Journal of Mathematics
Volume 2015 (2015), Article ID 728298, 6 pages
http://dx.doi.org/10.1155/2015/728298

Research Article

On -Symmetric -Paracontact Metric Manifolds

D. G. Prakasha and K. K. Mirji

Department of Mathematics, Karnatak University, Dharwad 580 003, India

Received 30 July 2015; Revised 11 October 2015; Accepted 13 October 2015

Academic Editor: Mauro Nacinovich

Copyright © 2015 D. G. Prakasha and K. K. Mirji. 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 notions of -symmetric, 3-dimensional locally -symmetric, -Ricci symmetric, and 3-dimensional locally -Ricci symmetric -paracontact metric manifolds have been introduced and properties of these structures have been discussed.

1. Introduction

The study of paracontact geometry was initiated by Kaneyuki and Williams [1]. A systematic study of paracontact metric manifolds and their subclasses were started out by Zamkovoy [2]. Since then, several geometers studied paracontact metric manifolds and obtained various important properties of these manifolds ([3–10], etc.). The geometry of paracontact metric manifolds can be related to the theory of Legendre foliations. In [11], the authors introduced the class of paracontact metric manifolds for which the characteristic vector field belongs to the -nullity condition (or distribution) for some real constants and . Such manifolds are known as -paracontact metric manifolds. If , then the notion of -nullity distribution reduces to -nullity distribution. A paracontact metric manifold with belonging to -nullity distribution is called -paracontact metric manifold.

In [12], Takahashi introduced the notion of locally -symmetric Sasakian manifold as a weaker version of local symmetry of such manifolds. In the context of contact geometry, the notion of -symmetry was introduced and studied by Boeckx et al. [13] with examples. In ([14, 15]), they studied the notion of -symmetry and discussed several examples for Kenmotsu manifolds and almost contact metric manifolds of dimension 3. In [16, 17], S. S. Shukla and M. K. Shukla, studied -Ricci symmetric Kenmotsu manifolds and -symmetric para-Sasakian manifolds.

In the present work, we study -symmetry and Ricci -symmetry on -paracontact metric manifolds. In Section 2, we give a brief account of the -paracontact metric manifolds. In Section 3, we study the properties of -symmetric -paracontact metric manifolds. Section 4 deals with 3-dimensional locally -symmetric -paracontact metric manifolds. In this section, we prove that scalar curvature is constant. Section 5 is devoted to studying the Ricci -symmetric -paracontact metric manifolds. Finally, we study the properties of 3-dimensional locally Ricci -symmetric -paracontact metric manifolds in Section 6.

2. Preliminaries

A -dimensional smooth manifold has an almost paracontact structure if it admits a tensor field of type , a vector field , a 1-form , and a Riemannian metric satisfying the following conditions ([2, 18]):

for every vector field on .

In a paracontact metric manifold , we define a tensor field by , where denotes the operator of Lie differentiation. Then, is symmetric and satisfies

If denotes the Levi-Civita connection of , then we have the following relation:

A paracontact metric manifold is said to be a -space if its curvature tensor satisfies

for all tangent vector fields , where are smooth functions on .

Here, the characteristic vector field belongs to the -nullity distribution. A paracontact metric manifold with belonging to -nullity distribution is called a -paracontact metric manifold. In particular, if , then the notion of -nullity distribution reduces to -nullity distribution. A paracontact metric manifold such that belongs to -nullity distribution is called -paracontact metric manifold. Then, curvature tensor reduces to the following form:

For -paracontact metric manifold , the following identities hold:

for any vector fields on , where and denote the Ricci operator and Ricci tensor of , respectively.

-paracontact metric manifold is called an Einstein manifold if it satisfies

where is any scalar.

Definition 1. -paracontact metric manifold is said to be -symmetric if

for arbitrary vector fields .

Definition 2. -paracontact metric manifold is said to be locally -symmetric if

for all vector fields orthogonal to .

3. -Symmetric -Paracontact Metric Manifolds

Let us consider -symmetric -paracontact metric manifold. Then, by virtue of (1) and (13), we have

Taking the inner product of (15) by , we have

Let , , be an orthonormal basis of the tangent space at any point of the manifold. Then, putting in (16) and taking summation over , , we have

Considering the second term of (17) and setting , we have

Next,

Since is an orthonormal basis, . Using (6), we have

Using (20) in (19), we have

Since , we have

Using (22) in (21), we have

Using (4) in (23), we have

Putting in (17) and using (24), it follows that

We know that

Using (4), (9), (10), and (25) in (26), we have

Putting in (27) and using (1), (2), (3), and (10), we have

Again, putting in (28) and using (1) and (7), we obtain

By virtue of (28) and (29), we have

Thus, we can state the following theorem.

Theorem 3. A -dimensional -symmetric -paracontact metric manifold is an Einstein manifold.

4. Three-Dimensional Locally -Symmetric -Paracontact Metric Manifolds

For a three-dimensional semi-Riemannian manifold, the conformal curvature tensor is given by

for arbitrary vector fields .

If , then (31) reduces to the following form:

Putting in (32) and using (6), we get

Again, putting in (33) and using (11), we get

Taking the inner product of (34) with , we obtain

Using (34) and (35) in (32), we have

where is Riemannian curvature tensor on the 3-dimensional -paracontact metric manifold.

Taking the covariant differentiation of (36) with respect to , we have

Applying to both sides of (37), we have

Now, taking orthogonal to and using (14), we get

Hence, we can state the following theorem.

Theorem 4. A 3-dimensional -paracontact metric manifold is locally -symmetric if the scalar curvature tensor of is constant.

5. -Ricci Symmetric -Paracontact Metric Manifolds

Definition 5. -paracontact metric manifold is said to be -Ricci symmetric if the Ricci operator satisfies

for all vector fields on .
If are orthogonal to , then manifold is said to be locally -Ricci symmetric.

Using (1) in (40), we have

Taking the inner product of (41) with , we have

Further simplification of (42) gives the following:

Putting in (43), we have

Using (4), (10), and (11) in (44), we have

Putting in (45), we have

Again, putting , in (46) and then using (1), (3), and (10), we have

Replace in (47), and using (1), (7), and symmetric property of , we have

By virtue of (47) and (48), we have

Hence, we can state the following theorem.

Theorem 6. A -dimensional -paracontact metric manifold is -Ricci symmetric if is an Einstein manifold.

6. Three-Dimensional -Ricci Symmetric -Paracontact Metric Manifolds

On a 3-dimensional -paracontact metric manifold, the Ricci operator is given by (34).

Now, taking the covariant differentiation of (34) with respect to , we have

Applying to both sides of (50), we have

Taking orthogonal to in (51), we get the following form:

In view of the above equation, we are able to state the following theorem.

Theorem 7. A 3-dimensional -paracontact metric manifold is locally -Ricci symmetric if the scalar curvature tensor of is constant.

7. Example of 3-Dimensional Locally -Symmetric -Paracontact Metric Manifolds with

We consider the manifold with the usual cartesian coordinates . The vector fields

are linearly independent at each point of . We can compute

We define the semi-Riemannian metric as the nondegenerate one, whose only nonvanishing components are , and the 1-form as , which satisfies , . Let be the -tensor field defined by , , and . Then,

Therefore, is a paracontact metric structure on .

Moreover, , , and . Hence, and, given , if and if .

Let be the Levi-Civita connection. Using the properties of paracontact metric structure and Koszul’s formula

we can compute

Hence, is -paracontact metric manifold with .

Using the following definition of Riemannian curvature tensor

we obtain

From this, it follows that for all vector fields , , and are orthogonal to . Thus, the three-dimensional -paracontact metric manifold with is locally -symmetric.

Also from the above expressions for the curvature tensor, we obtain that the scalar curvature tensor is constant. Therefore, from Theorem 4, it follows that the manifold under consideration is locally -symmetric.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are thankful to the Referee and the Editorial Board for the valuable comments and suggestions that helped them to improve the paper.

References

S. Kaneyuki and F. L. Williams, “Almost paracontact and parahodge structures on manifolds,” Nagoya Mathematical Journal, vol. 99, pp. 173–187, 1985.

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

D. V. Alekseevsky, C. Medori, and A. Tomassini, “Maximally homogeneous para-CR manifolds,” Annals of Global Analysis and Geometry, vol. 30, no. 1, pp. 1–27, 2006.

D. V. Alekseevsky, V. Cortés, A. S. Galaev, and T. Leistner, “Cones over pseudo-Riemannian manifolds and their holonomy,” Journal für die Reine und Angewandte Mathematik, vol. 2009, no. 635, pp. 23–69, 2009.

G. Calvaruso and D. Perrone, “Geometry of -paracontact metric manifolds,” http://arxiv.org/abs/1307.7662

G. Calvaruso, “Homogeneous paracontact metric three-manifolds,” Illinois Journal of Mathematics, vol. 55, no. 2, pp. 697–718, 2011.

S. Erdem, “On almost (para)contact (hyperbolic) metric manifolds and harmonicity of -holomorphic maps between them,” Houston Journal of Mathematics, vol. 28, no. 1, pp. 21–45, 2002.

S. Ivanov, D. Vassilev, and S. Zamkovoy, “Conformal paracontact curvature and the local flatness theorem,” Geometriae Dedicata, vol. 144, no. 1, pp. 79–100, 2010.

C. Murathan and I. K. Erken, “The harmonicity of the Reeb vector field on paracontact metric 3-manifolds,” Pacific Journal of Mathematics, vol. 234, no. 2, pp. 325–344, 2008.

D. G. Prakasha and K. K. Mirji, “On (k, μ)-paracontact metric manifolds,” General Mathematics Notes, vol. 25, no. 2, pp. 68–77, 2014.

B. C. Montano, I. K. Erken, and C. Murathan, “Nullity conditions in paracontact geometry,” Differential Geometry and Its Application, vol. 30, no. 6, pp. 665–693, 2012.

T. Takahashi, “Sasakian -symmetric spaces,” Tohoku Mathematical Journal, vol. 29, no. 1, pp. 91–113, 1977.

E. Boeckx, P. Bueken, and L. Vanhecke, “-symmetric contact metric spaces,” Glasgow Mathematical Journal, vol. 41, no. 3, pp. 409–416, 1999.

U. C. De, “On -symmetric Kenmotsu manifold,” International Electronic Journal of Geometry, vol. 1, no. 1, pp. 33–38, 2008.

U. C. De, A. Yildiz, and A. F. Yalınız, “Locally -symmetric normal almost contact metric manifolds of dimension 3,” Applied Mathematics Letters, vol. 22, no. 5, pp. 723–727, 2009.

S. S. Shukla and M. K. Shukla, “On φ-Ricci symmetric Kenmotsu manifolds,” Novi Sad Journal of Mathematics, vol. 39, no. 2, pp. 89–95, 2009.

S. S. Shukla and M. K. Shukla, “On -symmetric para-sasakian manifolds,” International Journal of Mathematical Analysis, vol. 4, no. 16, pp. 761–769, 2010.

M. Manev and M. Staikova, “On almost paracontact Riemannian manifolds of type (n, n),” Journal of Geometry, vol. 72, no. 1-2, pp. 108–114, 2001.