministry of Science and Technology, Government of India, to the second author Amruthalakshmi M. R. (AMR) by providing financial assistance in the form of DST-INSPIRE Ferllowship(N0:DST/INSPIRE Fellowship/[IF 190869]) and also the fifth author Young Jin Suh(YJS) was supported by NRF-2018-R1D1A1B from National Research Foundation of Korea.

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