Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds

Abstract

The aim of this work is to study on quasi conformally flat quasi Einstein-Weyl manifolds. In this book chapter, firstly, an interesting relationship between complementary vector field and generator of the quasi Einstein-Weyl manifold is obtained and supported by an example. Then, it is investigated that quasi conformally flat quasi Einstein-Weyl manifolds are of quasi constant curvature, recurrent and semi-symmetric under which conditions after obtaining the expression of the curvature tensor of the quasi conformally flat quasi Einstein-Weyl manifold. Furthermore, some equivalences are obtained between to be of quasi constant curvature and to be semi-symmetric in quasi conformally flat quasi Einstein-Weyl manifolds.

1. Introduction

In 1918, H. Weyl generalized Riemannian geometry as a new way to formulate the unified field theory in physics and defined Weyl manifolds with conformal metric and symmetric connection [1]. After this study, Weyl manifolds attracted the attention of many mathematicians. In 1943, E. Cartan defined Einstein-Weyl manifolds and studied three-dimensional Einstein-Weyl spaces [2]. In 1985, P.E. Jones and K.P. Tod have studied Einstein-Weyl spaces, and then they have done many studies on this subject [3]. Although Weyl’s theory did not attract much attention in physics, it attracted the attention of mathematicians and studies have been carried out on this subject until today.

An n-dimensional Weyl manifold M is defined as a manifold with a torsion-free connection Γ and a conformal metric tensor $g_{\mathit{ij}}$, if the compatible condition is in the form of

$${\nabla }_k{g}_{\mathit{ij}}-2g_{\mathit{ij}}{\mathrm{\Phi }}_k=0$$E1

which is equivalent to

$${\nabla }_k{g}^{\mathit{ij}}+2g^{\mathit{ij}}{\mathrm{\Phi }}_k=0,$$E2

where ${\mathrm{\Phi }}_k$ is a complementary covariant vector field [4]. Such a Weyl manifold is denoted by $\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$ and (1) tells us that a Riemannian manifold is obtained if ${\mathrm{\Phi }}_k=0$ or ${\mathrm{\Phi }}_k$ is gradient.

${\mathrm{\Phi }}_k$ changes by

$${\stackrel{\sim }{\mathrm{\Phi }}}_k={\mathrm{\Phi }}_k+{\partial }_k\left(log\lambda \right)$$E3

under the transformation of the metric tensor $g_{\mathit{ij}}$ in the form of

$${\stackrel{\sim }{g}}_{\mathit{ij}}={\lambda }^2{g}_{\mathit{ij}}$$E4

where λ is a point function [4]. With reference to this transformation, the quantity A is called a satellite of $g_{\mathit{ij}}$ with the weight of {p} if it changes by [5]

$$\stackrel{\sim }{A}={\lambda }^{p}A$$E5

and the quantity ${\stackrel{·}{\nabla }}_{k}A$ is called prolonged covariant derivative of the satellite A of $g_{\mathit{ij}}$ with the weight of {p} if it is defined by [5]

$${\stackrel{·}{\nabla }}_{k}A={\nabla }_{k}A-p{\mathrm{\Phi }}_{k}A.$$E6

From (1), (4) and (6), we have

$${\stackrel{·}{\nabla }}_k{g}_{\mathit{ij}}=0$$E7

which $g_{\mathit{ij}}$ is with the weight of {2}.

The coefficients ${\mathrm{\Gamma }}_{\mathit{jk}}^i$’s of a torsion-free connection Γ on the Weyl manifold $\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$ are given by

$${\mathrm{\Gamma }}_{\mathit{jk}}^i=\left\{\begin{array}{c}i\\ \mathit{jk}\end{array}\right\}-\left({\delta }_j^i{\mathrm{\Phi }}_k+{\delta }_k^i{\mathrm{\Phi }}_j-g_{\mathit{jk}}{g}^{\mathit{ih}}{\mathrm{\Phi }}_h\right)$$E8

where $\left\{\begin{array}{c}i\\ \mathit{jk}\end{array}\right\}$’s are the Christoffel symbols of second kind [4].

The curvature tensor $R_{\mathit{ijk}}^h$ of the symmetric connection Γ on the Weyl manifold is defined by

$$R_{\mathit{ijk}}^h={\partial }_j{\mathrm{\Gamma }}_{\mathit{ik}}^h-{\partial }_k{\mathrm{\Gamma }}_{\mathit{ij}}^h+{\mathrm{\Gamma }}_{\mathit{rj}}^h{\mathrm{\Gamma }}_{\mathit{ik}}^r-{\mathrm{\Gamma }}_{\mathit{rk}}^h{\mathrm{\Gamma }}_{\mathit{ij}}^r.$$E9

The Ricci tensor $R_{\mathit{ij}}$, which is defined by $R_{\mathit{ij}}=R_{\mathit{ijh}}^h$, satisfies

$$R_{\left[\mathit{ij}\right]}=n{\nabla }_i{\mathrm{\Phi }}_{j]}=\frac{1}{2}{R}_{\mathit{rji}}^r.$$E10

With the help of (9), the conformal curvature tensor $C_{\mathit{ijk}}^h$ and the concircular curvature tensor ${\stackrel{\sim }{C}}_{\mathit{ijk}}^h$ of a torsion-free connection Γ on the Weyl manifold are expressed by

$$C_{\mathit{ijk}}^h=\begin{array}{l}{R}_{\mathit{ijk}}^h-\frac{1}{n}{\delta }_i^h{R}_{\mathit{rjk}}^r+\frac{1}{n-2}\left({\delta }_j^h{R}_{\mathit{ik}}-{\delta }_k^h{R}_{\mathit{ij}}+g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\mathit{mj}}-g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\mathit{mk}}\right)\\ -\frac{1}{n\left(n-2\right)}\left({\delta }_j^h{R}_{\mathit{rki}}^r-{\delta }_k^h{R}_{\mathit{rji}}^r+g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\mathit{rjm}}^r-g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\mathit{rkm}}^r\right)\\ +\frac{R}{\left(n-1\right)\left(n-2\right)}\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right),\end{array}$$E11

$${\stackrel{\sim }{C}}_{\mathit{ijk}}^h=R_{\mathit{ijk}}^h-\frac{R}{n\left(n-1\right)}\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right),$$E12

where $R_{\mathit{ijk}}^h$, $R_{\mathit{ij}}$ and R denote the curvature tensor, the Ricci tensor and the scalar curvature of Γ, respectively [6, 7].

In 1968, Yano and Sawaki defined and studied a new curvature tensor called quasi conformal curvature tensor on a Riemannian manifold [8]. Similarly, the notion of quasi conformal curvature tensor $W_{\mathit{ijk}}^h$ of type $\left(1,3\right)$ on a Weyl manifold of dimension n ($n>3$) is introduced by [9]

$$W_{\mathit{ijk}}^h=-\left(n-2\right){\mathit{bC}}_{\mathit{ijk}}^h+\left[a+\left(n-2\right)b\right]{\stackrel{\sim }{C}}_{\mathit{ijk}}^h,$$E13

where a, b are arbitrary constants not simultaneously zero, $C_{\mathit{ijk}}^h$ and ${\stackrel{\sim }{C}}_{\mathit{ijk}}^h$ are conformal curvature tensor and concircular curvature tensor of type $\left(1,3\right)$, respectively.

By substituting (11) and (12) in (13) the quasi conformal curvature tensor can be expressed by

$$W_{\mathit{ijk}}^h=\begin{array}{l}{\mathit{aR}}_{\mathit{ijk}}^h+b\left\{{\delta }_k^h{R}_{\mathit{ij}}-{\delta }_j^h{R}_{\mathit{ik}}+g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\mathit{mk}}-g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\mathit{mj}}\right\}\\ +\frac{b}{n}\{\left(n-2\right){\delta }_i^h{R}_{\mathit{rjk}}^r+{\delta }_j^h{R}_{\mathit{rki}}^r-{\delta }_k^h{R}_{\mathit{rji}}^r+g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\mathit{rjm}}^r-g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\mathit{rkm}}^r\}\\ -\frac{R}{n}\left\{\frac{a}{n-1}+2b\right\}\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right).\end{array}$$E14

2. The concept of quasi conformally flatness on quasi Einstein-Weyl manifolds

Quasi Einstein manifolds occupy a large place in the mathematical literature. For instance, research on quasi-Einstein manifolds helps us to understand the global character of topological spaces. Beside mathematics, studies on quasi-Einstein manifolds gain meaning with applications to general relativity.

The concept of quasi Einstein manifold was firstly introduced by M. C. Chaki and R. K. Maity as follows [10]:

A non-flat Riemannian manifold $\left(M_n,g_{\mathit{ij}}\right)$$\left(n>2\right)$ is defined to be a quasi Einstein manifold if its Ricci tensor $R_{\mathit{ij}}$ of type $\left(0,2\right)$ is not identically zero and satisfies the condition

$$R_{\mathit{ij}}=\alpha g_{\mathit{ij}}+\beta A_i{A}_j,$$E15

where α, β are scalars of which $\beta \ne 0$ and $A_i$ is a non-zero unit covariant vector field. In such an n-dimensional manifold which is denoted by ${\left(\mathit{QE}\right)}_n$; α, β are called associated scalars and $A_i$ is called the generator of the manifold.

After Chaki and Maity, quasi Einstein manifolds are studied by many other authors. Moreover, in the articles [11, 12, 13] that inspired this study, conformal flatness and quasi conformal flatness were examined on quasi Einstein manifolds.

In this study, the concept of quasi conformal flatness on quasi Einstein manifolds were adapted to quasi Einstein-Weyl manifolds which was introduced by İ. Gül and E.Ö. Canfes as follows [14]:

Definition 1.A non-flat Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$of dimension$n\left(n>2\right)$is said to be a quasi Einstein-Weyl manifold if the symmetric part of its Ricci tensor$R_{\mathit{ij}}$of type (0,2) is not identically zero and satisfies the condition

$$R_{\left(\mathit{ij}\right)}=\alpha g_{\mathit{ij}}+\beta A_i{A}_j$$E16

where α and β are scalars of weight {−2} with$\beta \ne 0$. The scalars α, β are called “associated scalars” and the unit covariant vector$A_i$of weight {1} is called “generator of the manifold”. Such a manifold is denoted${\left(\mathit{QEW}\right)}_n$.

Therefore the aim of the present book chapter is to examine quasi conformally flat quasi Einstein-Weyl manifolds. It is organized as follows: In Section 1, the general information about Weyl manifolds are given. In Section 2, a theorem which shows the relationship between complementary vector field ${\mathrm{\Phi }}_k$ and generator $A_k$ of quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$ is proved and the expression of the curvature tensor of the quasi conformally flat quasi Einstein-Weyl manifold is obtained. In Section 3, three basic concepts are defined on quasi conformally flat quasi Einstein-Weyl manifolds and the necessary and sufficient conditions for these concepts are emphasized.

By means of (10) and (16), Ricci tensor $R_{\mathit{ij}}$ of ${\left(\mathit{QEW}\right)}_n$ is expressed by

$$R_{\mathit{ij}}=\alpha g_{\mathit{ij}}+\beta A_i{A}_j+n{\nabla }_i{\mathrm{\Phi }}_{j]}$$E17

which implies

$$R=\mathit{\alpha n}+\beta .$$E18

From (17), we have

$$\begin{array}{ccc}{R}_{\mathit{ij},l}-{\mu }_l{R}_{\mathit{ij}}& =& g_{\mathit{ij}}\left\{{\alpha }_{,l}-\alpha \left({\mu }_l-2{\mathrm{\Phi }}_l\right)\right\}+\left({\beta }_{,l}-{\mathit{\beta \mu }}_l\right)A_i{A}_j+\beta \left(A_{i,l}{A}_j+A_i{A}_{j,l}\right)+\\ & & \frac{n}{2}\left[\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)\right].\hfill \end{array}$$E19

Since

$$R_{,l}={\left(g^{\mathit{ij}}{R}_{\mathit{ij}}\right)}_{,l}=-2{\mathrm{\Phi }}_l{g}^{\mathit{ij}}{R}_{\mathit{ij}}+g^{\mathit{ij}}{R}_{{\mathit{ij}}_{,l}}$$E20

and $A_i$ is normalized by the condition

$$g^{\mathit{ij}}{A}_i{A}_j=1\iff A^j{A}_j=1$$E21

it is found that by multiplying (19) by $g^{\mathit{ij}}$

$$R_{,l}-\left({\mu }_l-2{\mathrm{\Phi }}_l\right)R=n\left\{{\alpha }_{,l}-\alpha \left({\mu }_l-2{\mathrm{\Phi }}_l\right)\right\}+\left({\beta }_{,l}-{\mathit{\beta \mu }}_l\right)+\beta \left(A_{i,l}{A}^i+A^j{A}_{j,l}\right).$$E22

By means of (18),

$$R_{,l}-{\mu }_{l}R=n\left({\alpha }_{,l}-{\mu }_l\alpha \right)+\left({\beta }_{,l}-{\mu }_l\beta \right).$$E23

We obtain that

$${\mathrm{\Phi }}_l\left(R-\mathit{\alpha n}\right)=\beta A_{i,l}{A}^i$$E24

where $\beta =R-\mathit{\alpha n}$.

Hence we have the following:

Theorem 1.The complementary vector field${\mathrm{\Phi }}_i$and the generator$A_i$of the quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$are related by

$$A_{i,l}{A}^i={\mathrm{\Phi }}_l.$$E25

Although the first part of the following example was given to prove the existence of the quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>2\right)$ in [14], the verification of Theorem 1 is made by the author of the present book chapter in the second part of the example.

Example 1. A three dimensional Weyl manifold$M_3$is equipped with a metric$g_{\mathit{ij}}$by

$${\mathit{ds}}^2=g_{\mathit{ij}}{\mathit{dx}}^i{\mathit{dx}}^j=e^{x^1}\left[{\left({\mathit{dx}}^1\right)}^2+{\left({\mathit{dx}}^2\right)}^2\right]+{\left({\mathit{dx}}^3\right)}^2$$

and a 1-form Φ whose components${\mathrm{\Phi }}_k$given by$\mathrm{\Phi }=e^{x^1}{\mathit{dx}}^2+{\mathit{dx}}^3$. The nonzero coefficients${\mathrm{\Gamma }}_{\mathit{jk}}^i$of a torsion-free connection Γ are [14]

$${\mathrm{\Gamma }}_{11}^1=\frac{1}{2},{\mathrm{\Gamma }}_{12}^1={\mathrm{\Gamma }}_{21}^1=-e^{x^1},{\mathrm{\Gamma }}_{13}^1={\mathrm{\Gamma }}_{31}^1=-1,{\mathrm{\Gamma }}_{22}^1=-\frac{1}{2},$$

$${\mathrm{\Gamma }}_{11}^2=e^{x^1},{\mathrm{\Gamma }}_{12}^2={\mathrm{\Gamma }}_{21}^2=\frac{1}{2},{\mathrm{\Gamma }}_{22}^2=-e^{x^1},{\mathrm{\Gamma }}_{23}^2={\mathrm{\Gamma }}_{32}^2=-1,{\mathrm{\Gamma }}_{33}^2=1,$$

$${\mathrm{\Gamma }}_{11}^3={\mathrm{\Gamma }}_{22}^3=e^{x^1},{\mathrm{\Gamma }}_{23}^3={\mathrm{\Gamma }}_{32}^3=-e^{x^1},{\mathrm{\Gamma }}_{33}^3=-1.$$

It is clear that$\left(M_3,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is a Weyl manifold with the connection Γ satisfying thecondition (1). An elementary calculation gives the following nonzero components of the Ricci tensor [14]:

$$R_{11}=e^{x^1}\left(1+e^{x^1}\right),R_{12}=-R_{21}=\frac{3}{2}{e}^{x^1},R_{22}=e^{x^1},$$

$$R_{23}=R_{32}=-e^{x^1},R_{33}=e^{x^1}.$$

Moreover, the components of the symmetric parts of the Ricci tensor$R_{\mathit{ij}}$and the scalar curvature R are [14]

$$R_{\left(11\right)}=e^{x^1}\left(1+e^{x^1}\right),R_{\left(22\right)}=e^{x^1},R_{\left(23\right)}=-e^{x^1},R_{\left(33\right)}=e^{x^1},R=2\left(1+e^{x^1}\right).$$

Therefore, by considering (15), we find that [14]

$$\alpha =1+e^{x^1},\beta =-\left(1+e^{x^1}\right),$$

$$A_1=0,A_2=\frac{e^{x^1}}{\sqrt{1+e^{x^1}}},A_3=\frac{1}{\sqrt{1+e^{x^1}}}$$

where$g^{\mathit{ij}}{A}_i{A}_j=1$. Thus,$\left(M_3,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is a quasi-Einstein Weyl manifold. Now covariant derivatives of$A_1,$$A_2$and$A_3$with respect to$x^k\left(k=\mathrm{1,2,3}\right)$are as follows:

$$A_{1,1}=0,A_{1,2}=0,A_{1,3}=0;$$

$$A_{2,1}=\frac{e^{x^1}}{2{\left(1+e^{x^1}\right)}^{\frac{3}{2}}},A_{2,2}=\frac{e^{2x^1}-e^{x^1}}{\sqrt{1+e^{x^1}}},A_{2,3}=0;$$E26

$$A_{3,1}=\frac{-e^{x^1}}{2{\left(1+e^{x^1}\right)}^{\frac{3}{2}}},A_{3,2}=\frac{2e^{x^1}}{\sqrt{1+e^{x^1}}},A_{3,3}=\frac{1-e^{x^1}}{\sqrt{1+e^{x^1}}}$$

On the other hand, the reciprocals of$A_i$‘s are

$$A^1=0,A^2=A^3=\frac{1}{\sqrt{1+e^{x^1}}}$$E27

By substituting(26)and(27)in(25),

$$A_{1,1}{A}^1+A_{2,1}{A}^2+A_{3,1}{A}^3={\mathrm{\Phi }}_1,$$

$$A_{1,2}{A}^1+A_{2,2}{A}^2+A_{3,2}{A}^3={\mathrm{\Phi }}_2,$$

$$A_{1,3}{A}^1+A_{2,3}{A}^2+A_{3,3}{A}^3={\mathrm{\Phi }}_3$$

are obtained.

A Weyl manifold $\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$$\left(n>3\right)$ is called quasi conformally flat, if the quasi conformal curvature tensor $W_{\mathit{ijk}}^h$ satisfy the condition

$$W_{\mathit{ijk}}^h=0.$$E28

Now, let us suppose that ${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$ is quasi conformally flat with $a\ne 0$ and $b\ne 0$. Then from (14),

$$R_{\mathit{ijk}}^h=\begin{array}{l}\frac{-b}{a}\left\{{\delta }_k^h{R}_{\mathit{ij}}-{\delta }_j^h{R}_{\mathit{ik}}+g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\mathit{mk}}-g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\mathit{mj}}\right\}\\ +\frac{-2b}{\mathit{\text{an}}}\{\left(n-2\right){\delta }_i^h{R}_{\left[\mathit{kj}\right]}+{\delta }_j^h{R}_{\left[\mathit{ik}\right]}-{\delta }_k^h{R}_{\left[\mathit{ij}\right]}+g_{\mathit{ik}}{g}^{\mathit{hm}}{R}_{\left[\mathit{mj}\right]}-g_{\mathit{ij}}{g}^{\mathit{hm}}{R}_{\left[\mathit{mk}\right]}\}\\ +\frac{R}{\mathit{\text{an}}}\left\{\frac{a}{n-1}+2b\right\}\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right)\end{array}$$E29

On the other hand, since it is assumed that the manifold is ${\left(\mathit{QEW}\right)}_n$, its Ricci tensor $R_{\mathit{ij}}$ can be written as (17) which satisfies (18).

Substituting (10), (17) and (18) in (29), the curvature tensor $R_{\mathit{ijk}}^h$ is obtained as

$$R_{\mathit{ijk}}^h=\begin{array}{l}P\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right)+Q\left\{{\delta }_k^h{A}_i{A}_j-{\delta }_j^h{A}_i{A}_k+g_{\mathit{ij}}{g}^{\mathit{hm}}{A}_m{A}_k-g_{\mathit{ik}}{g}^{\mathit{hm}}{A}_m{A}_j\right\}\\ -\frac{\left(n-2\right)b}{a}\left\{2{\delta }_i^h{\nabla }_k{\mathrm{\Phi }}_{j]}+{\delta }_k^h{\nabla }_i{\mathrm{\Phi }}_{j]}-{\delta }_j^h{\nabla }_i{\mathrm{\Phi }}_{k]}+g_{\mathit{ij}}{g}^{\mathit{hm}}{\nabla }_m{\mathrm{\Phi }}_{k]}-g_{\mathit{ik}}{g}^{\mathit{hm}}{\nabla }_m{\mathrm{\Phi }}_{j]}\right\}\end{array}$$E30

where $P=\left\{\frac{\mathit{\alpha n}+\beta }{n\left(n-1\right)}+\frac{2\mathit{b\beta }}{\mathit{\text{an}}}\right\}$ and $Q=\frac{-b}{a}\beta$ are scalars.

Ricci tensor $R_{\mathit{ij}}$ is obtained as

$$R_{\mathit{ij}}=\left\{P\left(n-1\right)+Q\right\}{g}_{\mathit{ij}}+Q\left(n-2\right)A_i{A}_j-\frac{n\left(n-2\right)b}{a}{\nabla }_i{\mathrm{\Phi }}_{j]}$$E31

by contracting on the indices h and k in (30) and the scalar curvature is found in the form of

$$R=\left\{P\left(n-1\right)+Q\right\}n+Q\left(n-2\right)$$E32

by transvecting (31) by $g^{\mathit{ij}}.$

Using (10), (30), (31) and (32) in (11), it is obtained that

$$C_{\mathit{ijk}}^h=0$$E33

leading us to following:

Corollary 1.Quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$is conformally flat.

3. Some necessary and sufficient conditions on quasi conformally flat quasi Einstein-Weyl manifolds

The concept of a space of quasi constant curvature was firstly introduced by Chen and Yano [15]. Similarly, we can define a Weyl manifold of quasi constant curvature as follows:

Definition 2.A Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$$\left(n>3\right)$is said to be of quasi constant curvature if it is conformally flat and its curvature tensor$R_{\mathit{ijk}}^h$of type (1,3) is in the form of

$$R_{\mathit{ijk}}^h=U\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right)+V\left\{{\delta }_k^h{A}_i{A}_j-{\delta }_j^h{A}_i{A}_k+g_{\mathit{ij}}{g}^{\mathit{hm}}{A}_m{A}_k-g_{\mathit{ik}}{g}^{\mathit{hm}}{A}_m{A}_j\right\},$$E34

where U and V are scalars with V$\ne 0$and$A_i$is a covariant vector.

On the other hand, Amur and Maralabhavi [16] proved that a quasi conformally flat Riemannian manifold is either conformally flat or Einstein. So, a quasi conformally flat quasi Einstein manifold, which is not Einstein, is conformally flat and its curvature tensor satisfies the condition in (32) with $a\ne 0$ and $b\ne 0$. Therefore, a quasi conformally flat quasi Einstein manifold with $a\ne 0$ and $b\ne 0$ is of quasi constant curvature.

However, the situation is more complicated for quasi conformally flat ${\left(\mathit{QEW}\right)}_n$. Because although quasi conformally flat ${\left(\mathit{QEW}\right)}_n$ is conformally flat, it does not meet the requirement in (34) automatically. Therefore, a quasi conformally flat ${\left(\mathit{QEW}\right)}_n$ will be of quasi constant curvature under special conditions.

Suppose that quasi conformally flat ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$ be of quasi constant curvature with the same definition in (2). Since $a\ne 0$ and $b\ne 0$, from (29),

$$2{\delta }_i^h{\nabla }_k{\mathrm{\Phi }}_{j]}+{\delta }_k^h{\nabla }_i{\mathrm{\Phi }}_{j]}-{\delta }_j^h{\nabla }_i{\mathrm{\Phi }}_{k]}+g_{\mathit{ij}}{g}^{\mathit{hm}}{\nabla }_m{\mathrm{\Phi }}_{k]}-g_{\mathit{ik}}{g}^{\mathit{hm}}{\nabla }_m{\mathrm{\Phi }}_{j]}=0$$E35

is obtained. By transvecting (35) by $g^{\mathit{ij}}$,

$$\left(n-4\right)g^{\mathit{hi}}{\nabla }_i{\mathrm{\Phi }}_{k]}=0$$E36

and transvecting one more time by $g_{\mathit{hj}}$ with the assumption of $n>4$, it is found that

$${\nabla }_j{\mathrm{\Phi }}_{k]}=0$$E37

which means that the covariant derivative ${\mathrm{\Phi }}_{k,j}$ is symmetric.

Conversely, let the covariant derivative ${\mathrm{\Phi }}_{k,j}$ be symmetric in a quasi conformally flat ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right).$ If (37) is substituted in (30), then (34) is obtained. Hence we get the following:

Theorem 2.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$to be of quasi constant curvature is that the covariant derivative${\mathrm{\Phi }}_{k,j}$is symmetric.

Now, let us consider in which cases the covariant derivative ${\mathrm{\Phi }}_{k,j}$ is symmetric in a quasi conformally flat ${\left(\mathit{QEW}\right)}_n$, remembering that ${\mathrm{\Phi }}_k$ is different from zero or non-gradient. So, let us give the definitions of some special vector fields in the Weyl manifold $\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$:

Definition 3.A vector field ξ in the Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is called torse-forming if it satisfies the condition${\nabla }_X\xi =\mathit{\rho X}+\lambda \left(X\right)\xi$, where$\xi$$\in \chi \left(M\right)$,$\lambda \left(X\right)$is a linear form and ρ is a function. In the local coordinates, it is expressed by${\nabla }_i{\xi }^h={\mathit{\rho \delta }}_i^h+{\xi }^h{\lambda }_i$, where${\delta }_i^h$is the Kronecker symbol,${\xi }^h$and${\lambda }_i$are the components of ξ and λ. A torse-forming vector field ξ is called concircular if${\nabla }_i{\xi }_j=\rho g_{\mathit{ij}}$with${\xi }_j=g_{\mathit{hj}}{\xi }^h$.

Definition 4.A vector field ϕ in the Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is called ϕ (Ric) vector field if it satisfies$\nabla \varphi =\mathit{\text{μRic}}$, where$\mu$is a constant and$\mathit{Ric}$is the Ricci tensor. In local coordinates, it is expressed by${\nabla }_i{\varphi }_j=\mu R_{\mathit{ij}}$, where${\varphi }_i$and$R_{\mathit{ij}}$are the components of ϕ and$\mathit{Ric}$.

Definition 5.The components${\varphi }^i$of a vector field ϕ in the Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is defined as parallel if${\varphi }_{,j}^i=0$and is defined concurrent if${\varphi }_{,j}^i=c{\delta }_j^i$, where$c$is a constant.

When we apply the above definitions to parallel, concurrent and concircular complementary vector field ${\mathrm{\Phi }}_k$ in a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$, the covariant derivatives ${\mathrm{\Phi }}_{k,j}$ of these vector fields are

$${\mathrm{\Phi }}_{k,j}=2{\mathrm{\Phi }}_k{\mathrm{\Phi }}_j,{\mathrm{\Phi }}_{k,j}=2{\mathrm{\Phi }}_k{\mathrm{\Phi }}_j+{\mathit{cg}}_{\mathit{kj}},{\mathrm{\Phi }}_{k,j}=\rho g_{\mathit{kj}},$$E38

respectively.

Now, let us consider ${\mathrm{\Phi }}_k$ as a $\varphi$ (Ric) vector field. From Definition 3.4 and (10),

$$\left(1-\mathit{\mu n}\right){\nabla }_j{\mathrm{\Phi }}_{k]}=0.$$E39

Finally, let us write the covariant derivative ${\mathrm{\Phi }}_{k,j}$ for a torse forming vector field ${\mathrm{\Phi }}_k$ defined by ${\nabla }_i{\mathrm{\Phi }}^h={\mathit{\rho \delta }}_i^h+{\mathrm{\Phi }}^h{A}_i$, where $A_i$ is the generator of ${\left(\mathit{QEW}\right)}_n$. By using Definition 3.3, we have

$${\mathrm{\Phi }}_{k,j}=2{\mathrm{\Phi }}_j{\mathrm{\Phi }}_k+\rho g_{\mathit{kj}}+A_j{\mathrm{\Phi }}_k.$$E40

By means of (38), (39) and (40), we can express the following:

Corollary 2.A quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$is of quasi constant curvature if the complementary vector field${\mathrm{\Phi }}_k$satisfies any one of the following:

1. ${\mathrm{\Phi }}_k$is a parallel, concurrent or concircular vector field,

2. ${\mathrm{\Phi }}_k$is a ϕ (Ric) vector field with$\mu \ne \frac{1}{n}$,

3. ${\mathrm{\Phi }}_k$is a torse forming vector field defined by${\nabla }_i{\mathrm{\Phi }}^h={\mathit{\rho \delta }}_i^h+{\mathrm{\Phi }}^h{A}_i$, where$A_j{\mathrm{\Phi }}_k-A_k{\mathrm{\Phi }}_j=0.$

Now, we seek a necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$ to be recurrent. So, firstly, let us define the concept of recurrency in the quasi Einstein-Weyl manifolds by analogy to A.G.Walker’s definition [17]:

Definition 6.A non-flat Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is called recurrent if there exists a non-zero covariant vector${\mu }_l$such that

$$R_{\mathit{ijk},l}^h-{\mu }_l{R}_{\mathit{ijk}}^h=0.$$E41

Suppose that quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$, whose associated scalars α and β satisfy

$$\frac{{\alpha }_{,l}}{\alpha }=\frac{{\beta }_{,l}}{\beta }={\mu }_l-2{\mathrm{\Phi }}_l,$$E42

is recurrent. From (39), it follows that

$$R_{\mathit{ij},l}-{\mu }_l{R}_{\mathit{ij}}=0$$E43

by contracting on the indices h and k in (41) and transvecting (43) by $g^{\mathit{ij}}$ gives us

$$R_{,l}-\left({\mu }_l-2{\mathrm{\Phi }}_l\right)R=0$$E44

by means of (20).

By substituting (43) in (23), it is obtained that

$$n\left\{{\alpha }_{,l}-\left({\mu }_l-2{\mathrm{\Phi }}_l\right)\alpha \right\}+\left\{{\beta }_{,l}-\left({\mu }_l-2{\mathrm{\Phi }}_l\right)\beta \right\}=0$$E45

which is satisfied by associated scalars in the above hypothesis.

If (42) is substituted in (19) and transvecting by $A^i$

$$0=\beta \left(A_{j,l}-{\mathrm{\Phi }}_l{A}_j\right)+\frac{n}{2}\left[\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)\right]A^i$$E46

is obtained.

The conditions

$$A_{j,l}-{\mathrm{\Phi }}_l{A}_j=0\iff A_{j,l}={\mathrm{\Phi }}_l{A}_j$$E47

and

$$\left[\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)\right]A^i=0$$E48

are satisfied in order to provide (46) since $\beta \ne 0$ and $n>3$ for a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$.

If (47) is satisfied, then

$$A_{,l}^j=-{\mathrm{\Phi }}_l{A}^j,$$E49

where $A^j=g^{\mathit{jh}}{A}_h$.

Let us first compute first and second covariant derivatives of the complementary vector ${\mathrm{\Phi }}_j$ defined by (25) by considering (21), (47) and (48):

$${\mathrm{\Phi }}_{j,i}=A^k{A}_{k,i}-{\mathrm{\Phi }}_j{\mathrm{\Phi }}_i,$$E50

$${\mathrm{\Phi }}_{j,\mathit{il}}=2{\mathrm{\Phi }}_j{\mathrm{\Phi }}_i{\mathrm{\Phi }}_l-A^k\left(A_{k,\mathit{il}}{\mathrm{\Phi }}_j+A_{k,\mathit{jl}}{\mathrm{\Phi }}_i+A_{k,\mathit{ji}}{\mathrm{\Phi }}_l-A_{k,\mathit{jil}}\right).$$E51

By using (50) and (51), the expressions $\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)A^i$ and ${\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)A^i$ in (48) can be written as

$$\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)A^i=A^i{A}^k\left[{\mathrm{\Phi }}_l\left(A_{k,\mathit{ij}}-A_{k,\mathit{ji}}\right)+\left(A_{k,\mathit{jil}}-A_{k,\mathit{ijl}}\right)\right],$$E52

$${\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)A^i=A^i{A}^k\left[{\mu }_l\left(A_{k,\mathit{ji}}-A_{k,\mathit{ij}}\right)\right].$$E53

If (52) and (53) are substituted in (48), then

$$A^i{A}^k\left[\left(A_{k,\mathit{jil}}-A_{k,\mathit{ijl}}\right)-\left({\mu }_l+{\mathrm{\Phi }}_l\right)\left(A_{k,\mathit{ji}}-A_{k,\mathit{ij}}\right)\right]=0$$E54

is obtained. Since $A^i$ and $A^k$’s are linearly independent,

$$A_{k,\mathit{jil}}=\left({\mu }_l+{\mathrm{\Phi }}_l\right)A_{k,\mathit{ji}}.$$E55

Conversely, let (47) and (55) be satisfied in a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$ whose associated scalars α and β satisfying (42).

From (30),

$$\begin{array}{ll}{R}_{\mathit{ijk},l}^h-{\mu }_l{R}_{\mathit{ijk}}^h=& \left\{P_{,l}-\left({\mu }_l-2{\mathrm{\Phi }}_l\right)P\right\}\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right)\\ & +\left(Q_{,l}-{\mu }_{l}Q\right)\left\{{\delta }_k^h{A}_i{A}_j-{\delta }_j^h{A}_i{A}_k+g_{\mathit{ij}}{g}^{\mathit{hm}}{A}_m{A}_k-g_{\mathit{ik}}{g}^{\mathit{hm}}{A}_m{A}_j\right\}\\ & +Q\{{\delta }_k^h\left(A_{i,l}{A}_j+A_i{A}_{j,l}\right)-{\delta }_j^h\left(A_{i,l}{A}_k+A_i{A}_{k,l}\right)\\ & +g_{\mathit{ij}}{g}^{\mathit{hm}}\left(A_{m,l}{A}_k+A_m{A}_{k,l}\right)-g_{\mathit{ik}}{g}^{\mathit{hm}}\left(A_{m,l}{A}_j+A_m{A}_{j,l}\right)\}\\ & -\frac{\left(n-2\right)b}{a}\{{\delta }_i^h\left[\left({\mathrm{\Phi }}_{j,\mathit{kl}}-{\mathrm{\Phi }}_{k,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,k}-{\mathrm{\Phi }}_{k,j}\right)\right]\\ & +\frac{1}{2}{\delta }_k^h\left[\left({\mathrm{\Phi }}_{j,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,i}-{\mathrm{\Phi }}_{i,j}\right)\right]\\ & -\frac{1}{2}{\delta }_j^h\left[\left({\mathrm{\Phi }}_{k,\mathit{il}}-{\mathrm{\Phi }}_{i,\mathit{kl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{k,i}-{\mathrm{\Phi }}_{i,k}\right)\right]\\ & +\frac{1}{2}{g}_{\mathit{ij}}{g}^{\mathit{hm}}\left[\left({\mathrm{\Phi }}_{k,\mathit{ml}}-{\mathrm{\Phi }}_{m,\mathit{kl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{k,m}-{\mathrm{\Phi }}_{m,k}\right)\right]\\ & -\frac{1}{2}{g}_{\mathit{ik}}{g}^{\mathit{hm}}\left[\left({\mathrm{\Phi }}_{j,\mathit{ml}}-{\mathrm{\Phi }}_{m,\mathit{jl}}\right)-{\mu }_l\left({\mathrm{\Phi }}_{j,m}-{\mathrm{\Phi }}_{m,j}\right)\right]\}\end{array}$$E56

If (42), (47) and (55) are written in (56), then (41) is obtained. Hence we can state the following:

Theorem 3.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$whose recurrent scalars α and β having the same recurrency vector${\mu }_l-2{\mathrm{\Phi }}_l$to be recurrent is that the equations$A_{k,j}={\mathrm{\Phi }}_j{A}_k$and$A_{k,\mathit{jil}}=\left({\mu }_l+{\mathrm{\Phi }}_l\right)A_{k,\mathit{ji}}$are satisfied.

Let us dedicate the last part of this section to the concept of semi-symmetricness in a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$. Firstly, let us define semi-symmetric ${\left(\mathit{QEW}\right)}_n$ similar to the definition which is made by Szabo for Riemannian manifolds [18] as follows:

Definition 7.A non-flat Weyl manifold$\left(M,g_{\mathit{ij}},{\mathrm{\Phi }}_k\right)$is called semi-symmetric if its curvature tensor$R_{\mathit{ijk}}^h$of type (1,3) satisfies the condition

$$R_{\mathit{ijk},\mathit{lm}}^h-R_{\mathit{ijk},\mathit{ml}}^h=0.$$E57

It follows that

$$R_{\mathit{ij},\mathit{lm}}-R_{\mathit{ij},\mathit{ml}}=0$$E58

by contracting on the indices h and k in (57).

Let us suppose that a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$ with $\mathit{a\alpha n}\ne \mathit{b\beta }\left(n-2\right)$ is semi-symmetric. From (17) and (58),

$$0=\begin{array}{l}\beta \left[\left(A_{i,\mathit{lm}}-A_{i,\mathit{ml}}\right)A_j+A_i\left(A_{j,\mathit{lm}}-A_{j,\mathit{ml}}\right)\right]+2\alpha g_{\mathit{ij}}\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right)\\ +\frac{n}{2}\left[\left({\mathrm{\Phi }}_{j,\mathit{ilm}}-{\mathrm{\Phi }}_{i,\mathit{jlm}}\right)-\left({\mathrm{\Phi }}_{j,\mathit{iml}}-{\mathrm{\Phi }}_{i,\mathit{jml}}\right)\right].\end{array}$$E59

With the aid of the Ricci identity given as

$$v_{i,\mathit{jk}}-v_{i,\mathit{kj}}=v_h{R}_{\mathit{ijk}}^h,$$E60

where $v_i$‘s are the components of a covariant vector, it is obtained that

$$\begin{array}{ll}0=& \beta \left[\left(A_h{R}_{\mathit{ilm}}^h\right)A_j+A_i\left(A_h{R}_{\mathit{jlm}}^h\right)\right]+2\alpha g_{\mathit{ij}}\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right)\\ & +\frac{n}{2}\left[\left({\mathrm{\Phi }}_{j,\mathit{ilm}}-{\mathrm{\Phi }}_{i,\mathit{jlm}}\right)-\left({\mathrm{\Phi }}_{j,\mathit{iml}}-{\mathrm{\Phi }}_{i,\mathit{jml}}\right)\right].\end{array}$$E61

If (61) is transvected by $g^{\mathit{ij}}$, it is found that

$$0=2\beta \left(A_h{R}_{\mathit{ilm}}^h\right)A^i+2\mathit{\alpha n}\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right).$$E62

Substituting the following equation, resulted from (48),

$$\left(A_h{R}_{\mathit{ilm}}^h\right)A^i=-\frac{\left(n-2\right)b}{a}\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right)$$E63

which is valid in a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$ in (62) gives us

$$\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right)\left(\mathit{\alpha n}-\beta \frac{\left(n-2\right)b}{a}\right)=0.$$E64

Because of the restriction on α and β,

$${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}=0.$$E65

If we form the difference ${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}$ after taking covariant derivative of (25) with repect to $x^m$, we have

$${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}=A^i\left(A_{i,\mathit{lm}}-A_{i,\mathit{ml}}\right)+A_{i,l}{A}_{,m}^i-A_{i,m}{A}_{,l}^i.$$E66

If firstly rearranging the first term on the right hand side of the equation in (66) with the help of (60) and then using (63) and (64) in the resulting equation gives

$$A_{i,l}{A}_{,m}^i-A_{i,m}{A}_{,l}^i=0$$E67

or equivalently

$$A_{i,l}{A}_{j,m}-A_{j,l}{A}_{i,m}=0.$$E68

Conversely, let us assume that the generator $A_i$ of a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$ satisfies the condition (67) or equivalently (68). If (67) is substituted in (66), then (65) is satisfied by means of (60). In this case,

$$A^i\left(A_{i,\mathit{lm}}-A_{i,\mathit{ml}}\right)=A^i\left(A_h{R}_{\mathit{ilm}}^h\right)=0$$E69

which means that

$$A_{i,\mathit{lm}}-A_{i,\mathit{ml}}=0$$E70

since $A^i$‘s are linearly independent.

From (30),

$$\begin{array}{l}{R}_{\mathit{ijk},\mathit{ls}}^h=\left(P_{,\mathit{ls}}+2{\mathrm{\Phi }}_{l,s}P+2{\mathrm{\Phi }}_l{P}_{,s}+2{\mathrm{\Phi }}_s{P}_{,l}+4{\mathrm{\Phi }}_l{\mathrm{\Phi }}_{s}P\right)\left({\delta }_k^h{g}_{\mathit{ij}}-{\delta }_j^h{g}_{\mathit{ik}}\right)\\ +Q_{,\mathit{ls}}\left\{{\delta }_k^h{A}_i{A}_j-{\delta }_j^h{A}_i{A}_k+g_{\mathit{ij}}{g}^{\mathit{hm}}{A}_m{A}_k-g_{\mathit{ik}}{g}^{\mathit{hm}}{A}_m{A}_j\right\}\\ +Q_{,l}\left\{{\delta }_k^h\left(A_{i,s}{A}_j+A_i{A}_{j,s}\right)-{\delta }_j^h\left(A_{i,s}{A}_k+A_i{A}_{k,s}\right)+g_{\mathit{ij}}{g}^{\mathit{hm}}\left(A_{m,s}{A}_k+A_m{A}_{k,s}\right)\right\}\\ -g_{\mathit{ik}}{g}^{\mathit{hm}}\left(A_{m,s}{A}_j+A_m{A}_{j,s}\right)\\ +Q_{,s}\left\{{\delta }_k^h\left(A_{i,l}{A}_j+A_i{A}_{j,l}\right)\right.-{\delta }_j^h\left(A_{i,l}{A}_k+A_i{A}_{k,l}\right)+g_{\mathit{ij}}{g}^{\mathit{hm}}\left(A_{m,l}{A}_k+A_m{A}_{k,l}\right)\\ \left.-g_{\mathit{ik}}{g}^{\mathit{hm}}\left(A_{m,l}{A}_j+A_m{A}_{j,l}\right)\right\}+Q\left\{{\delta }_k^h\left(A_{i,\mathit{ls}}{A}_j+A_{i,l}{A}_{j,s}+A_{i,s}{A}_{j,l}+A_i{A}_{j,\mathit{ls}}\right)\right\}\\ -{\delta }_j^h\left(A_{i,\mathit{ls}}{A}_k+A_{i,l}{A}_{k,s}+A_{i,s}{A}_{k,l}+A_i{A}_{k,\mathit{ls}}\right)\\ +g_{\mathit{ij}}{g}^{\mathit{hm}}\left(A_{m,\mathit{ls}}{A}_k+A_{m,l}{A}_{k,s}+A_{m,s}{A}_{k,l}+A_m{A}_{k,\mathit{ls}}\right)\\ -g_{\mathit{ik}}{g}^{\mathit{hm}}\left.\left(A_{m,\mathit{ls}}{A}_j+A_{m,l}{A}_{j,s}+A_{m,s}{A}_{j,l}+A_m{A}_{j,\mathit{ls}}\right)\right\}\end{array}$$E71

is obtained. If necessary simplifications are made in the difference $R_{\mathit{ijk},\mathit{ls}}^h-R_{\mathit{ijk},\mathit{sl}}^h$ which is formed by means of (66), then it is found that

$$\begin{array}{ll}{R}_{\mathit{ijk},\mathit{ls}}^h-R_{\mathit{ijk},\mathit{sl}}^h& =Q\left\{{\delta }_k^h\left[\left(A_{i,\mathit{ls}}-A_{i,\mathit{sl}}\right)A_j+A_i\left(A_{j,\mathit{ls}}-A_{j,\mathit{sl}}\right)\right]\right.\\ & -{\delta }_j^h\left[\left(A_{i,\mathit{ls}}-A_{i,\mathit{sl}}\right)A_k+A_i\left(A_{k,\mathit{ls}}-A_{k,\mathit{sl}}\right)\right]\\ & +g_{\mathit{ij}}{g}^{\mathit{hm}}\left[\left(A_{m,\mathit{ls}}-A_{m,\mathit{sl}}\right)A_k+A_m\left(A_{k,\mathit{ls}}-A_{k,\mathit{sl}}\right)\right]\\ & \left.-g_{\mathit{ik}}{g}^{\mathit{hm}}\left[\left(A_{m,\mathit{ls}}-A_{m,\mathit{sl}}\right)A_j+A_m\left(A_{j,\mathit{ls}}-A_{j,\mathit{sl}}\right)\right]\right\}\end{array}$$E72

If (70) is written in (72), then we have

$$R_{\mathit{ijk},\mathit{ls}}^h-R_{\mathit{ijk},\mathit{sl}}^h=0$$E73

which tells us that quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$ is semi-symmetric. Therefore we can express the following:

Theorem 4.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>3\right)$with$\mathit{\alpha an}\ne \mathit{\beta b}\left(n-2\right)$to be semi-symmetric is that the equation$A_{i,l}{A}_{,m}^i-A_{i,m}{A}_{,l}^i=0$is satisfied.

In the last part of this section, let us take a look at the relationships between to be of quasi constant curvature and to be semi-symmetric in a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$.

If we combine Theorem 1 with Theorem 3 we get the following:

Corollary 3.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$with$\mathit{\alpha an}\ne \mathit{\beta b}\left(n-2\right)$to be semi-symmetric is that the manifold is of quasi constant curvature.

Now, we will examine two special cases of the generator $A^i$ of a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$:

I.case: For a parallel generator $A^i$; since

$$A_{i,l}=2A_i{\mathrm{\Phi }}_l\text{and}{A}_{i,\mathit{lm}}-A_{i,\mathit{ml}}=2A_i\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right)$$E74

from Definition 5, it is clear that a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$, which the generator $A^i$ is parallel, is automatically semi-symmetric. If Definition 5 and the equations in (74) are used in (66), then (65) is obtained. This means that the manifold is of quasi constant curvature by means of Theorem 2.

Conversely, let us assume that quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$, which the generator $A^i$ is parallel, is of quasi constant curvature. In this case, if (70), which is implied by (64), is substituted in (72), then (73) is achieved which means that the manifold is semi-symmetric. Hence we can state the following:

Theorem 5.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$which the generator$A^i$is parallel to be semi-symmetric is that the manifold is of quasi constant curvature.

II.case: Let us consider a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$ with $2\beta +\mathit{\alpha n}\ne 0$, which the generator $A^i$ is concurrent, is semi-symmetric. From Definition 5, it follows that

$$A_{i,l}=2A_i{\mathrm{\Phi }}_l+{\mathit{cg}}_{\mathit{il}},A_{i,\mathit{lm}}-A_{i,\mathit{ml}}=2A_i\left({\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}\right),$$

$$A_{i,l}{A}_{,m}^i-A_{i,m}{A}_{,l}^i=c\left(A_{m,l}-A_{l,m}\right).$$E75

If Definition 5 and the equations in (75) are used in (66), then

$${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}=c\left(A_{l,m}-A_{m,l}\right).$$E76

Using (75) and (76) in (62) gives

$$c\left(4\beta +2\mathit{\alpha n}\right)\left(A_{l,m}-A_{m,l}\right)=0.$$E77

Because of the assumption on $\alpha$ and $\beta$, $A_{l,m}-A_{m,l}=0$ and therefore ${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}=0$ by (76) which tells us that the manifold is of quasi constant curvature.

Conversely, a quasi conformally flat quasi Einstein-Weyl manifold ${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$, which the generator $A^i$ is concurrent, is of quasi constant curvature. Then, by Theorem 1, ${\mathrm{\Phi }}_{l,m}-{\mathrm{\Phi }}_{m,l}=0$ which is equivalent to $A_{l,m}-A_{m,l}=0$ by (76). If the last equation is substituted in (72), then (73) is obtained which means that the manifold is semi-symmetric. Hence we can state the following:

Theorem 6.A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold${\left(\mathit{QEW}\right)}_n$$\left(n>4\right)$with$2\beta +\mathit{\alpha n}\ne 0$which the generator$A^i$is concurrent to be semi-symmetric is that the manifold is of quasi constant curvature.

Acknowledgments

The author is grateful to the referee for his/her valuable comments and suggestions for the improvement of the book chapter.