Abstract
The object of the paper is to study some properties of the generalized Einstein tensor which is recurrent and birecurrent on pseudo-Ricci symmetric manifolds . Considering the generalized Einstein tensor as birecurrent but not recurrent, we state some theorems on the necessary and sufficient conditions for the birecurrency tensor of 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 -dimensional Riemannian manifold , is called a pseudo-Ricci symmetric manifold if the Ricci tensor of type (0,2) is not identically zero and satisfies the condition [4] where is a nonzero 1-form, is a vector field by
for all vector fields , and denotes the operator of covariant differentiation with respect to the metric . Such a manifold is denoted by . The 1-form is called the associated 1-form of the manifold. If , 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]).
A nonflat Riemannian manifold , , is called generalized recurrent if the Ricci tensor is nonzero and satisfies the condition where and are nonzero 1-forms [9]. If the associated 1-form becomes zero, then the manifold reduces to Ricci recurrent, i.e.,
A Riemannian manifold , , is said to be an Einstein manifold if the following condition: holds on , where and denote the Ricci tensor and scalar curvature of , 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 realizing the following relation: where , are real numbers and is a nonzero 1-form such that
for all vector fields .
A nonflat Riemannian manifold , is defined to be a quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the condition (7).
2. Recurrent Generalized Einstein Tensor in
It is well known that the Einstein tensor for a Riemannian manifold is defined by where and 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 is constant [12].
Now, we assume that our manifold has nonzero -Einstein tensor. By taking the covariant derivative of (10), in the local coordinates, we get
If we contract (1) over and , then we obtain
Substituting (1) and (12) into (11), we achieve
Now, contracting (13) with respect to and , we obtain
If we assume that is conservative [13], i.e., then from (14), we have where .
If is an eigenvalue of the Ricci tensor corresponding to the eigenvector , then is an eigenvalue of the Ricci tensor corresponding to the eigenvector . Conversely, if equation (15) holds, then the form (14) the generalized Einstein tensor is conservative. We have thus proved the following.
Theorem 1. For amanifold, the necessary and sufficient condition of the generalized Einstein tensorbe conservative is thatandbe eigenvalues of the Ricci tensorcorresponding to the eigenvectorsand, respectively.
Let be recurrent, i.e., from (5),
Substituting equations (10) and (13) into equation (16) yields
If we contract (17) over and , then we have
This leads to the following result:
Theorem 2. In amanifold, let us assume that the generalized Einstein tensoris recurrent with the recurrence vector generated by the 1-form. Then, the recurrency vectorsatisfies equation ((18)).
Now, we assume that the generalized Einstein tensor is conservative. From (15) and (18), we get where .
Then, the following theorem holds true:
Theorem 3. In amanifold, let the generalized Einstein tensobe recurrent with the recurrence vector generated by the 1-form. If the generalized Einstein tensoris also conservative, then the vectorsandare linearly dependent.
Let be a generalized recurrent. Then from (4),
If we contract (21) over and , then we have
If , then
This leads to the following result:
Theorem 4. If, amanifold admitting the generalized Einstein tensorwhich is the generalized recurrent cannot exist.
3. Birecurrent Generalized Einstein Tensor in
In this section, we examine some properties of the generalized Einstein tensor in which is birecurrent. If the generalized Einstein tensor satisfies the condition
for some nonzero covariant tensor field , then we call as birecurrent generalized Einstein tensor
It is easy to see that a recurrent generalized Einstein tensor is birecurrent. In fact, by taking the covariant derivative of (16) with respect to , we get
with .
Now, we assume that admitting the generalized Einstein tensor satisfies (24), but not (16). Changing the order of indices and in (23) and subtracting the expression so obtained from (23), we have where the bracket indicates antisymmetrization. If is a symmetric tensor, then , and vice versa.
Thus, we have the following result:
Lemma 5. The birecurrency tensor of the generalized Einstein tensoris symmetric if and only if the equationholds.
Now, by taking the covariant derivative of (13), we obtain where
The covariant derivative of 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 and , we get
Substituting (31) into (33) yields
If , the generalized Einstein tensor reduces to the Einstein tensor . So, we can state the following:
Theorem 6. In, the birecurrency tensor of Einstein tensoris symmetric if and only if the vector fieldsandare linearly dependent.
Let us now recall that a vector field was introduced by Hinterleitner and Kiosak as a vector field satisfying the condition [14], where is some constant, is the Ricci tensor, and is the Levi-Civita connection.
If , then it follows from (34) that
It is evident that and are closed or and vector fields.
Therefore, we have
Theorem 7. In, the birecurrency tensor of generalized Einstein tensorwithis symmetric if and only if the vector fieldsandare closed orand.
Theorem 8. In, the birecurrency tensor of generalized Einstein tensorwithis symmetric if and only if the vector fieldsandare linearly dependent, and the vector fieldis closed or.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.