• Communications of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.613-623
• /
• 2020

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.

• Korean Journal of Mathematics
• /
• v.29 no.3
• /
• pp.547-554
• /
• 2021

The object of the present paper is to characterize Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton. It is shown that a Sasakian 3-manifold M with constant scalar curvature admitting a proper gradient Ricci-Yamabe soliton is Einstein and locally isometric to a unit sphere. Also, the potential vector field is an infinitesimal automorphism of the contact metric structure. In addition, if M is complete, then it is compact.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.1
• /
• pp.101-110
• /
• 2022

In this article, the notion of *-conformal Ricci soliton is defined as a self similar solution of the *-conformal Ricci flow. A Sasakian 3-metric satisfying the *-conformal Ricci soliton is completely classified under certain conditions on the soliton vector field. We establish a relation with Fano manifolds and proves a homothety between the Sasakian 3-metric and the Berger Sphere. Also, the potential vector field V is a harmonic infinitesimal automorphism of the contact metric structure.

• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.235-242
• /
• 2023

In this paper, we studied several geometrical aspects of a perfect fluid spacetime admitting a Ricci ρ-soliton and an η-Ricci ρ-soliton. Beside this, we consider the velocity vector of the perfect fluid space time as a gradient vector and obtain some Poisson equations satisfied by the potential function of the gradient solitons.

• Honam Mathematical Journal
• /
• v.40 no.2
• /
• pp.325-346
• /
• 2018

The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci pseudosymmetric, $W_3$-Ricci pseudosymmetric, conharmonic Ricci pseudosymmetric, conformal Ricci pseudosymmetric $(LCS)_n$-manifolds to be shrinking, steady and expanding. We also construct an example of concircular Ricci pseudosymmetric $(LCS)_3$-manifold whose metric is Ricci soliton.

• Korean Journal of Mathematics
• /
• v.29 no.2
• /
• pp.445-453
• /
• 2021

The purpose of the paper is to study of Para-Kenmotsu metric as a 𝜂-Ricci soliton. The paper is organized as follows: • If an 𝜂-Einstein para-Kenmotsu metric represents an 𝜂-Ricci soliton with flow vector field V, then it is Einstein with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with the flow vector field V being an infinitesimal paracontact transformation, then V is strict and the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with non-zero flow vector field V being collinear with 𝜉, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Finally, we cited few examples to illustrate the results obtained.

• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.825-837
• /
• 2022

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric P-connection. At first, it is proven that if the metric of such a manifold is a gradient m-quasi-Einstein metric, then either the gradient of the potential function 𝜓 is collinear with the vector field P or, λ = -(m + 2) and the manifold is of constant sectional curvature -1, provided P𝜓 ≠ m. Next, it is shown that if the metric of the manifold under consideration is a gradient 𝜌-Einstein soliton, then the gradient of the potential function is collinear with the vector field P. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric P-connection is a gradient 𝜔-Ricci soliton, then the manifold is of constant sectional curvature -1 and λ + 𝜇 = -2. Finally, we consider an example to verify our results.

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

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

• Kyungpook Mathematical Journal
• /
• v.61 no.4
• /
• pp.781-790
• /
• 2021

The goal of this paper is to characterize a class of almost Kenmotsu manifolds admitting *-conformal Ricci solitons. It is shown that if a (2n + 1)-dimensional (k, µ)'-almost Kenmotsu manifold M admits *-conformal Ricci soliton, then the manifold M is *-Ricci flat and locally isometric to ℍⁿ⁺¹(-4) × ℝⁿ. The result is also verified by an example.