The object of the paper is to study some properties of the generalized Einstein tensor GX,Y which is recurrent and birecurrent on pseudo-Ricci symmetric manifolds PRSn. Considering the generalized Einstein tensor GX,Y as birecurrent but not recurrent, we state some theorems on the necessary and sufficient conditions for the birecurrency tensor of GX,Y to be symmetric.

1. Introduction

In the late twenties, because of the important role of symmetric spaces in differential geometry, Cartan [1], who, in particular, obtained a classification of those spaces, established Riemannian symmetric spaces. The notion of the pseudosymmetric manifold was introduced by Chaki [2] and Deszcz [3]. Recently, some necessary and sufficient conditions for a Chaki pseudosymmetric (respectively, pseudo-Ricci symmetric [4]) manifold to be Deszcz pseudosymmetric (respectively, Ricci-pseudo symmetric [5]) have been examined in [6].

A nonflat n-dimensional Riemannian manifold M,g, n>3 is called a pseudo-Ricci symmetric manifold if the Ricci tensor S of type (0,2) is not identically zero and satisfies the condition [4]
(1)∇ZSX,Y=2πZSX,Y+πXSZ,Y+πYSX,Z,where π is a nonzero 1-form, ρ is a vector field by
(2)gX,ρ=πX

for all vector fields X, and ∇ denotes the operator of covariant differentiation with respect to the metric g. Such a manifold is denoted by PRSn. The 1-form π is called the associated 1-form of the manifold. If π=0, then the manifold reduces to a Ricci symmetric manifold or covariantly constant
(3)∇ZSX,Y=0.

The notion of pseudo-Ricci symmetry is different from that of Deszcz [3].

The pseudo-Ricci symmetric manifolds have some importance in the general theory of relativity. So, pseudo-Ricci symmetric manifolds on some structures have been studied by many authors (see, e.g., [7, 8]).

A nonflat Riemannian manifold M,g, n>2, is called generalized recurrent if the Ricci tensor S is nonzero and satisfies the condition
(4)∇ZSX,Y=AZSX,Y+BZgX,Y,where A and B are nonzero 1-forms [9]. If the associated 1-form B becomes zero, then the manifold reduces to Ricci recurrent, i.e.,
(5)∇ZSX,Y=AZSX,Y.

A Riemannian manifold M,g, n≥2, is said to be an Einstein manifold if the following condition:
(6)S=rng,holds on M, where S and r denote the Ricci tensor and scalar curvature of M,g, respectively. According to [10], equation (6) is called the Einstein metric condition. Also, Einstein manifolds form a natural subclass of various classes of Riemannian manifolds by a curvature condition imposed on their Ricci tensor [10]. For instance, every Einstein manifold belongs to the class of Riemannian manifolds M,g realizing the following relation:
(7)SX,Y=agX,Y+bAXAY,where a, b are real numbers and A is a nonzero 1-form such that
(8)gX,U=AX

for all vector fields X.

A nonflat Riemannian manifold M,g, n>2, is defined to be a quasi-Einstein manifold if its Ricci tensor S of type 0,2 is not identically zero and satisfies the condition (7).

2. Recurrent Generalized Einstein Tensor <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M46"><mml:mi>G</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula> in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M47"><mml:msub><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mtext>PRS</mml:mtext></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

It is well known that the Einstein tensor EX,Y for a Riemannian manifold is defined by
(9)EX,Y=SX,Y−rgX,Y,where SX,Y and r are, respectively, the Ricci tensor and the scalar curvature of the manifold, playing an important part in Einstein’s theory of gravitation as well as in proving some theorems in Riemannian geometry [10]. Moreover, the Einstein tensor can be obtained from Yano’s tensor of concircular curvature. In [11], by using this approach, some generalizations of the Einstein tensor were achieved.

In this section, we consider the generalized Einstein tensor
(10)GX,Y=SX,Y−κrgX,Y,where κ is constant [12].

Now, we assume that our manifold PRSn has nonzero GX,Y-Einstein tensor. By taking the covariant derivative of (10), in the local coordinates, we get
(11)∇kGij=∇kSij−κgij∇kr.

If we contract (1) over X and Y, then we obtain
(12)∇kr=2πkr+2πhShk.

Substituting (1) and (12) into (11), we achieve
(13)∇kGij=2πkSij+πiSkj+πjSik−2πkr+2πhShkκgij.

Now, contracting (13) with respect to i and k, we obtain
(14)divGij=∇kGjk=3−2κπhShj+1−2κrπj.

If we assume that GX,Y is conservative [13], i.e., divG=0, then from (14), we have
(15)3−2κPj+1−2κrπj=0,where Pj=πhShj.

If 1−2κr is an eigenvalue of the Ricci tensor S corresponding to the eigenvector πX, then 3−2κ is an eigenvalue of the Ricci tensor S corresponding to the eigenvector Pj. Conversely, if equation (15) holds, then the form (14) the generalized Einstein tensor GX,Y is conservative. We have thus proved the following.

Theorem 1.

For aPRSnmanifold, the necessary and sufficient condition of the generalized Einstein tensorGX,Ybe conservative is that1−2κrand3−2κbe eigenvalues of the Ricci tensorScorresponding to the eigenvectorsπjandPj=πhShj, respectively.

Let GX,Y be recurrent, i.e., from (5),
(16)∇kGij=AkGij.

Substituting equations (10) and (13) into equation (16) yields
(17)2πkSij+πiSjk+πjSik−2πkr+2πhShkκgij=AkSij−rκgij.

If we contract (17) over i and k, then we have
(18)3−2κPj+1−2κrπj=AkSjk−rκAj.

This leads to the following result:

Theorem 2.

In aPRSnmanifold, let us assume that the generalized Einstein tensorGX,Yis recurrent with the recurrence vector generated by the 1-formA. Then, the recurrency vectorAsatisfies equation ((18)).

Now, we assume that the generalized Einstein tensor GX,Y is conservative. From (15) and (18), we get
(19)Qj−rκAj=0,where Qj=AkSjk.

Then, the following theorem holds true:

Theorem 3.

In aPRSnmanifold, let the generalized Einstein tensoGX,Ybe recurrent with the recurrence vector generated by the 1-formA. If the generalized Einstein tensorGX,Yis also conservative, then the vectorsQjandAjare linearly dependent.

Let GX,Y be a generalized recurrent. Then from (4),
(20)∇kGij=AkGij+Bkgij.

Using (1) and (10), we get
(21)2πkSij+πiSkj+πjSik+κgij2πkr+2πhSkh=AkSij+κrgij+Bkgij.

If we contract (21) over i and j, then we have
(22)1+κn2πkr+2Pk−Akr=nBk.

If 1+κn=0, then Bk=0.

This leads to the following result:

Theorem 4.

Ifκ=−1/n, aPRSnmanifold admitting the generalized Einstein tensorGX,Ywhich is the generalized recurrent cannot exist.

3. Birecurrent Generalized Einstein Tensor <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M112"><mml:mi>G</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula> in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M113"><mml:msub><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>P</mml:mi><mml:mi>R</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this section, we examine some properties of the generalized Einstein tensor GX,Y in PRSn which is birecurrent. If the generalized Einstein tensor GX,Y satisfies the condition
(23)∇l∇kGij=μlkGij

for some nonzero covariant tensor field μlk, then we call Gij as birecurrent generalized Einstein tensor.

It is easy to see that a recurrent generalized Einstein tensor GX,Y is birecurrent. In fact, by taking the covariant derivative of (16) with respect to Ul, we get
(24)∇l∇kGij=∇lAk+AkAlGij

with μlk=∇lAk+AkAl.

Now, we assume that PRSn admitting the generalized Einstein tensor GX,Y satisfies (24), but not (16). Changing the order of indices l and k in (23) and subtracting the expression so obtained from (23), we have
(25)∇l∇kGij=μlkGij,where the bracket indicates antisymmetrization. If μlk is a symmetric tensor, then ∇l∇kGij=0, and vice versa.

Thus, we have the following result:

Lemma 5.

The birecurrency tensor of the generalized Einstein tensorGX,Yis symmetric if and only if the equation(26)∇l∇kGij=0holds.

Now, by taking the covariant derivative of (13), we obtain
(27)∇l∇kGij=4πkπl+2∇lπkSij−rκgij+3πkπiSjl+3πkπjSil+2πiπjSlk+∇lπi+2πiπlSjk+∇lπj+2πlπjSik−4Plπkκgij−2∇lPkκgij,where Pk=πiSki.

The covariant derivative of Pk is
(28)∇lPk=∇lπhSkh=∇lπhSkh+πh∇lSkh.

Writing (1) as
(29)∇lSki=2πlSki+πkSki+πiSkl,using (28) and (29), we achieve
(30)∇lPk=∇lπhSkh+2πlPk+πkPl+πSkl,(31)∇lπhSkh=∇lPk−2πlPk−πkPl−πSkl.

Now, we apply Lemma 5, and by using equation (26), we obtain
(32)2∇lπk−∇kπlSij−rκgij+πiπk−∇kπiSjl−πlπi−∇lπiSjk+3πkπj−∇kπjSil−3πlπj−∇lπjSik+4πlPk−πkPlκgij−2∇lPk−∇kPlκgij=0.

Contracting (32) with respect to i and j, we get
(33)r∇lπk−∇kπl1−κn+21+κnπlPk−πkPl+∇lπhSkh−∇kπhSlh−∇lPk−∇kPlκn=0.

Substituting (31) into (33) yields
(34)1−κnr∇lπk−∇kπl+∇lPk−∇kPl+1+2κnπlPk−πkPl=0.

If κ=1/n, the generalized Einstein tensor GX,Y reduces to the Einstein tensor EX,Y. So, we can state the following:

Theorem 6.

InPRSn, the birecurrency tensor of Einstein tensorEX,Yis symmetric if and only if the vector fieldsπkandPkare linearly dependent.

Let us now recall that a φRic vector field was introduced by Hinterleitner and Kiosak as a vector field satisfying the condition ∇φ=μRic [14], where μ is some constant, Ric is the Ricci tensor, and ∇ is the Levi-Civita connection.

If κ=−1/2n, then it follows from (34) that
(35)r∇lπk−∇kπl+∇lPk−∇kPl=0.

It is evident that πk and Pk are closed or πRic and PRic vector fields.

Therefore, we have

Theorem 7.

InPRSn, the birecurrency tensor of generalized Einstein tensorGX,Ywithκ=−1/2nis symmetric if and only if the vector fieldsπkandPkare closed orπRicandPRic.

Theorem 8.

InPRSn, the birecurrency tensor of generalized Einstein tensorGX,Ywithκ≠−1/2nis symmetric if and only if the vector fieldsπkandPkare linearly dependent, and the vector fieldπkis closed orπRic.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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