On -Symmetric -Paracontact Metric Manifolds

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 satisfiesIf 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 satisfiesfor 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 iffor arbitrary vector fields .

Definition 2. -paracontact metric manifold is said to be locally -symmetric iffor 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 haveTaking the inner product of (15) by , we haveLet , , be an orthonormal basis of the tangent space at any point of the manifold. Then, putting in (16) and taking summation over , , we haveConsidering the second term of (17) and setting , we haveNext,Since is an orthonormal basis, . Using (6), we haveUsing (20) in (19), we haveSince , we haveUsing (22) in (21), we haveUsing (4) in (23), we havePutting in (17) and using (24), it follows thatWe know thatUsing (4), (9), (10), and (25) in (26), we havePutting in (27) and using (1), (2), (3), and (10), we haveAgain, putting in (28) and using (1) and (7), we obtainBy virtue of (28) and (29), we haveThus, 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 byfor arbitrary vector fields .

If , then (31) reduces to the following form:Putting in (32) and using (6), we getAgain, putting in (33) and using (11), we getTaking the inner product of (34) with , we obtainUsing (34) and (35) in (32), we havewhere is Riemannian curvature tensor on the 3-dimensional -paracontact metric manifold.

Taking the covariant differentiation of (36) with respect to , we haveApplying to both sides of (37), we haveNow, taking orthogonal to and using (14), we getHence, 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 satisfiesfor all vector fields on .
If are orthogonal to , then manifold is said to be locally -Ricci symmetric.

Using (1) in (40), we haveTaking the inner product of (41) with , we haveFurther simplification of (42) gives the following:Putting in (43), we haveUsing (4), (10), and (11) in (44), we havePutting in (45), we haveAgain, putting , in (46) and then using (1), (3), and (10), we haveReplace in (47), and using (1), (7), and symmetric property of , we haveBy virtue of (47) and (48), we haveHence, 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 haveApplying to both sides of (50), we haveTaking 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 fieldsare linearly independent at each point of . We can computeWe 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 formulawe 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.