Almost <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 9.91493 8.14084" width="9.91493pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg>-</span>Cosymplectic Pseudo Metric Manifolds

Öztürk, Sermin; Öztürk, Hakan

doi:https://doi.org/10.1155/2021/4106025

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion and Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 4106025 | https://doi.org/10.1155/2021/4106025

Almost -Cosymplectic Pseudo Metric Manifolds

Sermin Öztürk¹and Hakan Öztürk²

Academic Editor: Ljubisa Kocinac

Received21 Apr 2021

Accepted25 Jun 2021

Published30 Jul 2021

Abstract

The main purpose of this paper is to study almost -cosymplectic pseudo metric manifold satisfying certain -parallel tensor fields. We first focus on the concept of almost -cosymplectic pseudo metric manifold and its curvature properties. Then, we obtain some results related to the -parallelity of , , and . Moreover, the deformation of almost -Kenmotsu pseudo metric structure is given. We conclude the paper with an illustrative example of almost -cosymplectic pseudo metric manifold.

1. Introduction

Manifolds known as almost contact metric manifolds have been studied in [1–3]. The class of almost contact metric manifolds which are called almost Kenmotsu manifolds is firstly introduced by Kenmotsu. These manifolds appear for the first time in [4], where they have been locally classified. Kenmotsu defined a structure closely related to the warped product which was characterized by tensor equations.

Recently, Kim and Pak have introduced a wide subclass of almost contact metric manifolds called almost -cosymplectic manifolds [5]. The authors investigated canonical foliations of an almost -cosymplectic manifold. Later, most of the research is devoted to this topic [6–10]. However, the classical papers related to almost contact metric manifolds are assumed to have a Riemannian metric, and we notice that the almost contact manifolds furnished with a pseudo Riemannian metric are introduced in [11–14].

On that account, Wang and Liu introduced the geometry of almost Kenmotsu pseudo metric manifolds [12]. They emphasized the analogies and differences in connection with the Riemann metric tensor and obtained certain classification results related to locally symmetry and nullity condition. Also, Naik et al. studied Kenmotsu pseudo metric manifolds. In particular, the authors established necessary and sufficient conditions for Kenmotsu pseudo metric manifolds satisfying certain tensor conditions [13].

Furthermore, Boeckx and Cho studied -parallel contact metric spaces in [15]. They considered a milder condition that is -parallel, i.e.,in contact metric manifolds for all .

In [16], Ghosh et al. studied the -parallelity of the torsion tensor for a contact metric manifold . The torsion tensor field defined asfor any vector fields and on was firstly introduced by Hamilton and Chern [17].

In this paper, we consider the almost -cosymplectic pseudo metric manifold which is a wide subclass of almost contact pseudo metric manifolds. We first give the concept of almost -cosymplectic pseudo metric manifolds and state general curvature properties. We derive several formulas on almost -cosymplectic pseudo metric manifolds. These formulas would enable us to find the geometrical properties of almost -cosymplectic pseudo metric manifolds with -parallel tensor and . We study the -parallelity of the tensor fields and . Next, we obtain some results related to the -parallelity and -cyclic parallelity of the torsion tensor . Moreover, we investigate the deformation of almost -Kenmotsu pseudo metric structure. Finally, we give an illustrative example of almost -cosymplectic pseudo metric manifolds.

2. Preliminaries

Let be a -dimensional differentiable manifold equipped with a triple , where is a type of (1, 1) tensor field, is a vector field, and is a 1-form on such thatwhich implies

A pseudo Riemannian metric on is said to be compatible with the almost contact structure if where .

A smooth manifold furnished with an almost contact structure and a compatible pseudo Riemannian metric is called an almost contact pseudo metric manifold which is denoted by . It is clear that , , and .

On such a manifold, the fundamental 2-form of is defined by for any vector fields on [18]. An almost contact pseudo metric manifold satisfying the conditions and is said to be an almost -Kenmotsu pseudo metric manifold for and . It is well known that the normality of almost contact structure is expressed by the vanishing of the tensor as follows:where is the Nijenhuis tensor of [19].

An almost contact pseudo metric manifold is said to be almost cosymplectic pseudo metric manifold if and , where d is the exterior differential operator.

If we join these two classes, we obtain the notion of an almost -cosymplectic pseudo metric manifold, defined by , for any real number [5]. When an almost -cosymplectic pseudo metric manifold has a normal almost contact structure, we can say that is an -cosymplectic pseudo metric manifold. In this paper, we shall denote by and the Lie algebra of all tangent vector fields on and the Levi Civita connection of pseudo Riemannian metric , respectively.

3. Certain Properties

In this section, we give the basic relations on almost -cosymplectic pseudo metric manifolds.

Proposition 1. Let be an almost contact metric manifold and be the Riemannian connection. Then, the following equations are held [3]:Here, denotes the cyclic sum over the vector fields , and [1].

Lemma 1. Let be an almost contact pseudo metric manifold. Then, the following equation is held:for any tangent vector fields where are defined byrespectively. Here, denotes the Lie derivative in the direction of [20].

Proposition 2. Let be an almost -cosymplectic pseudo metric manifold. Then, we havefor any tangent vector fields .

Proof. Considering the Koszul formula (11), we haveIn view of (6), (7), and (10) for , we deduceTaking into account of (12) and (13), we getThen, making use of (22) in (21) and (20) reduces toHere, if we choose the symmetric (1, 1)-type tensor field as follows:for any vector fields , and Equation (24) takes the formwhich completes the proof of (15). From (14) and (15), the first equation of (16) is obvious. Moreover, using (20) for and putting and in (13), we obtain the following:This means that for any nonzero vector field . Now, considering the sum of and for any vector field , we haveEquation (27) shows that the sum of and vanishes identically. In addition, from (8), we getwhere . Thus, we can complete the proof of (18). Also, from (14) and (15), we can easily obtain (19). Here, if is defined by , then we have , and .
Now, we investigate the curvature properties of almost contact pseudo metric manifolds. First, we have the following propositions.

Proposition 3. Let be an almost -cosymplectic pseudo metric manifold. Then, we havefor any tangent vector fields .

Proof. Making use of the Riemannian curvature tensor and (15), we obtain (29) such that .

Proposition 4. Let be an almost -cosymplectic pseudo metric manifold. Then, the following relations are held:for any tangent vector fields .

Proof. By the hypothesis, using (29) with and considering the following equations:we obtain (30). Applying to (30) and remarking that , we get (31). Also, with the help of (30) for , we haveThen, we getwhich reduces to (32) where .
Now, we may take a local orthonormal -basis as follows:From (29) and the Ricci curvature tensor, we havewhere . It follows thatThen, we havesuch thatSince , we deduceThus, the proof of (33) completes. Moreover, putting in (33), we obtain (34) where . This proof can also be given in another way. Consider the local orthonormal -basis on . The sectional curvatures of nondegenerate planes spanned by and , respectively, are defined aswhere for all indices and is the Jacobi operator defined by . Thus, we haveand from (46) and (47), it follows thatIt is well known thatFrom Equation (50), we obtainwhich completes the proof.

4. Main Results

In this section, we consider some certain parallel tensor conditions on almost -cosymplectic pseudo metric manifolds. Also, we study the deformation of almost -Kenmotsu pseudo metric manifolds with . Firstly, we study the -parallelity of the tensor fields and on almost -cosymplectic pseudo metric manifolds. As we know that we can take where is tangentially part of and is the normal part of . So, the symmetric (1, 1)-type tensor field on a Riemannian manifold is said to be a -parallel tensor if it holds the following:for all tangent vectors , , and orthogonal to [15].

Proposition 5. Let be an almost -cosymplectic pseudo metric manifold. If satisfies the -parallelity condition, then we havefor any tangent vector fields .

Proof. Assume that is -parallel. From (52), we havefor any . It follows thatFrom Equation (55), we deduceThus, it completes the proof.

Proposition 6. Let be an almost -cosymplectic pseudo metric manifold. If satisfies the -parallelity condition, then we havefor any .

Proof. By the hypothesis, we suppose that is -parallel. Then, we havefor any . By a straightforward computation, we obtainWith the help of (15) and (17), Equation (59) reduces toFrom Equation (60) and , (57) is easily seen. Then, the proof is completed.

Theorem 1. An almost -cosymplectic pseudo metric manifold with -parallel tensor holds the following equation:for any where is the Jacobi operator with respect to .

Proof. Making use of (29), we haveThen, simplifying Equation (62), we obtainwhich is desired result.

Theorem 2. Let be an almost -cosymplectic pseudo metric manifold with -parallel tensor . Then, is the eigenvector of Ricci operator on .

Proof. Let be an orthonormal basis of the tangent space at any point. Taking the inner product of both sides of (61) with respect to and contracting (61) for with , we havefor any . This means thatThus, it completes the proof.

Theorem 3. Let be an almost -cosymplectic pseudo metric manifold. If satisfies the -parallelity condition and , then the eigenvalues of are constant.

Proof. Let be an eigen unit vector field such that where is an eigen function corresponding to the vector field . Then, (53) can be written asfor . Also, we haveTaking into account of (66) and (67), we also getFurthermore, since , we obtainFollows from (68) and (69), we have . Thus, it completes the proof.

Proposition 7. Let be an almost -cosymplectic pseudo metric manifold. Then, the torsion tensor field holds the following:for any .

Proof. From the definition of , we getwhich completes the proof.

Proposition 8. Let be an almost -cosymplectic pseudo metric manifold. If is -parallel, then we havefor any .

Proof. The hypothesis is essentially same asfor all tangent vectors orthogonal to . Putting and using the definition of , we obtainIt follows thatPutting in (75), we haveAlso, it is noted thatFinally, taking into account of (74)–(77), (72) holds. Then, we complete the proof.

Theorem 4. Let be an almost -cosymplectic pseudo metric manifold. If is -parallel, then is the eigenvector of Ricci operator on .

Proof. From (72), we haveSimplifying Equation (78), we getWith the help of (29), (31), and (79), we obtainBy a direct calculation, the desired result is achieved.

Theorem 5. Let be an almost -cosymplectic pseudo metric manifold. If is cyclically -parallel, then is the eigenvector of Ricci operator on .

Proof. According to the hypothesis, it means thatfor all tangent vectors orthogonal to . Simplifying Equation (81), we obtainwhere denotes the cyclic sum over . It follows thatContracting (83) with respect to and , we haveFrom (33), (84) reduces toFrom (85), the proof is clearly seen.
Now, we investigate the deformation of almost contact pseudo metric manifold. Here, our main goal is to study the relationship between pseudo Riemannian metrics with different signatures associated to the same almost contact pseudo metric manifold.
Let be an almost contact pseudo metric structure associated to a compatible pseudo Riemannian metric where on a smooth manifold . With the help of [20], we have the following pseudo Riemannian metric formula:for any where , . This means that (86) is still compatible pseudo metric with the same almost contact pseudo metric structure .
Thus, we give the following results. Here, we denote by and as the semi Riemannian connection and the curvature tensor of on almost -Kenmotsu pseudo metric manifold, respectively.

Proposition 9. Let be an almost -Kenmotsu pseudo metric manifold. Then, is also an almost -Kenmotsu pseudo metric manifold in the sense of (86).

Proof. According to (86), we denote by the fundamental 2-form with respect to and then we have with . Furthermore, from , we obtain for any . Thus, it completes the proof.

Proposition 10. Let be an almost -Kenmotsu pseudo metric manifold and the pseudo Riemannian metric given by (86). Then, the following equations are held:for any .

Proof. According to the Koszul formula, we haveUsing (86) and (89), we getwhere .
On the other hand, making use of (86) in (90), it follows thatThen, taking into account of (90) and (91), we obtain (87). Moreover, using the definition of Riemannian curvature tensor and (87), we haveFrom (92), we obtainFinally, substituting (15) into (92), it reduces to (88). Then, we complete the proof.

5. An Example

Consider the manifold such that where are the standard coordinates in . The vector fields are

Moreover, the following equations hold:

From Equation (95), there exists an almost contact pseudo metric structure on . In order to check, whether it is almost -cosymplectic pseudo metric or not, we verify the condition . On the other hand, all s vanish except for . Hence, we have

It follows that

Since , we have

Here, it is noted that . Therefore, is an -cosymplectic pseudo metric manifold.

6. Conclusion and Discussion

Since Kenmotsu introduced the notion of Kenmotsu structures in [4] which can be regarded as an analogy of almost contact metric structures, numerous authors studied such structures under some certain conditions [1–3, 7–10]. In particular, Dileo and Pastore studied certain parallel tensors, local symmetry, and nullity distribution on almost Kenmotsu manifolds [21]. Also, Kim and Pak introduced a new definition which combines almost Kenmotsu and almost cosymplectic manifold called almost cosymplectic manifold [5].

On the other hand, a systematic study of almost -cosymplectic pseudo manifolds has not been undertaken yet. The main purpose of this paper is to contribute to future studies on this subject. We introduce the geometry of almost -cosymplectic pseudo metric manifolds and underline the differences and similarities in the sense of Riemannian metric tensor. For this purpose, many results are given in the third and fourth sections. This study will shed light on our future investigations. Our further studies will be devoted to nullity distributions, local symmetry, semisymmetric conditions, and the other curvature tensor fields on almost -cosymplectic pseudo metric manifolds.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this study.

Acknowledgments

This work was supported by Afyon Kocatepe University Scientific Research Coordination Unit with the project no. 17.FEN.BİL.11.

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Copyright © 2021 Sermin Öztürk and Hakan Öztürk. 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.

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