$ \eta $-Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection

Yusuf Dogru; Yusuf Dogru

doi:10.3934/math.2023603

AIMS Mathematics

2023, Volume 8, Issue 5: 11943-11952. doi: 10.3934/math.2023603

Previous Article Next Article

Research article Special Issues

$\eta$ -Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection

Yusuf Dogru ^,

Air Training Command, Konak, İzmir, Turkey

Received: 23 January 2023 Revised: 10 March 2023 Accepted: 16 March 2023 Published: 20 March 2023
MSC : 35Q51, 53C07, 53C25

We consider a generalization of a Ricci soliton as $\eta$ -Ricci-Bourguignon solitons on a Riemannian manifold endowed with a semi-symmetric metric and semi-symmetric non-metric connection. We find some properties of $\eta$ -Ricci-Bourguignon soliton on Riemannian manifolds equipped with a semi-symmetric metric and semi-symmetric non-metric connection when the potential vector field is torse-forming with respect to a semi-symmetric metric and semi-symmetric non-metric connection.

Keywords:

Citation: Yusuf Dogru. $\eta$ -Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection[J]. AIMS Mathematics, 2023, 8(5): 11943-11952. doi: 10.3934/math.2023603

Related Papers:

[1]	Tong Wu, Yong Wang . Super warped products with a semi-symmetric non-metric connection. AIMS Mathematics, 2022, 7(6): 10534-10553. doi: 10.3934/math.2022587
[2]	Yanlin Li, Aydin Gezer, Erkan Karakaş . Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection. AIMS Mathematics, 2023, 8(8): 17335-17353. doi: 10.3934/math.2023886
[3]	Rajesh Kumar, Sameh Shenawy, Lalnunenga Colney, Nasser Bin Turki . Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds. AIMS Mathematics, 2024, 9(11): 30364-30383. doi: 10.3934/math.20241465
[4]	Shahroud Azami, Mehdi Jafari, Nargis Jamal, Abdul Haseeb . Hyperbolic Ricci solitons on perfect fluid spacetimes. AIMS Mathematics, 2024, 9(7): 18929-18943. doi: 10.3934/math.2024921
[5]	Rongsheng Ma, Donghe Pei . $ \ast $-Ricci tensor on $ (\kappa, \mu) $-contact manifolds. AIMS Mathematics, 2022, 7(7): 11519-11528. doi: 10.3934/math.2022642
[6]	Yanlin Li, Dipen Ganguly, Santu Dey, Arindam Bhattacharyya . Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds. AIMS Mathematics, 2022, 7(4): 5408-5430. doi: 10.3934/math.2022300
[7]	Xiaosheng Li . New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces. AIMS Mathematics, 2023, 8(10): 23062-23086. doi: 10.3934/math.20231174
[8]	Mohd Danish Siddiqi, Fatemah Mofarreh . Schur-type inequality for solitonic hypersurfaces in $ (k, \mu) $-contact metric manifolds. AIMS Mathematics, 2024, 9(12): 36069-36081. doi: 10.3934/math.20241711
[9]	Shahroud Azami, Rawan Bossly, Abdul Haseeb . Riemann solitons on Egorov and Cahen-Wallach symmetric spaces. AIMS Mathematics, 2025, 10(1): 1882-1899. doi: 10.3934/math.2025087
[10]	Yanlin Li, Mohd Danish Siddiqi, Meraj Ali Khan, Ibrahim Al-Dayel, Maged Zakaria Youssef . Solitonic effect on relativistic string cloud spacetime attached with strange quark matter. AIMS Mathematics, 2024, 9(6): 14487-14503. doi: 10.3934/math.2024704

Abstract

1. Introduction

A semi-symmetric connection is a linear connection on a Riemannian manifold $(M, g)$ whose torsion tensor $T$ is of the form

$\begin{equation*} T(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \phi (\mathcal{\zeta }_{2}) \mathcal{\zeta }_{1}-\phi (\mathcal{\zeta }_{1})\mathcal{\zeta }_{2}, \end{equation*}$

where $\phi$ is a $1$ -form defined by $\phi (\mathcal{\zeta }_{1}) = g(\mathcal{\zeta }_{1}, U)$ and $U$ is a vector field on $M$ ^[11].

If $\nabla$ is the Levi-Civita connection of a Riemannian manifold $(M, g),$ then the semi-symmetric metric connection $\widetilde{\nabla }$ is defined by

$\begin{equation} \widetilde{\nabla }_{\mathcal{\zeta }_{1}}\mathcal{\zeta }_{2} = \nabla _{ \mathcal{\zeta }_{1}}\mathcal{\zeta }_{2}+\phi (\mathcal{\zeta }_{2}) \mathcal{\zeta }_{1}-g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})U, \end{equation}$

(1.1)

where $\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}, U$ are vector fields on $M$ ^[20]. Let $\widetilde{\text{ }R}$ and $R$ denote Riemannian curvature tensor fields of $\widetilde{\nabla }$ and $\nabla$ , respectively. Then from (1.1), it is easy to see that

$\begin{eqnarray} \widetilde{R}(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\mathcal{\zeta }_{3} & = &R(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\mathcal{\zeta }_{3}-\alpha ( \mathcal{\zeta }_{2}, \mathcal{\zeta }_{3})\mathcal{\zeta }_{1}+\alpha ( \mathcal{\zeta }_{1}, \mathcal{\zeta }_{3})\mathcal{\zeta }_{2} \\ &&-g(\mathcal{\zeta }_{2}, \mathcal{\zeta }_{3})B\mathcal{\zeta }_{1}+g( \mathcal{\zeta }_{1}, \mathcal{\zeta }_{3})B\mathcal{\zeta }_{2}, \end{eqnarray}$

(1.2)

where

$\begin{equation} \alpha (\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = g(B\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = (\nabla _{\mathcal{\zeta }_{1}}\phi )\mathcal{\zeta } _{2}-\phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2})+\frac{1}{2}g( \mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}). \end{equation}$

(1.3)

Denote by $\widetilde{Ric}$ and $Ric$ the Ricci tensor fields of the connections $\widetilde{\nabla }$ and $\nabla,$ respectively. Then from (1.2), it is easy to see that

$\begin{equation} \widetilde{Ric} = Ric-(n-2)\alpha -(tr\alpha) g, \text{(see [20]).} \end{equation}$

(1.4)

The semi-symmetric non-metric connection $\overset{\circ }{ \widetilde{\nabla }}$ is defined by

$\begin{equation} \overset{\circ }{\widetilde{\nabla }}_{\mathcal{\zeta }_{1}}\mathcal{\zeta } _{2} = \nabla _{\mathcal{\zeta }_{1}}\mathcal{\zeta }_{2}+\phi (\mathcal{\zeta }_{2})\mathcal{\zeta }_{1}, \end{equation}$

(1.5)

where $\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}$ are vector fields on $M$ and $\nabla$ is the Levi-Civita connection of a Riemannian manifold $(M, g)$ ^[1]. Let $\overset{\circ }{\widetilde{R}}$ and $R$ denote the Riemannian curvature tensor fields of $\overset{\circ }{\widetilde{\nabla }}$ and $\nabla,$ respectively. Then from (1.5), it is easy to see that

$\begin{equation} \overset{\circ }{\widetilde{R}}(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) \mathcal{\zeta }_{3} = R(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\mathcal{ \zeta }_{3}-\sigma (\mathcal{\zeta }_{2}, \mathcal{\zeta }_{3})\mathcal{\zeta }_{1}+\sigma (\mathcal{\zeta }_{1}, \mathcal{\zeta }_{3})\mathcal{\zeta }_{2}, \end{equation}$

(1.6)

where

$\begin{equation} \sigma (\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = g(B\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = (\nabla _{\mathcal{\zeta }_{1}}\phi )\mathcal{\zeta } _{2}-\phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2}). \end{equation}$

(1.7)

Denote by $\overset{\circ }{\widetilde{Ric}}$ and $Ric$ the Ricci tensor fields of the connections $\overset{\circ }{\widetilde{\nabla }}$ and $\nabla,$ respectively. Then from (1.6), we have

$\begin{equation} \overset{\circ }{\widetilde{Ric}} = Ric-(n-1)\sigma , \text{(see [1]).} \end{equation}$

(1.8)

Let $(M, g)$ be a Riemannian manifold. R. S. Hamilton ^[12] presented the Ricci flow for the first time as

$\begin{equation*} \begin{array}{c} \frac{\partial }{\partial t}g(t) = -2Ric(g(t)). \end{array} \end{equation*}$

The Ricci flow is an evolution equation for Riemannian metrics. Ricci solitons correspond to self-similar solutions of Ricci flow. In the recent years, the geometry of Ricci solitons has been studied by many geometers. See, for example, ^[3,8,15,17].

Another generalization of Ricci soliton is $\eta$ -Ricci-Bourguignon soliton. An $\eta$ -Ricci-Bourguignon soliton (see ^[18]) is defined by

$\begin{equation} \frac{1}{2}{\rm{£}} _{\lambda }g+Ric = (\alpha ^{\ast }+\beta \tau)g+\gamma \eta \otimes \eta , \end{equation}$

(1.9)

where $\lambda$ is the potantial vector field, $\eta$ is a $1$ -form on $M,$ ${\rm{£}} _{\lambda }g$ denotes the Lie derivative of $g$ in the direction of $\lambda$ , $Ric$ is the Ricci curvature, $\tau$ is scalar curvature and $\alpha ^{\ast }, \beta, \gamma$ are real numbers. $\eta$ -Ricci-Bourguignon solitons on submanifolds were studied in ^[5].

In the present study, we consider some properties of $\eta$ - Ricci-Bourguignon soliton on Riemannian manifolds equipped with a semi-symmetric metric connection and semi-symmetric non-metric connection when the potential vector field is torse-forming with respect to a semi-symmetric metric connection and semi-symmetric non-metric connection. As recent studies on torse-forming vector fields see ^[4,14,15].

The paper is organized as follows: In Section 2, $\eta$ -Ricci-Bourguignon solitons on Riemannian manifolds with a semi-symmetric metric connection are studied. In Section 3, $\eta$ -Ricci-Bourguignon solitons on Riemannian manifolds endowed with a semi-symmetric non-metric connection is considered.

2. $\eta$ -Ricci-Bourguignon solitons on Riemannian manifolds equipped with a semi-symmetric metric connection

In this section, we consider Ricci solitons on Riemannian manifolds endowed with a semi-symmetric metric connection.

The Euclidean 3-space, hyperbolic 3-space and Minkowski motion group are included in the following 3-parameter family of Riemannian homogeneous spaces ( $\mathbb{R}^{3}, g\left[\mu _{1, }\mu _{2}, \mu _{3}\right]$ ) with left invariant metric

$\begin{equation*} g\left[ \mu _{1, }\mu _{2}, \mu _{3}\right] = e^{-2\mu _{1}t}dx^{2}+e^{-2\mu _{2}t}dy^{2}+\mu _{3}^{2}dt^{2}. \end{equation*}$

Here $\mu _{1, }\mu _{2}$ are real constants and $\mu _{3}$ is a positive constant.

The Lie group $G(\mu _{1, }\mu _{2}, \mu _{3})$ can be realised as a closed subgroup of affine transformation group $GL_{3}\mathbb{R}\ltimes \mathbb{R}^{3}$ of $\mathbb{R}^{3}.$

The Levi-Civita connection $\nabla$ of $G(\mu _{1, }\mu _{2}, \mu _{3})$ is given by the following formula:

$\begin{array}{c} \nabla _{E_{1}}E_{1} = \frac{\mu _{1}}{\mu _{3}}E_{3}, \text{ }\nabla _{E_{1}}E_{2} = 0, \text{ }\nabla _{E_{1}}E_{3} = -\frac{\mu _{1}}{\mu _{3}}E_{1}, \\ \nabla _{E_{2}}E_{1} = 0, \text{ }\nabla _{E_{2}}E_{2} = \frac{\mu _{2}}{\mu _{3}} E_{3}, \text{ }\nabla _{E_{2}}E_{3} = -\frac{\mu _{2}}{\mu _{3}}E_{2}, \\ \nabla _{E_{3}}E_{1} = \text{ }\nabla _{E_{3}}E_{2} = \text{ }\nabla _{E_{3}}E_{3} = 0. \end{array}$

(2.1)

The Ricci tensor field $Ric$ of $G$ is given by

$\begin{equation*} R_{11} = -\frac{\mu _{1}(\mu _{1}+\mu _{2})}{\mu _{3}^{2}}, \text{ }R_{22} = - \frac{\mu _{2}(\mu _{1}+\mu _{2})}{\mu _{3}^{2}}, R_{33} = -\frac{\mu _{1}^{2}+\mu _{2}^{2}}{\mu _{3}^{2}}\text{ } \end{equation*}$

and the scalar curvature $\tau$ of $G$ is given by

$\begin{equation*} \tau = -\frac{2}{\mu _{3}^{2}}\left( \mu _{1}^{2}+\mu _{2}^{2}+\mu _{1}\mu _{2}\right).\; \text{(see [13]).} \end{equation*}$

Using (2.1), the Levi-Civita connection $\nabla$ of $G(-1, 1, 1)$ is given by the following formula:

$\begin{equation*} \nabla _{E_{1}}E_{1} = -E_{3}, \text{ }\nabla _{E_{1}}E_{2} = 0, \text{ }\nabla _{E_{1}}E_{3} = E_{1}, \end{equation*}$

$\begin{equation*} \nabla _{E_{2}}E_{1} = 0, \text{ }\nabla _{E_{2}}E_{2} = E_{3}, \text{ }\nabla _{E_{2}}E_{3} = -E_{2}, \end{equation*}$

$\begin{equation*} \nabla _{E_{3}}E_{1} = \text{ }\nabla _{E_{3}}E_{2} = \text{ }\nabla _{E_{3}}E_{3} = 0. \end{equation*}$

Then we can state the following example:

Example 1. Assume that $\lambda = 2\sqrt{2}E_{2}+4E_{3}$ is the potential vector field. If $\eta$ is the $1$ -form corresponding to the vector field $P = \sqrt{2} E_{2}+2E_{3}$ , then $G(-1, 1, 1)$ is an $\eta$ -Ricci-Bourguignon soliton with respect to a semi-symmetric metric connection.

Using the Eq (1.1), we get the Lie derivative as follows

$\begin{eqnarray} ({\rm{£}} _{\lambda }g)(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) & = &g(\widetilde{\nabla }_{\mathcal{\zeta }_{1}}\lambda , \mathcal{ \zeta }_{2})+g(\mathcal{\zeta }_{1}, \widetilde{\nabla }_{\mathcal{\zeta } _{2}}\lambda )-2\phi (\lambda )g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) \\ &&+g(\mathcal{\zeta }_{1}, \lambda )\phi (\mathcal{\zeta }_{2})+g(\mathcal{ \zeta }_{2}, \lambda )\phi (\mathcal{\zeta }_{1}). \end{eqnarray}$

(2.2)

Therefore, using Eq (2.2), the soliton Eq (1.9) with respect to a semi-symmetric metric connection can be written as

$\begin{array}{c} \frac{1}{2}\left( g(\widetilde{\nabla }_{\mathcal{\zeta }_{1}}\lambda , \mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{1}, \widetilde{\nabla }_{\mathcal{ \zeta }_{2}}\lambda )\right) -\phi (\lambda )g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\\ +\frac{1}{2}\left( g(\mathcal{\zeta }_{1}, \lambda )\phi (\mathcal{\zeta } _{2})+g(\mathcal{\zeta }_{2}, \lambda )\phi (\mathcal{\zeta }_{1})\right) +Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) \\ = (\alpha ^{\ast }+\beta \tau)g(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2})+\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}). \end{array}$

(2.3)

A vector field $\lambda$ on a Riemannian manifold $\left(M, g\right)$ is called torse-forming ^[19], if

$\nabla _{\mathcal{\zeta }_{1}}\lambda = c\mathcal{\zeta }_{1}+\varpi ( \mathcal{\zeta }_{1})\lambda ,$

where $c$ is a smooth function, $\varpi$ is a $1$ -form and $\nabla$ is the Levi-Civita connection of $g.$

Specifically, if $\varpi = 0$ , then $\lambda$ is called a concircular vector field ^[10] and if $c = 0$ , then $\lambda$ is called a recurrent vector field ^[17].

A non-flat Riemannian manifold $(M, g)$ $(n\geq 3)$ is called a hyper-generalized quasi-Einstein manifold ^[16], if its Ricci tensor field is not likewise zero and provides

$Ric = b_{1}g+b_{2}\mathcal{\omega }_{1}\otimes \mathcal{\omega } _{1}+b_{3}\left( \mathcal{\omega }_{1}\otimes \mathcal{\omega }_{2}+\mathcal{ \omega }_{2}\otimes \mathcal{\omega }_{1}\right) +b_{4}\left( \mathcal{ \omega }_{1}\otimes \mathcal{\omega }_{3}+\mathcal{\omega }_{3}\otimes \mathcal{\omega }_{1}\right) ,$

where $b_{1}, b_{2}, b_{3}$ and $b_{4}$ are functions and $\mathcal{\omega } _{1}, \mathcal{\omega }_{2}$ and $\mathcal{\omega }_{3}$ are non-zero $1$ -forms. If $b_{4} = 0$ , then $M$ is called a generalized quasi-Einstein manifold in the sense of Chaki ^[7]. If $b_{3} = b_{4} = 0$ , then $M$ is called a quasi-Einstein manifold ^[6]. Suppose that $b_{2} = b_{3} = b_{4} = 0,$ then $(M, g)$ is an Einstein manifold ^[2]. The functions $b_{1}, b_{2}, b_{3}$ and $b_{4}$ are called associated functions.

A non-flat Riemannian manifold $(M, g)$ $(n\geq 3)$ is called a generalized quasi-Einstein manifold in the sense of De and Ghosh ^[9], if its Ricci tensor field is not identically zero and satisfies

$Ric = b_{1}g+b_{2}\mathcal{\omega }_{1}\otimes \mathcal{\omega }_{1}+b_{3} \mathcal{\omega }_{2}\otimes \mathcal{\omega }_{2},$

where $b_{1}, b_{2}$ and $b_{3}$ are functions. The functions $b_{1}, b_{2}$ and $b_{3}$ are called associated functions.

Now let $(M, g)$ be a Riemannian manifold equipped with a semi-symmetric metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric metric connection on $M$ . Then $\widetilde{\nabla }_{\mathcal{\zeta } _{1}}\lambda = c\mathcal{\zeta }_{1}+\varpi (\mathcal{\zeta }_{1})\lambda.$ So by (2.3), we can write

$Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})$

$-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta }_{1})+g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta } _{2})\right\}$

$-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\phi (\mathcal{\zeta } _{2})+g(\mathcal{\zeta }_{2}, \lambda )\phi (\mathcal{\zeta }_{1})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}).$

Thus, the following theorem can be stated:

Theorem 1. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric metric connection on $M$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if

$\begin{eqnarray*} && \;\;Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\\&&-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{ \zeta }_{1})+g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta } _{2})\right\} \\ &&-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\phi (\mathcal{\zeta } _{2})+g(\mathcal{\zeta }_{2}, \lambda )\phi (\mathcal{\zeta }_{1})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}). \end{eqnarray*}$

(2.4)

If $\lambda$ is a concircular potential vector field with respect to a semi-symmetric metric connection, then the following corollaries can be stated:

Corollary 1. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection and $\lambda$ a concircular potential vector field with respect to a semi-symmetric metric connection on $M$ . If $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton and $\phi$ is the $g$ dual of $\lambda$ , then $M$ is a generalized quasi Einstein manifold in the sense of De and Ghosh with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\left\Vert \lambda \right\Vert ^{2}\right), -1$ and $\gamma.$

Corollary 2. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection and $\lambda$ a concircular potential vector field with respect to a semi-symmetric metric connection on $M$ . If $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton and $\eta$ is the $g$ dual of $\lambda$ , then $M$ is a generalized quasi Einstein manifold in the sense of Chaki with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\phi (\lambda)\right), \gamma$ and $-\frac{1}{2}.$

Now assume that $\lambda$ is a torse-forming potential vector field and the $1$ -form $\eta$ is the $\ g$ -dual of $\lambda.$ Then from (2.4), we have

$Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})$

$-\frac{1}{2}\left\{ \eta (\mathcal{\zeta }_{1})\varpi (\mathcal{\zeta } _{2})+\eta (\mathcal{\zeta }_{2})\varpi (\mathcal{\zeta }_{1})\right\}$

$-\frac{1}{2}\left\{ \eta (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta } _{2})+\eta (\mathcal{\zeta }_{2})\phi (\mathcal{\zeta }_{1})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}).$

Then we obtain:

Theorem 2. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric metric connection on $M$ . Suppose that the $1$ -form $\eta$ is the $g$ dual of $\lambda$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if $M$ is a hyper-generalized quasi-Einstein manifold with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\phi (\lambda)\right), \gamma, -\frac{1}{2}$ and $-\frac{1}{2}.$

Using (1.4), the Eq (2.4) can be written as

$\begin{array}{c}\widetilde{Ric}(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )-tr\alpha \right) g(\mathcal{\zeta } _{1}, \mathcal{\zeta }_{2})\\ -\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta } _{1})\right\} -\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\phi ( \mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\phi (\mathcal{\zeta } _{1})\right\} \\ +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2})-(n-2)\alpha ( \mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}). \end{array}$

(2.5)

Thus, the following corollary can be expressed:

Corollary 3. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric metric connection on $M$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if the Ricci tensor field of a semi-symmetric metric connection is of the form (2.5).

Now assume that $U$ is a parallel unit vector field with respect to the Levi-Civita connection, i.e., $\nabla U = 0$ and $\Vert U\Vert = 1$ . Then

$(\nabla _{\mathcal{\zeta }_{1}}\phi )\mathcal{\zeta }_{2} = \nabla _{\mathcal{ \zeta }_{1}}\phi (\mathcal{\zeta }_{2})-\phi (\nabla _{\mathcal{\zeta }_{1}} \mathcal{\zeta }_{2}) = 0.$

So from (1.3), $\alpha (\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) = -\phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2})+\frac{1}{2}g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})$ and $tr\alpha = \frac{n}{2}-1.$ Thus by (2.5), we have

$\widetilde{Ric}(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = (\alpha ^{\ast }+\beta \tau-c+\phi (\lambda )-n+2)g(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2})$

$-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta } _{1})\right\} -\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\phi ( \mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\phi (\mathcal{\zeta } _{1})\right\}$

$+\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2})+\left( n-2\right) \phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2}).$

If $\lambda$ is a concircular potential vector field and $\phi$ is the $g$ dual of $\lambda$ , then

$\widetilde{Ric}(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = (\alpha ^{\ast }+\beta \tau-c+\left\Vert \lambda \right\Vert ^{2}-n+2)g(\mathcal{\zeta } _{1}, \mathcal{\zeta }_{2})$

$+\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2})+\left( n-3\right) \phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2}).$

Hence we have:

Theorem 3. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric metric connection, $U$ a parallel unit vector field with respect to the Levi-Civita connection $\nabla$ and $\lambda$ a concircular potential vector field with respect to a semi-symmetric metric connection on $M$ . Suppose that the $1$ -form $\phi$ is the $g$ -dual of $\lambda$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if $M$ is a generalized quasi-Einstein manifold in the sense of De and Ghosh with respect to a semi-symmetric metric connection with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\left\Vert \lambda \right\Vert ^{2}-n+2\right),$ $\gamma,$ $\left(n-3\right).$

3. $\eta$ -Ricci-Bourguignon on Riemannian manifolds equipped with a semi-symmetric non-metric connection

In this section, we consider Ricci solitons on Riemannian manifolds endowed with a semi-symmetric non-metric connection.

Using (2.1), the Levi-Civita connection $\nabla$ of $G(-1, -1, 1)$ is given by the following formula:

$\begin{equation*} \nabla _{E_{1}}E_{1} = -E_{3}, \text{ }\nabla _{E_{1}}E_{2} = 0, \text{ }\nabla _{E_{1}}E_{3} = E_{1}, \end{equation*}$

$\begin{equation*} \nabla _{E_{2}}E_{1} = 0, \text{ }\nabla _{E_{2}}E_{2} = -E_{3}, \text{ }\nabla _{E_{2}}E_{3} = E_{2}, \end{equation*}$

$\begin{equation*} \nabla _{E_{3}}E_{1} = \text{ }\nabla _{E_{3}}E_{2} = \text{ }\nabla _{E_{3}}E_{3} = 0. \end{equation*}$

Then we can state the following example:

Example 2. Assume that $\lambda = 2E_{1}-3E_{2}+E_{3}$ is the potential vector field. If $\eta$ is the $1$ -form corresponding to the vector field $P = 2E_{3}$ , then $G(-1, -1, 1)$ is an $\eta$ -Ricci-Bourguignon soliton with respect to a semi-symmetric non-metric connection.

Using the Eq (1.5), we get the Lie derivative as follows

$\begin{equation} ({\rm{£}} _{\lambda }g)(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = g(\overset {\circ }{\widetilde{\nabla }}_{\mathcal{\zeta }_{1}}\lambda , \mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{1}, \overset{\circ }{\widetilde{\nabla }}_{ \mathcal{\zeta }_{2}}\lambda )-2\phi (\lambda )g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}). \end{equation}$

(3.1)

Therefore, using Eq (3.1), the soliton Eq (1.9) with respect to a semi-symmetric non-metric connection can be written as

$\begin{array}{c} \frac{1}{2}\left( g(\overset{\circ }{\widetilde{\nabla }}_{\mathcal{\zeta } _{1}}\lambda , \mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{1}, \overset{\circ }{ \widetilde{\nabla }}_{\mathcal{\zeta }_{2}}\lambda )\right) \\ +Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = (\alpha ^{\ast }+\beta \tau+\phi (\lambda ))g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})+\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}). \end{array}$

(3.2)

Let us suppose that $U$ is a parallel unit vector field with respect to the Levi-Civita connection $\nabla.$ Using (1.5), we get

$\overset{\circ }{\widetilde{\nabla }}_{\mathcal{\zeta }_{1}}U = \mathcal{\zeta }_{1}.$

Thus we have:

Proposition 1. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection. If $U$ is a parallel unit vector field with respect to the Levi-Civita connection $\nabla,$ then $U$ is a torse-forming potential vector field with respect to a semi-symmetric non-metric connection of the form $\overset{\circ }{\widetilde{\nabla }}_{\mathcal{\zeta }_{1}}U = \mathcal{\zeta }_{1}$ .

Now let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric non-metric connection on $M$ . Then $\overset{ \circ }{\widetilde{\nabla }}_{\mathcal{\zeta }_{1}}\lambda = c\mathcal{\zeta }_{1}+\varpi (\mathcal{\zeta }_{1})\lambda.$ So by (3.2), we can write

$\begin{array}{c} Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) \\ -\frac{1}{2}\left\{ g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta }_{1})+g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta } _{2})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}). \end{array}$

(3.3)

Thus, the following theorem can be expressed:

Theorem 4. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection and $\lambda$ a torse-forming potential vector field with respect a semi-symmetric non-metric connection on $M$ . Then $(M, g)$ is an $\eta$ - Ricci-Bourguignon soliton if and only if the Ricci tensor field of the Levi-Civita connection is of the form (3.3).

If $\lambda$ is a concircular potential vector field with respect a semi-symmetric non-metric connection, then the following corollary can be stated:

Corollary 4. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection and $\lambda$ a concircular potential vector field with respect to a semi-symmetric non-metric connection on $M$ . If $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton, then $M$ is a generalized quasi-Einstein manifold in the sense of De and Ghosh with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\phi (\lambda)\right)$ and $\gamma$ .

Now let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric non-metric connection on $M$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton and $\eta$ is the $g$ dual of $\lambda$ if and only if

$\begin{array}{c} Ric(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})\\ -\frac{1}{2}\left\{ \eta (\mathcal{\zeta }_{2})\varpi (\mathcal{\zeta } _{1})+\eta (\mathcal{\zeta }_{1})\varpi (\mathcal{\zeta }_{2})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2}). \end{array}$

(3.4)

Thus, the following theorem can be stated:

Theorem 5. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection and $\lambda$ a torse-forming potential vector field with respect to a semi-symmetric non-metric connection on $M$ . Suppose that the $1$ -form $\eta$ is the $g$ dual of $\lambda$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if $M$ is a hyper-generalized quasi-Einstein manifold with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\phi (\lambda)\right), \gamma, -\frac{1}{2}$ and $0$ .

Using (1.8), the Eq (3.3) can be written as

$\begin{array}{c} \overset{\circ }{\widetilde{Ric}}(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) = \left( \alpha ^{\ast }+\beta \tau-c+\phi (\lambda )\right) g(\mathcal{ \zeta }_{1}, \mathcal{\zeta }_{2}) \\ -\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta } _{1})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta } _{2})-(n-1)\sigma (\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2}). \end{array}$

(3.5)

Now assume that $U$ is a parallel unit vector field with respect to the Levi-Civita connection, i.e., $\nabla U = 0$ and $\Vert U\Vert = 1$ . Then

$(\nabla _{\mathcal{\zeta }_{1}}\phi )\mathcal{\zeta }_{2} = \nabla _{\mathcal{ \zeta }_{1}}\phi (\mathcal{\zeta }_{2})-\phi (\nabla _{\mathcal{\zeta }_{1}} \mathcal{\zeta }_{2}) = 0.$

So from (1.7), $\sigma (\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) = -\phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2}).$ Thus by (3.5), we have

$\overset{\circ }{\widetilde{Ric}}(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) = (\alpha ^{\ast }+\beta \tau-c+\phi (\lambda ))g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})$

$-\frac{1}{2}\left\{ g(\mathcal{\zeta }_{1}, \lambda )\varpi (\mathcal{\zeta }_{2})+g(\mathcal{\zeta }_{2}, \lambda )\varpi (\mathcal{\zeta } _{1})\right\} +\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta } _{2})+\left( n-1\right) \phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta } _{2}).$

If $\lambda$ is a concircular potential vector field, then

$\overset{\circ }{\widetilde{Ric}}(\mathcal{\zeta }_{1}, \mathcal{\zeta } _{2}) = (\alpha ^{\ast }+\beta \tau-c+\phi (\lambda ))g(\mathcal{\zeta }_{1}, \mathcal{\zeta }_{2})$

$+\gamma \eta (\mathcal{\zeta }_{1})\eta (\mathcal{\zeta }_{2})+\left( n-1\right) \phi (\mathcal{\zeta }_{1})\phi (\mathcal{\zeta }_{2}).$

Hence we get:

Theorem 6. Let $(M, g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection, $U$ a parallel unit vector field with respect to the Levi-Civita connection $\nabla$ and $\lambda$ a concircular potential vector field with respect a semi-symmetric non-metric connection on $M$ . Then $(M, g)$ is an $\eta$ -Ricci-Bourguignon soliton if and only if $M$ is a generalized quasi-Einstein manifold in the sense of De and Ghosh with respect to a semi-symmetric non-metric connection with associated functions $\left(\alpha ^{\ast }+\beta \tau-c+\phi (\lambda)\right),$ $\gamma$ and $\left(n-1\right).$

Conflict of interest

The author declares no conflict of interest.

References

[1]	N. Agashe, M. Chafle, A semi-symmetric nonmetric connection on a Riemannian manifold, Indian J. Pure Appl. Math., 23 (1992), 399–409.
[2]	A. Besse, Einstein manifolds, Berlin: Springer-Verlag, 2008. http://dx.doi.org/10.1007/978-3-540-74311-8
[3]	A. Blaga, Solutions of some types of soliton equations in $\mathbb{R}^{3}$ , Filomat, 33 (2019), 1159–1162. http://dx.doi.org/10.2298/FIL1904159B doi: 10.2298/FIL1904159B
[4]	A. Blaga, C. Özgür, Almost $\eta$ -Ricci and almost $\eta$ -Yamabe solitons with a torse-forming potential vector field, Quaest. Math., 45 (2022), 143–163. http://dx.doi.org/10.2989/16073606.2020.1850538 doi: 10.2989/16073606.2020.1850538
[5]	A. Blaga, C. Özgür, Remarks on submanifolds as almost $\eta$ -Ricci-Bourguignon solitons, Facta Univ. Ser. Math. Inform., 37 (2022), 397–407. http://dx.doi.org/10.22190/FUMI220318027B doi: 10.22190/FUMI220318027B
[6]	M. Chaki, R. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen, 57 (2000) 297–306.
[7]	M. Chaki, On generalized quasi Einstein manifolds, Publ. Math. Debrecen, 58 (2001), 683–691.
[8]	B. Chen, A survey on Ricci solitons on Riemannian submanifolds, Contemporary Mathematics, 674 (2016), 27–39. http://dx.doi.org/10.1090/conm/674/13552 doi: 10.1090/conm/674/13552
[9]	U. De, G. Ghosh, On generalized quasi Einstein manifolds, Kyungpook Math. J., 44 (2004), 607–615.
[10]	A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc, 45 (1939) 443–473.
[11]	A. Friedmann, J. Schouten, Über die Geometrie der halbsymmetrischen Übertragungen, Math. Z, 21 (1924), 211–223. http://dx.doi.org/10.1007/BF01187468 doi: 10.1007/BF01187468
[12]	R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255–306. http://dx.doi.org/10.4310/jdg/1214436922 doi: 10.4310/jdg/1214436922
[13]	J. Inoguchi, Minimal surfaces in 3-dimensional solvable Lie groups, Chinese Ann. Math. B, 24 (2003), 73–84. http://dx.doi.org/10.1142/S0252959903000086 doi: 10.1142/S0252959903000086
[14]	A. Mihai, I. Mihai, Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications, J. Geom. Phys., 73 (2013), 200–208. http://dx.doi.org/10.1016/j.geomphys.2013.06.002 doi: 10.1016/j.geomphys.2013.06.002
[15]	C. Özgür, On Ricci solitons with a semi-symmetric metric connection, Filomat, 35 (2021), 3635–3641. http://dx.doi.org/10.2298/FIL2111635O doi: 10.2298/FIL2111635O
[16]	A. Shaikh, C. Özgür, A. Patra, On hyper-generalized quasi-Einstein manifolds, International Journal of Mathematical Sciences and Engineering Applications, 5 (2011), 189–206.
[17]	J. Schouten, Ricci-calculus: an introduction to tensor analysis and its geometrical applications, Berlin: Springer-Verlag, 1954. http://dx.doi.org/10.1007/978-3-662-12927-2
[18]	M. Siddiqi, M. Akyol, $\eta$ -Ricci-Yamabe soliton on Riemannian submersions from Riemannian manifolds, arXiv: 2004.14124.
[19]	K. Yano, On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad., 20 (1944), 340–345. http://dx.doi.org/10.3792/pia/1195572958 doi: 10.3792/pia/1195572958
[20]	K. Yano, On semi-symmetric metric connection, Rev. Roum. Math. Pures et Appl., 15 (1970), 1579–1586.

This article has been cited by:

1.	Mancho Manev, Ricci–Bourguignon Almost Solitons with Special Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds, 2024, 12, 2227-7390, 2100, 10.3390/math12132100
2.	Tarak Mandal, Uday Chand De, Avijit Sarkar, η $\eta$ ‐Ricci‐Bourguignon solitons on three‐dimensional (almost) coKähler manifolds, 2024, 0170-4214, 10.1002/mma.10505
3.	Tarak Mandal, Uday Chand De, Meraj Ali Khan, Mohammad Nazrul Islam Khan, Dengfeng Lü, A Study on Contact Metric Manifolds Admitting a Type of Solitons, 2024, 2024, 2314-4629, 10.1155/2024/8516906

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1445) PDF downloads(90) Cited by(3)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

$\eta$ -Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection

Related Papers:

Abstract

1. Introduction

2. $\eta$ -Ricci-Bourguignon solitons on Riemannian manifolds equipped with a semi-symmetric metric connection

3. $\eta$ -Ricci-Bourguignon on Riemannian manifolds equipped with a semi-symmetric non-metric connection

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

η \eta -Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection

Related Papers:

Abstract

1. Introduction

2. η \eta -Ricci-Bourguignon solitons on Riemannian manifolds equipped with a semi-symmetric metric connection

3. η \eta -Ricci-Bourguignon on Riemannian manifolds equipped with a semi-symmetric non-metric connection

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

$\eta$ -Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection

2. $\eta$ -Ricci-Bourguignon solitons on Riemannian manifolds equipped with a semi-symmetric metric connection

3. $\eta$ -Ricci-Bourguignon on Riemannian manifolds equipped with a semi-symmetric non-metric connection