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,
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
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
(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
manifold if and only if [30]
for any vector field
From (4.2) it follows that
for any vector field
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
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
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
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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