#### Abstract

The object of the present paper is to study -manifolds with vanishing quasi-conformal curvature tensor. -manifolds satisfying Ricci-symmetric condition are also characterized.

#### 1. Introduction

Recently, in [1], Shaikh introduced and studied Lorentzian concircular structure manifolds (briefly -manifold) which generalizes the notion of LP-Sasakian manifolds, introduced by Matsumoto [2].

Generalizing the notion of LP-Sasakian manifold in 2003 [1], Shaikh introduced the notion of -manifolds along with their existence and applications to the general theory of relativity and cosmology. Also, Shaikh and his coauthors studied various types of -manifolds by imposing the curvature restrictions (see [3–6]). In [7, 8], the authors also studied -manifolds.

The submanifold of an -manifold is studied by Atceken and Hui [9, 10] and Shukla et al. [11]. In [12], Yano and Sawaki introduced the quasi-conformal curvature tensor, and later it was studied by many authors with curvature restrictions on various structures [13].

After then, the same author studied weakly symmetric -manifolds by several examples and obtain various results in such manifolds. In [7], authors shown that a pseudo projectively flat and pseudo projectively recurrent manifolds are -Einstein manifold.

On the other hand, in [5], authors proved the existence of -recurrent manifold which is neither locally symmetric nor locally -symmetric by nontrivial examples. Furthermore, they also give the necessary and sufficient conditions for a -manifold to be locally -recurrent.

In this study, we have investigated the quasi-conformal flat -manifolds satisfying properties such as Ricci-symmetric, locally symmetric, and -Einstein. Finally, we give an example for -Einstein manifolds.

#### 2. Preliminaries

An -dimensional Lorentzian manifold is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric tensor , that is, admits a smooth symmetric tensor field of the type such that, for each , is a nondegenerate inner product of signature . In such a manifold, a nonzero vector is said to be timelike (resp., nonspacelike, null, and spacelike) if it satisfies the condition (resp., ≤0, =0, >0). These cases are called casual character of the vectors.

*Definition 1. *In a Lorentzian manifold , a vector field defined by
for any is said to be a concircular vector field if
for , where is a nonzero scalar function, is a 1-form, is also closed 1-form, and denotes the Levi-Civita connection on [7].

Let be a Lorentzian manifold admitting a unit timelike concircular vector field , called the characteristic vector field of the manifold. Then we have Since is a unit concircular unit vector field, there exists a nonzero 1-form such that The equation of the following form holds: for all , where is a nonzero scalar function satisfying being a certain scalar function given by .

Let us put then from (6) and (8), we can derive which tell us that is a symmetric -tensor. Thus the Lorentzian manifold together with the unit timelike concircular vector field , its associated 1-form , and -type tensor field is said to be a Lorentzian concircular structure manifold.

A differentiable manifold of dimension is called -manifold if it admits a -type tensor field , a covariant vector field , and a Lorentzian metric which satisfy for all . Particularly, if we take , then we can obtain the -Sasakian structure of Matsumoto [2].

Also, in an -manifold , the following relations are satisfied (see [3–6]): for all vector fields on , where and denote the Riemannian curvature tensor and Ricci curvature, respectively, is also the Ricci operator given by .

Now let be an -dimensional Riemannian manifold; then the concircular curvature tensor , the Weyl conformal curvature tensor , and the pseudo projective curvature tensor are, respectively, defined by where and are constants such that , and is also the scalar curvature of [7].

For an -dimensional -manifold the quasi-conformal curvature tensor is given by for all [14].

The notion of quasi-conformal curvature tensor was defined by Yano and Swaki [12]. If and , then quasi-conformal curvature tensor reduces to conformal curvature tensor.

#### 3. Quasi-Conformally Flat -Manifolds and Some of Their Properties

For an -dimensional quasi-conformally flat -manifold, we know for from (23), Here, taking into account of (16), we have Let be in (25); then also by using (18) we obtain Taking the inner product on both sides of the last equation by , we obtain that is, Now we are in a proposition to state the following.

Theorem 2. *If an -dimensional -manifold is quasi-conformally flat, then is an -Einstein manifold.*

Now, let be an orthonormal basis of the tangent space at any point of the manifold. Then putting in (28), and taking summation for , we have In view of (28) and (29), we obtain which is equivalent to for any .

By using (29) and (31) in (23) for a quasi-conformally flat -manifold , we get for all . If we consider Schur's Theorem, we can give the following the theorem.

Theorem 3. *A quasi-conformally flat -manifold M is a manifold of constant curvature provided that .*

Now let us consider an -manifold which is conformally flat. Thus we have from (21) that for all vector fields tangent to . Setting in (33) and using (16), (18) we have If we put in (34) and also using (18), we obtain

Corollary 4. *A conformally flat -manifold is an -Einstein manifold.*

Generalizing the notion of a manifold of constant curvature, Chen and Yano [15] introduced the notion of a manifold of quasi-constant curvature which can be defined as follows:

*Definition 5. *A Riemannian manifold is said to be a manifold of quasi-constant curvature if it is conformally flat and its curvature tensor of type is of the form
for all , where are scalars of which and is a nonzero 1-form (for more details, we refer to [13, 16]).

Thus we have the following theorem for -conformally flat manifolds.

Theorem 6. *A conformally flat -manifold is a manifold of quasi-constant curvature.*

*Proof. *From (33) and (35), we obtain
This implies (36) for
This proves our assertion.

Next, differentiating the (19) covariantly with respect to , we get for any . Making use of the definition of and (8), we have Thus we have Here taking account of (17), we arrive at Again, by using (13), (18), and (19), we reach Thus we have the following theorem.

Theorem 7. *If an -manifold is Ricci-symmetric; then is constant.*

*Proof. *If -dimensional -manifold is Ricci-symmetric, then from (43) we conclude that
It follows that
from which
which is equivalent to
that is,
which proves our assertion.

Since implies that , we can give the following corollary.

Corollary 8. *If an -dimensional -manifold is locally symmetric, then is constant.*

Now, taking the covariant derivation of the both sides of (18) with respect to , we have From the definition of the covariant derivation of Ricci-tensor, we have If an -manifold Ricci symmetric, then Theorem 7 and (43) imply that This leads us to state the following.

Theorem 9. *If an -manifold is Ricci symmetric, then it is an Einstein manifold.*

Corollary 10. *If an -manifold is locally symmetric, then it is an Einstein manifold.*

In this section, an example is used to demonstrate that the method presented in this paper is effective. But this example is a special case of Example 6.1 of [6].

*Example 11. *Now, we consider the 3-dimensional manifold
where denote the standard coordinates in . The vector fields
are linearly independent of each point of . Let be the Lorentzian metric tensor defined by
for . Let be the 1-form defined by for any . Let be the (1,1)-tensor field defined by
Then using the linearity of and , we have ,
for all . Thus for , defines a Lorentzian paracontact structure on .

Now, let be the Levi-Civita connection with respect to the Lorentzian metric , and let be the Riemannian curvature tensor of . Then we have
Making use of the Koszul formulae for the Lorentzian metric tensor , we can easily calculate the covariant derivations as follows:
From the previously mentioned, it can be easily seen that is an -structure on , that is, is an -manifold with and . Using the previous relations, we can easily calculate the components of the Riemannian curvature tensor as follows:
By using the properties of and definition of the Ricci tensor, we obtain
Thus the scalar curvature of is given by
On the other hand, for any , and can be written as and , where and are smooth functions on . By direct calculations, we have
Since and and , we have
This tell us that is an -Einstein manifold.

#### Acknowledgment

The authors would like to thank the reviewers for the extremely carefully reading and for many important comments, which improved the paper considerably.