Abstract
This paper deals with generalized quasi-Einstein manifold satisfying certain conditions on conharmonic curvature tensor. Here we study some geometric properties of generalized quasi-Einstein manifold and obtain results which reveal the nature of its associated 1-forms.
1. Introduction
It is well known that a Riemannian or a semi-Riemannian manifold is an Einstein manifold if its Ricci tensor of type is of the form , being the (constant) scalar curvature of the manifold. Let . Then the manifold is said to be a quasi-Einstein manifold [1–7] if, on , we have where is -form on and , are some smooth functions on . It is clear that the function and the -form are nonzero at every point on . The scalars , are the associated scalars of the manifold. Also the 1-form is called associated 1-form of the manifold defined by for any vector field , being a unit vector field called generator of the manifold. Such an -dimensional quasi-Einstein manifold is denoted by . The quasi-Einstein manifolds have also been studied in [8–11].
Generalizing the notion of quasi-Einstein manifold, in [12], De and Ghosh introduced the notion of generalized quasi-Einstein manifolds and studied its geometrical properties with the existence of such notion by several nontrivial examples. A Riemannian manifold is said to be generalized quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the condition: where , , and are scalars of which , and , are nowhere vanishing 1-forms such that , for any vector field . The unit vectors and corresponding to 1-forms and are orthogonal to each other. Also and are the generators of the manifold. Such an -dimensional manifold is denoted by . The generalized quasi-Einstein manifolds have also been studied in [13–16].
In 2008, De and Gazi [17] introduced the notion of nearly quasi-Einstein manifolds. A nonflat Riemannian manifold is called a nearly quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the condition: where and are nonzero scalars and is a nonzero symmetric tensor of type . An -dimensional nearly quasi-Einstein manifold was denoted by . The nearly quasi-Einstein manifolds have also been studied by Prakasha and Bagewadi [18].
The present paper is organized as follows. Section 2 deals with the preliminaries. Section 3 is concerned with conharmonic curvature tensor on . In this section, it is proved that a conharmonically flat is one of the manifold of generalized quasiconstant curvature. Also, it is proved that if in a the associated scalars are constants and the generators and are vector fields with the associated 1-forms and not being the 1-forms of recurrences, then the manifold is conharmonically conservative. In Section 4, we consider satisfying the condition . In the last section we study some geometrical properties of a .
2. Preliminaries
Consider a with associated scalars , , and and associated -forms , . Then from (2) we get where is the scalar curvature of the manifold. Since and are orthogonal unit vector fields, , , and . Again from (2), we have
Let be the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor . Then for all .
The rank-four tensor that remains invariant under conharmonic transformation for an -dimensional Riemannian manifold is given by [19] where and denotes the Riemannian curvature tensor of type defined by , where is the Riemannian curvature tensor of type .
3. Conharmonic Curvature Tensor on
A manifold of generalized quasiconstant curvature tensor is . But the converse is not true, in general. In this section, we enquire under what conditions the converse will be true.
A manifold whose conharmonic curvature tensor vanishes at every point of the manifold is called conharmonically flat manifold. Thus this tensor represents the deviation of the manifold from conharmonic flatness. It satisfies all the symmetric properties of the Riemannian curvature tensor . There are many physical applications of the tensor . For example, in [20], Abdussattar showed that sufficient condition for a space-time to be conharmonic to a flat is either empty in which case it is flat or filled with a distribution represented by energy momentum tensor possessing the algebraic structure of an electromagnetic field and conformal to a flat space-time [20].
Let us consider that the manifold under consideration is conharmonically flat. Then from (6) we have Using (2) in (7), we obtain
According to Chen and Yano [21], a Riemannian manifold is said to be of quasiconstant curvature if it is conformally flat and its curvature tensor of type has the form: where is a form defined by with as a unit vector field and , are scalars of which .
In [12], De and Ghosh generalize the notion of quasiconstant curvature and prove the existence of such a manifold. A Riemannian manifold is said to be a manifold of generalized quasiconstant curvature, if the curvature tensor of type satisfies the condition: where , , and are scalars and and are nonzero 1-forms.
We assume that the unit vector fields and defined by and are orthogonal; that is, . Now the relation (8) can be written as where , , and . Comparing (10) and (11), it follows that the manifold is of generalized quasiconstant curvature. Thus we have the following theorem.
Theorem 1. A conharmonically flat is one of generalized quasiconstant curvature.
Next, differentiating (6) covariantly and then contracting we obtain where denotes the divergence.
Again, it is known that in a Riemannian manifold we have Consequently by virtue of (13), the relation (12) takes the form: Now consider the associated scalars , , and as constants; then (4) yields that the scalar curvature is constant, and hence for all . Consequently, (14) reduces to Since , , and are constants, we have from (2) that By virtue of (16), we get from (15) that Next, if the generators and of the manifold under consideration are recurrent vector fields [22], then we have and , where and are the 1-forms of the recurrence such that and are different from and . Consequently, we get In view of (18), relation (17) reduces to Also, since , it follows that , and hence (18) reduces to for all . Similarly, we have . Hence, from (19), we have ; that is, the manifold under consideration is conharmonically conservative. Hence, we can state the following theorem.
Theorem 2. If, in a , the associated scalars are constants and the generators and corresponding to the associated 1-forms and are not being the recurrence 1-forms, then the manifold is conharmonically conservative.
4. Satisfying the Condition
Let us consider a , satisfying the condition . Then we have Setting and in (20), we have By using (5) in (21), we obtain Next, putting in (6) and then taking inner product with , we get where . Again from (5) we obtain from (23) Again, plugging in (6) and then taking inner product with , we have where . From (24) and (25), one can get By taking account of (26) in (22), we obtain This implies either or .
Now if , then, from (2), we have where . That is, the manifold is a .
On the other hand, if , then (24) gives Thus we can state the following theorem.
Theorem 3. If is a satisfying the condition , then either is or the curvature tensor of the manifold satisfies the property (29).
5. Some Geometric Properties of
In [23], Gray introduced two classes of Riemannian manifolds determined by the covariant differentiation of Ricci tensor. Class A consists of all Riemannian manifolds whose Ricci tensor satisfies the condition: that is, Ricci tensor is a Codazzi type tensor.
Class B consists of all Riemannian manifolds whose Ricci tensor satisfies the condition: that is, Ricci tensor is cyclic parallel.
First suppose that the associated scalars are constants and the Ricci tensor is of Codazzi type. Then from (2) we obtain Interchanging and in (32), we have Since is of Codazzi type, we have from (32) and (33) that Putting in (34) and using and , since is a unit vector, we get This shows that .
Again, putting in (34) and using and , since is a unit vector, we get This shows that .
Thus we can state the following theorem.
Theorem 4. In a , if the associated scalars are constants and the Ricci tensor is of Codazzi type, then the associated 1-forms and are closed.
Next, suppose that the generators and are Killing vector fields in a and the associated scalars are constants. Then where denotes the Lie derivative, which implies that From (38) it follows that since and .
Similarly, we have for all , , and . Now from (2) and using the relations (39)-(40) we have Therefore we can state the following theorem.
Theorem 5. In a , if the generators are Killing vector fields and the associated scalars are constants, then the Ricci tensor of the manifold is cyclic parallel.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
D. G. Prakasha is thankful to University Grants Commission, New Delhi, India, for financial support in the form of Major Research Project (no. 39-30/2010 (SR), dated December 23, 2012).