Some Properties of Generalized Einstein Tensor for a Pseudo-Ricci Symmetric Manifold

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 ðM, gÞ, ðn > 3Þ 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]


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 ðM, gÞ, ðn > 3Þ 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] where π is a nonzero 1-form, ρ is a vector field by for all vector fields X, and ∇ denotes the operator of covariant differentiation with respect to the metric g. Such a manifold is denoted by ðPRSÞ n . 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 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] where A and B are nonzero 1-forms [9]. If the associated 1form B becomes zero, then the manifold reduces to Ricci recurrent, i.e., A Riemannian manifold ðM, gÞ, ðn ≥ 2Þ, is said to be an Einstein manifold if the following condition: holds on M, where S and r denote the Ricci tensor and scalar curvature of ðM, gÞ, 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 ðM, gÞ realizing the following relation: where a, b are real numbers and A is a nonzero 1-form such that for all vector fields X. A nonflat Riemannian manifold ðM, gÞ, ðn > 2Þ, is defined to be a quasi-Einstein manifold if its Ricci tensor S of type ð0, 2Þ is not identically zero and satisfies the condition (7).

Recurrent Generalized Einstein Tensor
GðX, YÞ in ðPRSÞ n It is well known that the Einstein tensor EðX, YÞ for a Riemannian manifold is defined by where SðX, YÞ 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 where κ is constant [12]. Now, we assume that our manifold ðPRSÞ n has nonzero GðX, YÞ-Einstein tensor. By taking the covariant derivative of (10), in the local coordinates, we get If we contract (1) over X and Y, then we obtain Substituting (1) and (12) into (11), we achieve Now, contracting (13) with respect to i and k, we obtain If we assume that GðX, YÞ is conservative [13], i.e., div G = 0, then from (14), we have where P j = π h S hj . If ð1 − 2κÞr is an eigenvalue of the Ricci tensor S corresponding to the eigenvector πðXÞ, then ð3 − 2κÞ is an eigenvalue of the Ricci tensor S corresponding to the eigenvector P j . Conversely, if equation (15) holds, then the form (14) the generalized Einstein tensor GðX, YÞ is conservative. We have thus proved the following. Theorem 1. For a ðPRSÞ n manifold, the necessary and sufficient condition of the generalized Einstein tensor GðX, YÞ be conservative is that ð1 − 2κÞr and ð3 − 2κÞ be eigenvalues of the Ricci tensor S corresponding to the eigenvectors π j and P j = π h S hj , respectively.
Let GðX, YÞ be recurrent, i.e., from (5), Substituting equations (10) and (13) into equation (16) yields If we contract (17) over i and k, then we have This leads to the following result: In a ðPRSÞ n manifold, let us assume that the generalized Einstein tensor GðX, YÞ is recurrent with the recurrence vector generated by the 1-form A. Then, the recurrency vector A satisfies equation ((18)).

Advances in Mathematical Physics
Now, we assume that the generalized Einstein tensor GðX, YÞ is conservative. From (15) and (18), we get where Q j = A k S k j . Then, the following theorem holds true: Theorem 3. In a ðPRSÞ n manifold, let the generalized Einstein tenso GðX, YÞ be recurrent with the recurrence vector generated by the 1-form A. If the generalized Einstein tensor GðX, YÞ is also conservative, then the vectors Q j and A j are linearly dependent.
Let GðX, YÞ be a generalized recurrent. Then from (4), Using (1) and (10), we get If we contract (21) over i and j, then we have If 1 + κn = 0, then B k = 0: This leads to the following result: Theorem 4. If κ = −1/n, a ðPRSÞ n manifold admitting the generalized Einstein tensor GðX, YÞ which is the generalized recurrent cannot exist.

Birecurrent Generalized Einstein Tensor GðX, YÞ in ðPRSÞ n
In this section, we examine some properties of the generalized Einstein tensor GðX, YÞ in ðPRSÞ n which is birecurrent. If the generalized Einstein tensor GðX, YÞ satisfies the condition for some nonzero covariant tensor field μ lk , then we call G ij as birecurrent generalized Einstein tensor: It is easy to see that a recurrent generalized Einstein tensor GðX, YÞ is birecurrent. In fact, by taking the covariant derivative of (16) with respect to U l , we get with μ lk = ∇ l A k + A k A l . Now, we assume that ðPRSÞ n admitting the generalized Einstein tensor GðX, YÞ satisfies (24), but not (16). Changing the order of indices l and k in (23) and subtracting the expres-sion so obtained from (23), we have where the bracket indicates antisymmetrization. If μ lk is a symmetric tensor, then ∇ ½l ∇ k G ij = 0, and vice versa. Thus, we have the following result: The birecurrency tensor of the generalized Einstein tensor GðX, YÞ is symmetric if and only if the equation holds.
Now, by taking the covariant derivative of (13), we obtain where P k = π i S i k : The covariant derivative of P k is Writing (1) as using (28) and (29), we achieve Now, we apply Lemma 5, and by using equation (26), we obtain Contracting (32) with respect to i and j, we get r ∇ l π k − ∇ k π l ð Þ1 − κn ð Þ+ 2 1 + κn ð Þ π l P k − π k P l ð Þ

Advances in Mathematical Physics
Substituting (31) into (33) yields If κ = 1/n, the generalized Einstein tensor GðX, YÞ reduces to the Einstein tensor EðX, YÞ. So, we can state the following: Theorem 6. In ðPRSÞ n , the birecurrency tensor of Einstein tensor EðX, YÞ is symmetric if and only if the vector fields π k and P k are linearly dependent.
Let us now recall that a φðRicÞ 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 It is evident that π k and P k are closed or πðRicÞ and PðRicÞ vector fields.
Therefore, we have Theorem 7. In ðPRSÞ n , the birecurrency tensor of generalized Einstein tensor GðX, YÞ with κ = −1/2n is symmetric if and only if the vector fields π k and P k are closed or πðRicÞ and PðRicÞ.
Theorem 8. In ðPRSÞ n , the birecurrency tensor of generalized Einstein tensor GðX, YÞ with κ ≠ −1/2n is symmetric if and only if the vector fields π k and P k are linearly dependent, and the vector field π k is closed or πðRicÞ.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.