Kyungpook Mathematical Journal 2022; 62(2): 333-345
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
-Ricci Soliton on -almost Cosymplectic Manifolds
Savita Rani and Ram Shankar Gupta*
Received: June 22, 2021; Revised: October 10, 2021; Accepted: November 15, 2021
We study *-Ricci solitons on non-cosymplectic
Keywords: *-Ricci soliton, (κ, μ)-almost cosymplectic manifolds, Nullity distribution, Torse forming vector field
In the framework of Riemannian geometry, Blair et al.  introduced a
for smooth functions
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 
In 1959, Tachibana  introduced the notion of *-Ricci tensors on almost Hermitian manifolds. Later, Hamada  defined *-Ricci tensors of real hypersurfaces in non-flat complex space forms, and then Kaimakamis et al.  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
A smooth Riemannian manifold
On an acs manifold, we have 
Also, on an acs manifold, we have 
Endo  introduced
Also, on a Riemannian manifold
3. *-Ricci Solitons on
From (3.3), we find that
Lemma 3.1. Let
Similarly, we get
Differentiating (3.12) along
Theorem 3.2. Let
Contracting (3.15) over
From (3.4), we get
Lie-derivative of (3.19) along
Taking inner product of (3.22) with ξ, we get
Corollary 3.3. Let
Corollary 3.4. Let
Remark 3.5. In  the author proved that there do not exist *-Ricci soliton on non-cosymplectic
In page 5 of , the equation (3.12) should be corrected as
Thereafter, the argument given in  is not useful to obtain correct result.
4. *-Ricci Solitons on
-acs Manifolds with Some Particular Potential Vector Fields
In 1944, Yano  introduced a torse-forming vector field as a generalization of concircular, concurrent and parallel vector fields.
Definition 4.1. A vector field
The vector field
Recently, in 2017, Chen  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
Theorem 4.3. Let
Contracting (4.6) over
Theorem 4.4. Let
Corollary 4.5. Let
Theorem 4.6. Let
Theorem 4.7. Let
Theorem 4.8. Let
Corollary 4.9. Let
5. Examples of *-Ricci Soliton on
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
Further, from (3.5), we obtain
Also, we have
Now, we can see that
Example 5.2. Consider
Example 5.3. Consider
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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