Advances in Mathematical Physics
Volume 2016 (2016), Article ID 9615053, 8 pages
http://dx.doi.org/10.1155/2016/9615053

Research Article

On Pseudo-Petrov Symmetric Riemannian Manifolds

Sanjib Kumar Jana,¹ Fusun Nurcan,² Amit Kumar Debnath,³ and Joydeep Sengupta³

¹Department of Mathematics, Balurghat College, Balurghat, Dakshin Dinajpur, West Bengal 733 101, India
²Department of Mathematics, Marmara University, Goztepe Campus, 34722 Istanbul, Turkey
³Department of Mathematics, University of Gour Banga, Mokdumpur, Malda, West Bengal 732 103, India

Received 29 November 2015; Accepted 16 February 2016

Academic Editor: Luigi C. Berselli

Copyright © 2016 Sanjib Kumar Jana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The present paper deals with pseudo-Petrov symmetric Riemannian manifolds whose space-matter tensor satisfies a special condition. Firstly, basic results of pseudo-Petrov symmetric Riemannian manifolds are obtained. Then, pseudo-Petrov symmetric manifolds which are Einstein, quasi-Einstein, and locally decomposable are examined and some theorems involving these manifolds are proved. Finally, two examples proving the existence of pseudo-Petrov symmetric Riemannian manifolds are given.

1. Introduction

In the sense of Chaki, a nonflat Riemannian manifold is said to be pseudosymmetric [1] if its curvature tensor satisfies the following relation:

for all vector fields , where is a nonzero -form, denotes the Lie algebra of all smooth vector fields [over ] on manifold , and denotes the operator of covariant differentiation with respect to the metric tensor . The -form is called the associated -form of the manifold and an -dimensional manifold of this kind is denoted by .

In 1949, Petrov [2] introduced a tensor field of type and defined it by

where is the Riemannian curvature tensor of type , is the energy-momentum tensor of type , is a cosmological constant, is the energy density (scalar), is a tensor of type given by

for all , being the Lie algebra of smooth vector fields on , and the Kulkarni-Nomizu product [3] of two tensors and is defined by

for ,  . The tensor is known as the space-matter tensor of type of manifold .

Einstein’s field equation with cosmological constant [4] is given by

where is a cosmological constant, is the scalar curvature, and is the Ricci tensor of type . By virtue of (5), (2) takes the form

The present paper deals with a Riemannian manifold    whose space-matter tensor (nonvanishing identically) satisfies the condition

where is a nonzero -form such that for all . Such a manifold will be called a pseudo-Petrov symmetric manifold and denoted by .

In Section 2, we go through some basic results of [5]. In Section 3, we study Einstein and prove that scalar curvature can not be vanished in such a manifold. Section 4 is concerned about quasi-Einstein where we discuss which is locally decomposable in Section 5. Finally, the last section deals with nontrivial examples of and also of quasi-Einstein .

2. Some Basic Results of

In this section, we deal with some basic results of . Let be an orthonormal basis of the tangent space at any point of the manifold. Then, the Ricci tensor of type and the scalar curvature are given by

where is the symmetric endomorphism corresponding to the Ricci tensor . From (6), we obtain

for all and . Differentiating the above equation covariantly with respect to , we get

for all , , and . Then, using (6) in (7) and contracting with respect to and , we obtain, by virtue of (10), that

Setting in (11) and taking summation over ,  , we get

Assume that the scalar curvature and the energy density satisfy the relation

Then, (12) reduces to

From (5) and (15), we obtain

This leads to the following theorem.

Theorem 1. In a admitting Einstein’s field equation, is an eigenvalue of the energy-momentum tensor corresponding to the eigenvector defined by for all , provided that relation (13) holds.

3. Einstein

This section is concerned about a    which is also an Einstein manifold. The Ricci tensor of Einstein manifold [3] satisfies the condition

from which it follows that

In view of (17), (18), and (6), we find that

By virtue of (17), (19), and (7), we get

Setting in (20) and then taking summation over , , we find that

by virtue of (18). Replacing and by in (21) and taking summation over , , we have

Then, substituting and by in (21) and taking summation over , , we obtain

By virtue of (22) and (23), we have

Finally, in the view of (18) and (24), we get

Hence, we can state the following theorem.

Theorem 2. In an Einstein admitting Einstein’s field equation, the energy density scalar is constant and it is connected with the scalar curvature by the relation .

Now, in an Einstein admitting Einstein’s field equation, if possible, let the energy density scalar be equal to the cosmological constant . From (24), it follows that the scalar curvature vanishes identically. With the help of (20), we find that

which shows that the manifold is a . This leads to a contradiction, as an Einstein manifold can not accommodate a pseudosymmetric structure [1]. So our assumption that is not possible. Hence, we can state the following theorem.

Theorem 3. In an Einstein admitting Einstein’s field equation, the energy density satisfies the relation and the scalar curvature can not be vanished identically.

Definition 4. A vector field on a Riemannian manifold is said to be concurrent [6] if and only if there exists a nonzero constant such that for all .
is said to be parallel, in case for all .

Let us consider that an Einstein in which the vector field is defined by is parallel. Then, we have

If is concurrent, we have

Therefore, by using Ricci identity, we get which yields

From (17) and (29), it follows that

which is not possible according to our previous theorem. So our assumption is not true; that is, can not be a concurrent vector field. This leads to the following theorem.

Theorem 5. In an Einstein admitting Einstein’s field equation, the vector field defined by cannot be concurrent.

4. Quasi-Einstein

In this section, we discuss a quasi-Einstein manifold admitting a pseudosymmetric space-matter tensor. A Riemannian manifold    is said to be quasi-Einstein [7] if its Ricci tensor is not identically zero and satisfies that

where (≠ 0) are associated scalars and is a nonzero -form which is defined by for any vector field , with being a unit vector field, called the generator of the manifold. Relation (31) implies

Differentiating covariantly (31) with respect to , we obtain

In view of (33), (11) takes the following form:

Firstly, taking contraction of (34) with respect to and , we have

Then, also contracting (34) with respect to and , we have

Thus, (35) and (36) yield

Hence, we can state the following theorem.

Theorem 6. In a quasi-Einstein admitting Einstein’s field equation, the scalars and and the energy density are connected by relation (37).

Now, from (37), it is clear that will be also constant if is constant; that is, from (36), we have that

which implies

So, by the above relation and (5), we obtain

Thus, we get the following theorem.

Theorem 7. In a quasi-Einstein admitting Einstein’s field equation, is an eigenvalue of the energy-momentum tensor corresponding to the eigenvector defined by for all .

Let us consider that is a concurrent vector field; that is,

Hence, by the Ricci identity, it follows that which yields

From (31) and (42), we have

This leads to the following theorem.

Theorem 8. In a quasi-Einstein admitting Einstein’s field equation, if the vector field is concurrent then the associated scalars and are connected by relation (43).

5. Locally Decomposable

A differentiable manifold is said to be a product manifold [6] if and only if it can be expressed as , where . In this case, we say that admits both a local product structure and a separating coordinates system.

Moreover, if is a local product Riemannian manifold, then there exists a Riemannian metric given by

where and are local coordinates on and , respectively. Here and run over and and run over . Equivalently, the manifolds and are orthogonal. Further, if ’s are functions of ’s only and ’s are functions of ’s only, then such a manifold is called a locally decomposable Riemannian manifold. For a locally decomposable Riemannian manifold, we have

for all and and and . Let be a locally decomposable Riemannian manifold with a separating coordinates system such that we can express   . Then, the relations in (45) hold. We further assume that is a ; that is, there exists a smooth nonzero -form such that

for all Then, we find the following relations:

for and . From (48), we consider two cases, namely,

(1) on ;(2) in .

At first, we consider case (): from (51), it follows that

which implies

Also, from (54), we obtain

which yields that

where the scalar curvature and energy density on are denoted by and , respectively. From (56) and (59), it is easy to see that and satisfy

In view of (59) and (60), we find that for all , , and in . By virtue of this relation and (60), (58) reduces to

Secondly, we discuss case (): from in , we find

which yields

where is the scalar curvature on . Thus, we find

Using (64) in (63), we obtain

By virtue of the last relation, (62) reduces to

which shows that the manifold is of constant curvature. Hence, we have the following theorem.

Theorem 9. Let be a locally decomposable Riemannian manifold such that . If is a , then one gets that

(1) is a locally symmetric manifold if on ,(2) is a manifold of constant curvature if in .

Similarly, in view of (53), we can state the following theorem.

Theorem 10. Let be a locally decomposable Riemannian manifold such that   . If is a , then one obtains that

(1) is a locally symmetric manifold if on ,(2) is of constant curvature if in .

Now, contracting (49) with respect to and , we obtain

which yields that

where

This result leads to the following theorem.

Theorem 11. Let be a locally decomposable Riemannian manifold such that . If is a , then the relation

holds for any vector field on

Similarly, from (52), we can state the following theorem.

Theorem 12. Let be a locally decomposable Riemannian manifold such that . If is a , then the relation

holds for any vector field on .

6. Some Examples of

In this section, the existence of is proved by means of citing some examples.

Example 1. Let us consider the manifold with the metric

Then, the only nonvanishing components of the curvature tensor, the Ricci tensor, and the scalar curvature are given by

And, furthermore, considering , we find only nonvanishing component of the space-matter tensor and its covariant derivative as follows:

where “,” in between indices denotes covariant differentiation with respect to the coordinates. Let us consider the -form as

In the manifold under consideration, with the help of (75), (7) reduces to the following equation:

In the cases other than (76), either the result is trivial or both sides vanish identically. Now, we check the validity of the above equation:

Since the manifold under consideration is a , we can state the following theorem.

Theorem 13. Let be a Riemannian manifold endowed with the metric

Then, is a admitting Einstein’s field equation of nonvanishing scalar curvature with .

Example 2. Let be an open subset of endowed with the metric

where and run from to . Then, the only nonvanishing components of the curvature tensor, the Ricci tensor, and the scalar curvature are given by

We will now verify that is a . To verify this, let us consider that the -form and associated scalars and are as follows:

According to the definition of , (31) reduces to

since the components of (31) vanish identically and relation (31) holds trivially for the cases other than (82). By virtue of (79), (80), and (81), it follows that

for . By a similar argument, it can be easily shown that relation (82) also holds for . Therefore, is a .

Now, considering the energy density function as a function of only and taking its value equal to , we calculate only nonvanishing components of the space-matter tensor and its covariant derivatives in the form of

where “,” denotes the covariant differentiation with respect to the coordinates. Let us take the -form as follows:

In the manifold under consideration, (7) reduces to the following equations by (85):

In the cases other than (86), either the result is trivial or both sides vanish identically. With the help of (84) and (85), one can easily check the validity of the above relations. Therefore, the manifold under consideration is a . Hence, we can state the following theorem.

Theorem 14. Let be a Riemannian manifold endowed with the metric

Then, is a admitting Einstein’s field equation of nonvanishing scalar curvature with .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

References

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