• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.309-318
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• 2017

The object of the present paper is to study almost Ricci solitons and gradient almost Ricci solitons in 3-dimensional f-Kenmotsu manifolds.

• Korean Journal of Mathematics
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• v.28 no.4
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• pp.739-751
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• 2020

The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if (g, V, λ) be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold M, then it reduces to a Ricci soliton and the soliton is expanding for λ=-2. Besides these, in this section, we prove that if V is pointwise collinear with ξ, then V is a constant multiple of ξ and the manifold is of constant sectional curvature -1. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature -1 or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.539-552
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• 2020

In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Furthermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Finally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.161-174
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• 2018

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci solitons where the potential vector field X is point wise collinear with the Reeb vector field ${\xi}$ of the contact metric structure.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1101-1114
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• 2016

Let ($M^{2n+1}$, ${\phi}$, ${\xi}$, ${\eta}$, g) be a (k, ${\mu}$)'-almost Kenmotsu manifold with k < -1 which admits a gradient Ricci almost soliton (g, f, ${\lambda}$), where ${\lambda}$ is the soliton function and f is the potential function. In this paper, it is proved that ${\lambda}$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton ${\mathbb{H}}^{n+1}(-4){\times}{\mathbb{R}}^n$, and the soliton is expanding with ${\lambda}=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature -1 or the potential vector field is pointwise colinear with the Reeb vector field.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.623-638
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• 2023

In this article we study a 𝛿-Ricci-Yamabe almost soliton within the framework of paracontact metric manifolds. In particular we study 𝛿-Ricci-Yamabe almost soliton and gradient 𝛿-Ricci-Yamabe almost soliton on K-paracontact and para-Sasakian manifolds. We prove that if a K-paracontact metric g represents a 𝛿-Ricci-Yamabe almost soliton with the non-zero potential vector field V parallel to 𝜉, then g is Einstein with Einstein constant -2n. We also show that there are no para-Sasakian manifolds that admit a gradient 𝛿-Ricci-Yamabe almost soliton. We demonstrate a 𝛿-Ricci-Yamabe almost soliton on a (𝜅, 𝜇)-paracontact manifold.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.639-645
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• 2023

An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.691-705
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• 2019

In the present paper is to classify Beta-almost (${\beta}$-almost) Ricci solitons and ${\beta}$-almost gradient Ricci solitons on almost $CoK{\ddot{a}}hler$ manifolds with ${\xi}$ belongs to ($k,{\mu}$)-nullity distribution. In this paper, we prove that such manifolds with V is contact vector field and $Q{\phi}={\phi}Q$ is ${\eta}$-Einstein and it is steady when the potential vector field is pointwise collinear to the reeb vectoer field. Moreover, we prove that a ($k,{\mu}$)-almost $CoK{\ddot{a}}hler$ manifolds admitting ${\beta}$-almost gradient Ricci solitons is isometric to a sphere.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1315-1325
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• 2019

The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.

• Kyungpook Mathematical Journal
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• v.62 no.4
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• pp.715-728
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• 2022

The aim of this article is to study the h-almost Ricci solitons and h-almost gradient Ricci solitons on generalized Sasakian-space-forms. First, we consider h-almost Ricci soliton with the potential vector field V as a contact vector field on generalized Sasakian-space-form of dimension greater than three. Next, we study h-almost gradient Ricci solitons on a three-dimensional quasi-Sasakian generalized Sasakian-space-form. In both the cases, several interesting results are obtained.