#### Abstract

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is an -Einstein manifold. We also investigate some properties of curvature tensor, conformal curvature tensor, -curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor with respect to canonical paracontact connection on a para-Sasakian manifold. It is shown that a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection is of constant scalar curvature. We give some characterizations for pseudo-projectively flat para-Sasakian manifolds.

#### 1. Introduction

In 1976, Sato [1] introduced the almost paracontact structure satisfying and on a differentiable manifold. Although the structure is an analogue of the almost contact structure [2, 3], it is closely related to almost product structure (in contrast to almost contact structure, which is related to almost complex structure). It is well known that an almost contact manifold is always odd-dimensional but an almost paracontact manifold defined by Sato [1] could be even dimensional as well. Takahashi [4] defined almost contact manifolds equipped with an associated pseudo-Riemannian metric. In particular he studied Sasakian manifolds equipped with an associated pseudo-Riemannian metric. Also, in 1989, Matsumoto [5] replaced the structure vector field by in an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact structure. It is obvious that in a Lorentzian almost paracontact manifold, the pseudo-Riemannian metric has only signature 1 and the structure vector field is always timelike. These circumstances motivated the authors [6] to associate a pseudo-Riemannian metric, not necessarily Lorentzian, with an almost paracontact structure.

Kaneyuki and Konzai [7] defined the almost paracontact structure on pseudo-Riemannian manifold of dimension and constructed the almost paracomplex structure on . Zamkovoy [8] associated the almost paracontact structure introduced in [7] to a pseudo-Riemannian metric of signature and showed that any almost paracontact structure admits such a pseudo-Riemannian metric which is called compatible metric.

Tanaka-Webster connection has been introduced by Tanno [9] as a generalization of the well-known connection defined by Tanaka [10] and, independently, by Webster [11], in context of CR geometry. In a paracontact metric manifold Zamkovoy [8] introduced a canonical connection which plays the same role of the (generalized) Tanaka-Webster connection [9] in paracontact geometry. In this study we define a canonical paracontact connection on a para-Sasakian manifold which seems to be the paracontact analogue of the (generalized) Tanaka-Webster connection.

In the present paper we study canonical paracontact connection on a para-Sasakian manifold. Section 2 is devoted to preliminaries. In Section 3, we investigate the relation between curvature tensor (resp., Ricci tensor) with respect to canonical paracontact connection and curvature tensor (resp., Ricci tensor) with respect to Levi-Civita connection. In Section 4, conformal curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is obtained. Section 5 contains the expression of -curvature tensor. In Section 6, we study a para-Sasakian manifold satisfying the condition , where is considered as a derivation of the tensor algebra at each point of the manifold, is the curvature tensor, and is the conformal curvature tensor with respect to canonical paracontact connection. In Section 7, we obtain some equations in terms of Ricci tensor on a para-Sasakian manifold satisfying and , respectively. In Section 8 it is proved that a concircularly flat para-Sasakian manifold is of constant scalar curvature. Section 9 is devoted to pseudo-projectively flat para-Sasakian manifolds. In Section 10, we show that a para-Sasakian manifold satisfying with respect to canonical paracontact connection, is either of constant scalar curvature or an -Einstein manifold. Also, it is proved that if the condition holds on a para-Sasakian manifold with respect to canonical paracontact connection, then the scalar curvature is constant. In Section 11, we give some characterizations for para-Sasakian manifolds with canonical paracontact connection satisfying and , respectively. In the last section it is shown that a para-Sasakian manifold on which the condition holds, is either of constant scalar curvature or an -Einstein manifold.

#### 2. Preliminaries

A differentiable manifold of dimension is called almost paracontact manifold with the almost paracontact structure if it admits a tensor field of type , a vector field , a -form satisfying the following conditions [7]: where denotes the identity transformation. Moreover, the tensor field induces an almost paracomplex structure on the paracontact distribution ; that is, the eigendistributions corresponding to the eigenvalues of are both -dimensional. As an immediate consequence of the conditions (2.2) we have

If a -dimensional almost paracontact manifold with an almost paracontact structure admits a pseudo-Riemannian metric such that [8]
then we say that is an *almost paracontact metric manifold* with an *almost paracontact metric structure *, and such metric is called *compatible metric*. Any compatible metric is necessarily of signature .

From (2.4) it can be easily seen that [8] for any . The fundamental -form of is defined by An almost paracontact metric structure becomes a paracontact metric structure if , for all vector fields , where .

For a -dimensional manifold with an almost paracontact metric structure one can also construct a local orthonormal basis. Let be coordinate neighborhood on and any unit vector field on orthogonal to . Then is a vector field orthogonal to both and , and . Now choose a unit vector field orthogonal to , and . Then is also a vector field orthogonal to , , , and and . Proceeding in this way we obtain a local orthonormal basis called a -*basis* [8].

*Remark 2.1. *It is also known that a differentiable manifold has an almost paracontact metric structure if it admits a Riemannian metric such that (see [1]). But in our paper the metric is pseudo-Riemannian and satisfies condition (2.4).

Recall that an almost paracomplex structure [12] on a -dimensional manifold is a tensor field of type (1,1) such that and eigensubbundles , corresponding to the eigenvalue of , respectively, have equal dimensional . The Nijenhuis tensor of , given by
is the obstruction for the integrability of the eigensubbundles , . If , then the almost paracomplex structure is called *paracomplex* or* integrable* [13].

Let be an almost paracontact metric manifold with structure and consider the manifold . We denote a vector field on , by where is tangent to , is the coordinate on , and is a differentiable function on . An almost paracomplex structure on is defined in [14] by
If is integrable we say that the almost paracontact structure is *normal*.

A normal paracontact metric manifold is a *para-Sasakian manifold*. An almost paracontact metric structure on a is * para-Sasakian manifold* if and only if [8]
where and is Levi-Civita connection of .

From (2.10), it can be seen that

Also in a para-Sasakian manifold, the following relations hold [8]: for any vector fields . Here, is Riemannian curvature tensor and is Ricci tensor defined by , where is Ricci operator.

In the following we consider the connection defined by [9] where . If we use (2.11) in (2.17), then we obtain

*Definition 2.2. *We call the connection defined by (2.18) on a para-Sasakian manifold *the canonical paracontact connection on a para-Sasakian manifold*.

Proposition 2.3. *On a para-Sasakian manifold the connection has the following properties:
**
for all .*

*Proof. *Calculation is straightforward by using (2.18).

Furthermore, we define homeomorphisms , , and as follows: for all .

#### 3. Curvature Tensor

Let be a para-Sasakian manifold. The curvature tensor of with respect to the canonical paracontact connection is defined by for any .

By using (2.18) in (3.1) we obtain where is curvature tensor of with respect to Levi-Civita connection .

Let and be curvature tensors of type (0, 4) with respect to Levi-Civita connection and the canonical paracontact connection , respectively, given by for all .

Proposition 3.1. *In a para-Sasakian manifold one has,
**
where .*

*Proof. *Using (3.2) and first Bianchi identity with respect to Levi-Civita connection we obtain
From (2.5) we get (3.4).

From (3.2) we have
It is well known that
By taking into account the previous equations we get (3.5), (3.6), and (3.7), respectively.

Let be a local orthonormal -basis of a para-Sasakian manifold . Then the Ricci tensor and the scalar curvature of with respect to canonical paracontact connection are defined by where , and respectively.

Theorem 3.2. *In a -dimensional para-Sasakian manifold the Ricci tensor and scalar curvature of canonical paracontact connection are given by
**
where and and denote the Ricci tensor and scalar curvature of Levi-Civita connection , respectively. Consequently, is symmetric. *

*Proof. *Using (3.2) and (3.9), we have, for any ,
Since the Ricci tensor of Levi-Civita connection is given by
then (3.15) implies (3.13). Equation (3.14) follows from (3.13).

Also from (3.13), it is obvious that is symmetric.

Corollary 3.3. *If a para-Sasakian manifold is Ricci-flat with respect to canonical paracontact connection, then it is an -Einstein manifold. *

Lemma 3.4. *Let be a para-Sasakian manifold with canonical paracontact connection . Then
**
for all . *

*Proof. *Calculation is straightforward by using (2.2), (2.6), (2.13), and (2.16) in (3.2).

#### 4. Conformal Curvature Tensor

Let be a -dimensional para-Sasakian manifold. The conformal curvature tensor of with respect to canonical paracontact connection is defined by where . By using (3.2), (3.13), and (3.14) in (4.1) Since , then Moreover, from the first Bianchi identity we get

#### 5. -Curvature Tensor

In [15] Pokhariyal and Mishra have introduced a new tensor field, called -curvature tensor field, in a Riemannian manifold and studied its properties.

The -curvature tensor is defined by where and is a Ricci tensor of type . The -curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is defined by

From (3.2) and (3.13) we get Since , we have Furthermore, by using (5.3) and the first Bianchi identity we obtain

#### 6. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying

In this section we consider a para-Sasakian manifold satisfying the condition with respect to canonical paracontact connection. From (2.20) we get for any . Now putting in (6.2), we have Taking inner product with and using (3.18) in (6.3) we get Using (4.2) in (6.4), we obtain Again using (3.2), (3.13), and (3.14) in (6.5) we obtain

Let be an orthonormal basis of the tangent space at any point. Hence by suitable contracting of (6.6) we get

Theorem 6.1. *Let be a para-Sasakian manifold satisfying with respect to canonical paracontact connection . Then
**
for any . *

#### 7. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying and

In this section we consider a para-Sasakian manifold with canonical connection satisfying the condition From (2.21), we get where . Now if we put in (7.2), we have Taking inner product with and from (3.18), we obtain Using (5.3) in (7.4), we have Again using (3.2) and (3.13) in (7.5) we obtain

Let be an orthonormal basis of the tangent space at any point. So by a contraction of (7.6) with respect to and we get

Theorem 7.1. *Let be a para-Sasakian manifold satisfying with respect to canonical paracontact connection . Then
**
for any .*

Now let us consider a para-Sasakian manifold with canonical paracontact connection satisfying the condition From (2.22), we get

Now in (7.10), we have Taking inner product with and from (3.18) and (3.19) we obtain Using (5.3) in (7.12), we have

Let be an orthonormal basis of the tangent space at any point. Hence by suitable contracting (7.13) we get

Theorem 7.2. *Let be a para-Sasakian manifold satisfying with respect to canonical paracontact connection . Then
**
for any .*

#### 8. Concircularly Flat Para-Sasakian Manifold

The concircular curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is defined by for any .

By using (3.2) and (3.14) we obtain from (8.1)

If is a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection, then we have Hence using (2.2) in (8.3) we get Putting in (8.4) and using (2.2) and (2.14) we have From (8.5) we obtain Hence we have the following.

Theorem 8.1. *If a para-Sasakian manifold is concircularly flat with respect to canonical paracontact connection, then it is of constant scalar curvature.*

#### 9. Pseudo-projectively Flat Para-Sasakian Manifold

In 2002, Prasad [16] defined and studied a tensor field on a Riemannian manifold of dimension , which includes projective curvature tensor . This tensor field is known as *pseudo-projective curvature tensor*.

In this section, we study pseudo-projective curvature tensor with respect to canonical paracontact connection in a para-Sasakian manifold and we denote this curvature tensor with .

Pseudo-projective curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is defined by where and and are constants such that , .

If and , then (9.1) takes the form Using (3.2), (3.13) and (3.14) in (9.2), we get

If is a pseudo-projectively flat para-Sasakian manifold with respect to canonical paracontact connection, then for any . Hence using (2.2) in (9.4) we get Now putting in (9.5) and using (2.2), (2.14), and (2.16) we have From (9.6) we get Hence we have the following.

Theorem 9.1. *If a para-Sasakian manifold is pseudo-projectively flat with respect to canonical paracontact connection, then its scalar curvature is constant. *

#### 10. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying and

In this section we firstly consider a para-Sasakian manifold satisfying for any , with respect to canonical paracontact connection.

From (2.23), we have for all . Now by putting in (10.2), we get Using (3.18) in (10.3) we obtain Taking inner product with and from (8.2), we have Again using (3.2) in (10.5) we get

Let be an orthonormal basis of the tangent space at any point. So a contraction of (10.6) with respect to and gives Therefore we have the following.

Theorem 10.1. *Let be a para-Sasakian manifold satisfying the condition , for any . Then either*(i)*, that is, the scalar curvature is constant,* *or*(ii)* is an -Einstein manifold with equation
**for all . *

Now, let us consider a para-Sasakian manifold satisfying where and is the Ricci tensor of with respect to canonical paracontact connection.

From (10.9), for any , we obtain for all . Using (3.18) and (3.19) with (8.2) in (10.10) we get where .

Using (3.13) and (3.14) in (10.11), we get

Let be an orthonormal basis of the tangent space at any point. By a contraction of (10.12) with respect to and we obtain This gives which implies that manifold is of constant scalar curvature.

Theorem 10.2. *If the condition , for all , holds on a para-Sasakian manifold, then its scalar curvature is constant. *

#### 11. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying and

In this section we consider a para-Sasakian manifold with canonical paracontact connection satisfying the condition for all .

From (2.24), we get where . Now we take in (11.2), then we have