<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06576014pt" id="M1" height="7.93416pt" version="1.1" viewBox="-0.0657574 -7.8684 10.2422 7.93416" width="10.2422pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-43" d="M471 153C471 170 463 194 452 212C400 220 373 229 322 255C373 281 400 290 452 298C463 316 471 339 471 357C456 366 431 371 410 370C377 329 356 310 308 279C311 336 317 364 336 413C326 432 310 451 294 459C279 451 262 432 252 413C271 364 277 336 280 279C232 310 211 329 178 370C157 371 132 367 117 357C117 340 125 316 136 298C188 290 215 281 266 255C215 229 188 220 136 212C125 194 117 171 117 153C132 144 157 139 178 140C211 181 232 200 280 231C277 174 271 146 252 97C262 78 278 59 294 51C309 59 326 78 336 97C317 146 311 174 308 231C356 200 377 181 410 140C431 139 456 143 471 153Z"/></g></svg>-</span>Ricci Tensor on <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="M2" 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 Manifolds

Amruthalakshmi, M. R.; Prakasha, D. G.; Bin Turki, Nasser; Unal, Inan

doi:https://doi.org/10.1155/2022/7939654

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Theory and Applications of Riemannian Submersions

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Volume 2022 | Article ID 7939654 | https://doi.org/10.1155/2022/7939654

-Ricci Tensor on -Cosymplectic Manifolds

M. R. Amruthalakshmi,¹D. G. Prakasha,¹Nasser Bin Turki,²and Inan Unal³

Academic Editor: Meraj Ali Khan

Received19 Dec 2021

Accepted28 Jan 2022

Published21 Feb 2022

Abstract

In this paper, we study α-cosymplectic manifold admitting -Ricci tensor. First, it is shown that a -Ricci semisymmetric manifold is -Ricci flat and a -conformally flat manifold is an -Einstein manifold. Furthermore, the -Weyl curvature tensor on has been considered. Particularly, we show that a manifold with vanishing -Weyl curvature tensor is a weak -Einstein and a manifold fulfilling the condition is -Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold admitting -Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting -Ricci soliton and almost -Ricci soliton are drawn.

1. Introduction

In the last few years, theory of almost contact geometry and related topics are an active branch of research due to elegant geometry and applications to physics. Nowadays, many attentions have been drawn towards the study of almost cosymplectic manifolds which are a special class of almost contact manifolds. This notion was initiated by Goldberg and Yano [1], in 1969, and then, a very systematic approach for the study of almost cosymplectic manifolds was carried forward by many geometers. A smooth manifold of -dimension with the condition for a closed 1-form is a cosymplectic manifold. A simple example of almost cosymplectic manifolds is given by the products of almost Kaehler manifolds and the real line or the circle . At this moment, we refer the studies [2–5] and the references therein for a vast and exhaustive survey of the results on almost cosymplectic manifolds.

A new concept of the Ricci tensor named as -Ricci tensor has been defined by Tachibana [6] and Hamada [7] in complex geometry. Similar to a complex case, the -Ricci tensor of an almost contact metric manifold has been defined as follows:for all , where is the Riemannian curvature tensor. Naturally, Hamada also considered the notion of -Einstein manifold. An Hermitian manifold is -Einstein if we have , where is a constant. Also, in the same study of Hamada, a classification of -Einstein hypersurfaces was given. On the other hand, for an extension of Hamada’s work, we refer to Ivey and Ryan [8]. The concept of the -Ricci tensor has been studied in contact case. Venkatesha and his group ([9, 10]) recently studied some of the curvature properties on Sasakian manifold and contact metric generalized -space form using the -Ricci tensor.

In this study, the -Ricci tensor within the framework of -cosymplectic manifolds has been studied. In Section 2, we recall some basic formulas and results concerning -cosymplectic manifold and -Ricci tensor, which will be useful in further sections. An -cosymplectic manifold satisfying -Ricci semisymmetric and -conformally flat conditions are studied in Section 3 and shown that a -conformably flat -cosymplectic manifold is -Einstein and a -Ricci semisymmetric -cosymplectic manifold is -Ricci flat. In next section, the -Weyl curvature tensor has been studied in the background of -cosymplectic manifold, and several consequences are noticed. In the last section, we studied a special type of metric called -Ricci soliton. Here, we have proved some important results of -cosymplectic manifold admitting -Ricci soliton.

2. Preliminaries

Here, we are going to recall some general facts on -cosymplectic manifolds which are relevant to our work.

An almost contact metric manifold of -dimension is a -tuple with the following resources [11].for a tensor field , a characteristic vector field , a 1-form is dual of , and is a Riemannian metric. It is easily seen that

It is well known that the fundamental 2-form is defined by on .

For an almost contact metric manifold , we have the following classifications ([12, 13]):(1)If , then is a contact metric manifold(2)If and , then is an almost cosymplectic manifold [1](3)If and , then is an almost -Kenmotsu manifold for a nonzero scalar

In the contact geometry, the notion is normality that is a contact analogue of the integrability of an almost complex structure. An almost cosymplectic metric manifold being normal, if we have which is the Nijenhuis tensor of the tensor field , is defined byfor all . A normal almost cosymplectic manifold is a cosymplectic manifold.

Almost cosymplectic manifolds have been defined by Kim and Pak [14] by combining an almost -Kenmotsu and almost cosymplectic structures by the following formula:for a constant . On an -cosymplectic manifold, we havewhere denotes the Riemannian connection. From (6), it is easy to see thatand

On an -cosymplectic manifold of dimension , the following relationships are valid:where and are the curvature and Ricci tensors, respectively.

By the following lemma, we obtain some derivational features of -cosymplectic manifold.

Lemma 1. On an cosymplectic manifold of dimension , we have

Proof. Note that (11) implies , for defined by . Differentiating this along and using (7), we get (12). Next, differentiation of (10) with respect to givesLet be a local basis on . Replacing in the foregoing equation with summing over givesUsing second Bianchi’s identity leads toBy considering (16) in (17) and then with the help of (12), we concludewhich proves (13). Finally, contraction of (13) gives (14).

From Riemannian geometry, the covariant derivative of a -type of tensor field is given byfor all , where is stated for the divergence [15].

By following descriptions, we present some classification facts which come from the Ricci tensor and have been stated.(1)An cosymplectic manifold is called by weak -Einstein if we have for some function , where , and is defined by In other words, denotes the symmetric part of . If is constant, then is called -Einstein [16].(2) is called near quasi-Einstein manifold if the Ricci tensor is of the form where and are the nonzero scalars and is a nonzero tensor [17].(3) is called an -Einstein manifold if we have where and are the constants [18].

By decomposition of Riemannian curvature tensor , the Weyl conformal curvature tensor has been obtained in this way:for all [15]. It is noted that, Weyl conformal curvature tensor vanishes whenever the metric is conformally identical to a flat metric, and it is one of the important curvature properties on a manifold.

3. -Ricci Tensor on -Cosymplectic Manifold

We are in a situation to confer the equation of the -Ricci tensor in the framework of -cosymplectic manifolds and then study its various properties. In [19], authors derived the expression of the -Ricci tensor on -cosymplectic manifold which is of the following form:for all

Note that is not symmetric. By contraction of (25), the -scalar curvature is specified by

If the -Ricci tensor is a constant multiple of the Riemannian metric , then we say that the manifold is -Einstein. Moreover, the -scalar curvature is not constant on a -Einstein manifold.

3.1. -Ricci Semisymmetric -Cosymplectic Manifolds

An -cosymplectic manifold satisfying the condition for all is called Ricci semi symmetric, where acts as a derivation on . This notion was introduced by Mirjoyan [20] for Riemann spaces and then studied by many authors. Analogous to this, an -cosymplectic manifold is called -Ricci semisymmetric if its -Ricci tensor satisfies the condition .

Theorem 1. If a -dimensional -cosymplectic manifold is -Ricci semisymmetric, then is -Ricci flat. Moreover, it is an -Einstein manifold, and the Ricci tensor is to be exhibited as

Proof. Let us consider -Ricci semisymmetric -cosymplectic manifold . Then, condition is equivalent toPutting in (28) and then recalling (9), we haveIt is well known that . Making use of this in (29), we findAgain, plugging by in (30) shows that is -Ricci flat, that is, . Moreover, in view of (25) and (30), we have the required result.

3.2. -Conformally Flat -Cosymplectic Manifolds

An -cosymplectic manifold is said to be -conformally flat if we havefor all . Sasakian manifolds which are -conformally flat have been studied in [21]. In the following, we study a -conformally flat -cosymplectic manifold.

Theorem 2. An -conformally flat -cosymplectic manifold is -Einstein manifold. Moreover, is weak -Einstein.

Proof. Assume that an -cosymplectic manifold is -conformally flat. So, it is easy to see that carry if and only if . Hence, -conformally flat meansFor a local orthonormal basis of with , if we put in (32) and sum up with respect to , then we obtainand therefore,From (9), we obtain , and hence, from (34), we getThen, from (35), it follows from (32) thatIn an -cosymplectic manifold, in view of (3) and (9) for all , we can verify thatTaking instead of in (37), respectively, and making use of (36), we obtainBy the definition of , direct computation yieldswhere reveals that is -Einstein. In view of (3), we haveThus, , and hence, it is weak -Einstein. This completes the proof.

Next, for a constant scalar curvature of , in view of (40), we state the following.

Corollary 1. A -conformally flat -cosymplectic manifold of constant scalar curvature is a --Einstein manifold.

In an -cosymplectic manifold, the -Ricci tensor is given by (25), and so in view of (40), we state the following.

Corollary 2. A -conformally flat -cosymplectic manifold is -Einstein.

Furthermore, an -cosymplectic manifold is called to have the -parallel Ricci tensor if its Ricci tensor satisfies the condition . This notion was introduced in 1976 by Kon [22] in the framework of Sasakian manifolds and then studied by many authors. Analogous to this notion, we state the following:

Definition 1. An -cosymplectic manifold is said to have a -parallel -Ricci tensor if its -Ricci tensor satisfies the condition .

Replacing by and by in (39), we obtain . Covariant derivative of the foregoing equation with respect to , we get . Therefore, from Definition 1, we have the following.

Corollary 3. Let be a -dimensional -conformally flat -cosymplectic manifold. If admits a -parallel -Ricci tensor, then has a constant scalar curvature.

4. -Weyl Curvature Tensor on -Cosymplectic Manifolds

The notion of -Weyl curvature tensor on real hypersurfaces of complex space forms (particularly, nonflat) is defined recently by Kaimakamis and Panagiotidou [23] in the following way:for all , where is the -Ricci operator and is the -scalar curvature corresponding to .

Using (25), we can write

With the help of (41), (32), and (42), we obtain the expression for the -Weyl curvature tensor on -dimensional -cosymplectic manifold as

4.1. -Cosymplectic Manifold with Vanishing -Weyl Curvature Tensor

Theorem 3. An -cosymplectic manifold with vanishing -Weyl curvature tensor is an -Einstein manifold.

Proof. Let us consider an -cosymplectic manifold with vanishing -Weyl curvature tensor, that is, . Thus, (43) infers thatCovariant differentiation of above relation along and then contracting the resultant equation over yieldswhere “div” denotes the divergence. On the other side, differentiating covariantly along and then contracting with the aid of following well-known formulas,we easily obtainBy virtue of (45) and (47), we haveReplacing by in (48), we obtainRecalling Lemma 1 to findWriting instead of by in the foregoing equation and by making use of (11), we deriveMaking use of this equation in (50) yieldsUsing (14) in (52), we getThis proves our result.

Substituting (53) in (25), we havewhere shows that is -Einstein. In view of (3), we obtain

Thus, , and hence, it is weak -Einstein. Thus, we state the following.

Theorem 4. An -cosymplectic manifold with vanishing -Weyl curvature tensor is a weak -Einstein manifold.

4.2. -Cosymplectic Manifold Satisfying the Condition

An -cosymplectic manifold is called semisymmetric if its curvature tensor satisfies the condition . In [24], Szabo studied the intrinsic classification of semisymmetric spaces thoroughly. In this context, Venkatesha and Kumara [21] studied Sasakian manifolds satisfying condition . In this section, we make an attempt to study this condition in the framework of -cosymplectic manifolds and prove the following.

Theorem 5. An -cosymplectic manifold satisfying the condition is an -Einstein manifold.

Proof. Let be an -dimensional -cosymplectic manifold satisfying the condition . This infers thatPlugging in place of in the previous equation and then picking inner product with for the resultant equation, we obtainIn view of (9), it follows from (56) thatReplacing by in the above equation, we haveprovided . By virtue of (43), one can easily see thatwhere is an orthonormal basis of the tangent space at any point of the manifold. Taking in (59) and summing over and making use of (56)–(61), we getThis completes the proof.

5. -Cosymplectic Manifolds Admitting -Ricci Solitons

Hamilton [25] introduced the notion of Ricci solitons as fixed points of the Ricci flows on a Riemannian manifold, and they are also self-similar solutions. These self-similar solutions also generalize Einstein metrics. Ricci solitons also correspond to self-similar solutions of Hamilton’s Ricci flow. A Ricci soliton with a potential vector field is defined byfor some constant . The Ricci soliton is said to be shrinking, steady, and expanding accordingly as is negative, zero, and positive, respectively. The study of Ricci solitons and almost Ricci solitons on three-dimensional cosymplectic manifolds have been carried out by Wang [26] and De and Dey [27], respectively.

By taking the necessary modification (64), Kaimakamis and Panagiotidou [28] introduced the notion of a special type of metric called -Ricci soliton on real hypersurfaces of nonflat complex space forms. A Riemannian metric on is called -Ricci soliton, if the Lie derivative of a vector field on is given by

Recently, the study of -Ricci solitons within the context of almost contact and paracontact manifolds were carried out in the studies [18, 29–34] and drawn several interesting results. In this section, we intended to -Ricci soliton on a -cosymplectic manifold. Now, we prove the following result.

Theorem 6. Let be -cosymplectic manifold admitting a -Ricci soliton. If the potential vector field is pointwise collinear with , then is a near quasi-Einstein manifold.

Proof. Let be a pointwise collinear vector field with . Then, we have . From (7) and (65), we deriveLet be an -cosymplectic manifold admitting a -Ricci soliton. Then, from (61) and (62), we obtainLet be a gradient of smooth function on , that is, and . Then, by denoting the dual form of by , we writeBy taking account of foregoing equations in (67), we getThen, from (25), equation (65) reduces toLet us take a nonvanishing symmetric tensor in (66), such thatThen, equation (66) yieldswhere . So, is a near quasi-Einstein.

As an immediate outcome of Theorem 6, we have the following corollary.

Corollary 4. An -cosymplectic manifold admitting a -Ricci soliton is an -Einstein manifold if .

A near quasi-Einstein manifold is not a manifold of nearly quasiconstant curvature. But, it is noted (Theorem 3.1 of [35]) that, a conformally flat near quasi-Einstein manifold is a manifold of nearly quasiconstant curvature. Hence, as immediate consequence of this fact, we obtain the following corollary:

Corollary 5. A conformally flat -cosymplectic manifold admitting a -Ricci soliton is a manifold of near quasi-constant curvature if is a pointwise collinear with .

However, since a 3-dimensional Riemannian manifold is conformally flat, we have following.

Corollary 6. A 3-dimensional -cosymplectic manifold admitting -Ricci soliton is a manifold of nearly quasiconstant curvature if is a pointwise collinear with .

6. Conclusions

Einstein manifolds which are arisen from Einstein field equations are very important classes of Riemann manifolds. Some generalizations of Einstein manifolds have been defined in the literature, and there have been obtained some applications of these kinds of manifolds in theoretical physics. Contact manifolds are special Riemann manifolds with almost contact structures. In theoretical physics, there are valuable applications of contact manifolds. Contact manifolds divided into many subclasses via the certain properties of the structure. An important one is -cosymplectic manifold. This structure is also a generalization of some different contact structures. Many different characteristic properties of manifolds with structures have been arisen from their special structures. One of important notion is the -Ricci tensor. This notion carries significant curvature features, and this feature gives valuable information about the geometry of the manifold. In this study, -cosymplectic manifolds have been examined under the effect of the -Ricci tensor. Important results have been obtained on some generalized Einstein manifolds, which emerged with the effect of the -Ricci tensor. The notion of Ricci soliton comes from searching the solutions of Ricci flow equations. Ricci solitons have been effected from the structure of manifolds. We studied the concept of -Ricci soliton for -cosymplectic manifolds. By the way, important physical results have been stated in this study.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

M.R.A., D.G.P, N.B.T, and I.U. conceptualized the study, developed methodology, performed formal analysis. D.G.P., N.B.T., and I.U. wrote the original draft. M.R.A. and N.B.T. reviewed and edited the article. M.R.A., D.G.P., and I.U. supervised the study. D.G.P. administered the project. A.M.R. and N.B.T acquired fund .

Acknowledgments

This project was supported by the researchers supporting project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia. The first author M. R. Amruthalakshmi (MRA) is thankful to Department of Science and Technology, Ministry of Science and Technology, Government of India, for providing DST INSPIRE Fellowship (No: DST/INSPIRE Fellowship/(IF 190869)).

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