International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 395462, 24 pages

doi:10.1155/2012/395462

Research Article

Some Curvature Conditions on a Para-Sasakian Manifold with Canonical Paracontact Connection

Bilal Eftal Acet,¹ Erol Kılıç,² and Selcen Yüksel Perktaş¹

¹Department of Mathematics, Arts and Science Faculty, Adıyaman University, 02040 Adıyaman, Turkey

²Department of Mathematics, Arts and Science Faculty, İnönü University, 44280 Malatya, Turkey

Received 2 October 2012; Accepted 2 December 2012

Academic Editor: Jerzy Dydak

Copyright © 2012 Bilal Eftal Acet 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.

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

Taking inner product with and by using (3.18) and (9.3) we obtain

Again using (3.2) in (11.4), we have

Let be an orthonormal basis of the tangent space at any point. Hence by suitable contracting of (11.5) we get

Therefore we have the following.

Theorem 11.1. Let be a para-Sasakian manifold. If holds on , then either

(i) scalar curvature is constant with equation  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 (11.8) we obtain

Using (3.18) and (3.19) with (9.3) in (11.9) we have

where .

Using (3.13) and (3.14) in (11.10) we get

Let be an orthonormal basis of the tangent space at any point. So a contraction of (11.11) with respect to and gives

which implies that .

So we have the following.

Theorem 11.2. If the condition , for all , holds on a para-Sasakian manifold, then the scalar curvature of the manifold is constant.

12. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying

The projective curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is defined by

where . By using (3.2) and (3.13), from (12.1) we obtain

Let us consider the equation

holds on a para-Sasakian manifold. From (2.25), we get

for all . Using (12.2) with (3.18) and (3.19) we obtain

Taking inner product with and put , by using (3.19) we get

Using (9.3) in (12.6), we obtain

where .

From the last equation, with (3.13) we get

Hence by suitable contracting of (12.8), we obtain

Hence we have the following.

Theorem 12.1. Let be a -dimensional para-Sasakian manifold. If the condition holds on , then either

(i) scalar curvature is constant with equation  or(ii) is an -Einstein manifold with equation

for all .

Acknowledgments

This paper was supported by Adıyaman University, under Scientific Research Project no. FEFBAP/2012-005. The authors thank the referee for useful suggestions and remarks for the revised version.

References

I. Sato, “On a structure similar to the almost contact structure I.,” Tensor N. S., vol. 30, pp. 219–224, 1976.

S. Sasaki, “On differentiable manifolds with certain structures which are closely related to almost contact structure I,” The Tohoku Mathematical Journal, vol. 12, no. 2, pp. 459–476, 1960.

D. Blair, Contact Manifolds in Riemannian Geometry, vol. 509 of Lectures Notes in Mathematics, Springer, Berlin, Germany, 1976.

T. Takahashi, “Sasakian manifold with pseudo-Riemannian metric,” Tohoku Mathematical Journal, vol. 21, no. 2, pp. 644–653, 1969.

K. Matsumoto, “On Lorentzian paracontact manifolds,” Bulletin of the Yamagata University—Natural Science, vol. 12, no. 2, pp. 151–156, 1989.

M. M. Tripathi, E. Kılıç, S. Y. Perktaş, and S. Keleş, “Indefinite almost paracontact metric manifolds,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 846195, 19 pages, 2010.

S. Kaneyuki and M. Konzai, “Paracomplex structure and affine symmetric spaces,” Tokyo Journal of Mathematics, vol. 8, pp. 301–318, 1985.

S. Zamkovoy, “Canonical connections on paracontact manifolds,” Annals of Global Analysis and Geometry, vol. 36, no. 1, pp. 37–60, 2009.

S. Tanno, “Variational problems on contact Riemannian manifolds,” Transactions of the American Mathematical Society, vol. 314, no. 1, pp. 349–379, 1989.

N. Tanaka, “On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connection,” Japanese Journal of Mathematics, vol. 2, pp. 131–190, 1976.

S. M. Webster, “Pseudo-Hermitian structures on a real hypersurfaces,” Journal of Differential Geometry, vol. 13, pp. 25–41, 1979.

P. Libermann, “Sur le probleme d'equivalence de certaines structures infinitesimales,” Annali di Matematica Pura ed Applicata, vol. 36, pp. 27–120, 1954.

K. Yano, “Affine connexions in almost product spaces,” Kodai Mathematical Seminar Reports, vol. 11, pp. 1–24, 1959.

S. Kaneyuki and F. L. Willams, “Almost paracontact and parahodge structure on manifolds,” Nagoya Mathematics Journal, vol. 99, pp. 173–187, 1985.

G. P. Pokhariyal and R. S. Mishra, “The curvature tensor and their relativistic significance,” Yokohama Mathematical Journal, vol. 18, pp. 105–108, 1970.

B. Prasad, “A Pseudo-Projective curvature tensor on a Riemannian manifold,” Bulletin of Calcutta Mathematical Society, vol. 94, no. 3, pp. 163–166, 2002.