-Ricci Tensor on α-Cosymplectic Manifolds

In this paper, we study α-cosymplectic manifold M admitting ∗ -Ricci tensor. First, it is shown that a ∗ -Ricci semisymmetric manifold M is ∗ -Ricci flat and a φ-conformally flat manifold M is an η-Einstein manifold. Furthermore, the ∗ -Weyl curvature tensorW∗ on M has been considered. Particularly, we show that a manifold M with vanishing ∗ -Weyl curvature tensor is a weak φ-Einstein and amanifold M fulfilling the condition R(E1, E2) · W ∗ � 0 is η-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldM admitting ∗ -Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for threedimensional cosymplectic manifolds admitting ∗ -Ricci soliton and almost ∗ -Ricci soliton are drawn.


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. is 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 (2n + 1)-dimension with the condition η∧dη n ≠ 0 for a closed 1-form η is a cosymplectic manifold. A simple example of almost cosymplectic manifolds is given by the products of almost Kaehler manifolds and the real line R or the circle S 1 . At this moment, we refer the studies [2][3][4][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 E 1 , E 2 ∈ TM, where R is the Riemannian curvature tensor. Naturally, Hamada also considered the notion of * -Einstein manifold. An Hermitian manifold is * -Einstein if we have g(Q * E 1 , E 2 ) � λg(E 1 , E 2 ), where λ 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]. e 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 (κ, μ)-space form using the * -Ricci tensor. In this study, the * -Ricci tensor within the framework of α-cosymplectic manifolds has been studied. In Section 2, we recall some basic formulas and results concerning α-cosymplectic manifold and * -Ricci tensor, which will be useful in further sections. An α-cosymplectic manifold satisfying * -Ricci semisymmetric and ϕ-conformally flat conditions are studied in Section 3 and shown that a ϕ-conformably flat α-cosymplectic manifold is η-Einstein and a * -Ricci semisymmetric α-cosymplectic manifold is * -Ricci flat. In next section, the * -Weyl curvature tensor has been studied in the background of α-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 α-cosymplectic manifold admitting * -Ricci soliton.

Preliminaries
Here, we are going to recall some general facts on α-cosymplectic manifolds which are relevant to our work. An almost contact metric manifold of (2n + 1)-dimension is a 5-tuple (M, ϕ, ξ, η, g) with the following resources [11].
for a (1, 1)− tensor field ϕ, a characteristic vector field ξ, a 1form η is dual of ξ, and g is a Riemannian metric. It is easily seen that It is well known that the fundamental 2-form ω is defined by For an almost contact metric manifold M, we have the following classifications ( [12,13]): (1) If dη � ω, then M is a contact metric manifold (2) If dη � 0 and dω � 0, then M is an almost cosymplectic manifold [1] (3) If dη � 0 and dω � 2αη∧ω, then M is an almost α-Kenmotsu manifold for a nonzero scalar α 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 [ϕ, ϕ] � 0 which is the Nijenhuis tensor of the tensor field ϕ, is defined by for all E 1 , E 2 ∈ TM. A normal almost cosymplectic manifold is a cosymplectic manifold.
Almost α− cosymplectic manifolds have been defined by Kim and Pak [14] by combining an almost α-Kenmotsu and almost cosymplectic structures by the following formula: for a constant α. On an α-cosymplectic manifold, we have where ∇ denotes the Riemannian connection. From (6), it is easy to see that and On an α-cosymplectic manifold M of dimension 2n + 1, the following relationships are valid: where R and S are the curvature and Ricci tensors, respectively. By the following lemma, we obtain some derivational features of α-cosymplectic manifold.

Lemma 1.
On an α− cosymplectic manifold of dimension 2n + 1, we have Proof. Note that (11) implies Qξ � − 2nα 2 ξ, for Q defined by . Differentiating this along E 1 and using (7), we get (12). Next, differentiation of (10) with respect to T gives Let e i 2n+1 i�1 be a local basis on M. Replacing E 1 � T � e i in the foregoing equation with summing over i gives

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Using second Bianchi's identity leads to By considering (16) in (17) and then with the help of (12), we conclude which proves (13). Finally, contraction of (13) gives (14). □ From Riemannian geometry, the covariant derivative of a (1, s)-type of tensor field K is given by for all E 1 , E 1 , . . . , E 1 ∈ TM, where div is stated for the divergence [15].
By following descriptions, we present some classification facts which come from the Ricci tensor and have been stated.
for some function β, where g ϕ (E 1 , E 2 ) � g(ϕE 1 , ϕE 2 ), and S ϕ is defined by In other words, S ϕ denotes the symmetric part of S * . If β is constant, then M is called ϕ-Einstein [16].
where a and b are the nonzero scalars and E is a nonzero (0, 2) tensor [17].
where α and c are the constants [18].
By decomposition of Riemannian curvature tensor R, the Weyl conformal curvature tensor W has been obtained in this way: for all E 1 , E 2 , E 3 ∈ TM [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.

* -Ricci Tensor on α-Cosymplectic Manifold
We are in a situation to confer the equation of the * -Ricci tensor in the framework of α-cosymplectic manifolds and then study its various properties. In [19], authors derived the expression of the * -Ricci tensor on α-cosymplectic manifold which is of the following form: for all E 1 , E 2 ∈ TM. Note that S * is not symmetric. By contraction of (25), the * -scalar curvature is specified by If the * -Ricci tensor S * is a constant multiple of the Riemannian metric g, then we say that the manifold is * -Einstein. Moreover, the * -scalar curvature is not constant on a * -Einstein manifold.

* -Ricci Semisymmetric α-Cosymplectic
Manifolds. An α-cosymplectic manifold M satisfying the condition acts as a derivation on S. is notion was introduced by Mirjoyan [20] for Riemann spaces and then studied by many authors. Analogous to this, an α-cosymplectic manifold is called * -Ricci semisymmetric if its * -Ricci tensor satisfies the condition Theorem 1. If a (2n + 1)-dimensional α-cosymplectic manifold M is * -Ricci semisymmetric, then M is * -Ricci flat. Moreover, it is an η-Einstein manifold, and the Ricci tensor is to be exhibited as Proof. Let us consider * -Ricci semisymmetric α-cosym- Putting E 1 � ξ in (28) and then recalling (9), we have It is well known that Ric * (E 1 , ξ) � 0. Making use of this in (29), we find Again, plugging E 4 by ξ in (30) shows that M is * -Ricci flat, that is, (25) and (30), we have the required result.
Proof. Assume that an α-cosymplectic manifold is ϕ-conformally flat. So, it is easy to see that ϕ 2 C(ϕE 1 , Hence, ϕ-conformally flat means For a local orthonormal basis of TM with e 1 , . . . , e 2 n, ξ , if we put E 1 � E 4 � e i in (32) and sum up with respect to i, then we obtain and therefore,

Corollary 1. A ϕ-conformally flat α-cosymplectic manifold of constant scalar curvature is a ϕ--Einstein manifold.
In an α-cosymplectic manifold, the * -Ricci tensor is given by (25), and so in view of (40), we state the following.

* -Weyl Curvature Tensor on α-Cosymplectic Manifolds
e notion of * -Weyl curvature tensor W * on real hypersurfaces of complex space forms (particularly, nonflat) is defined recently by Kaimakamis and Panagiotidou [23] in the following way: for all E 1 , E 2 , E 3 ∈ TM, where Q * is the * -Ricci operator and r * is the * -scalar curvature corresponding to Q * .
Proof. Let us consider an α-cosymplectic manifold M with vanishing * -Weyl curvature tensor, that is, Covariant differentiation of above relation along E 4 and then contracting the resultant equation over E 4 yields 6 Advances in Mathematical Physics where "div" denotes the divergence. On the other side, differentiating W covariantly along E 4 and then contracting with the aid of following well-known formulas, we easily obtain By virtue of (45) and (47), we have Replacing E 2 by ξ in (48), we obtain Recalling Lemma 1 to find Writing ξ instead of E 3 by ξ in the foregoing equation and by making use of (11), we derive Making use of this equation in (50) yields Advances in Mathematical Physics 7 Using (14) in (52), we get is proves our result.

α-Cosymplectic Manifold Satisfying the Condition
R(X, Y) · W * � 0. An α-cosymplectic manifold M is called semisymmetric if its curvature tensor satisfies the condition R · R � 0. In [24], Szabo studied the intrinsic classification of semisymmetric spaces thoroughly. In this context, Venkatesha and Kumara [21] studied Sasakian manifolds satisfying condition R(E 1 , E 2 ) · W * � 0. In this section, we make an attempt to study this condition in the framework of α-cosymplectic manifolds and prove the following.
Proof. Let M be an (2n + 1)-dimensional α-cosymplectic manifold satisfying the condition Plugging ξ in place of E 1 in the previous equation and then picking inner product with ξ for the resultant equation, we obtain In view of (9), it follows from (56) that Replacing E 2 by U in the above equation, we have provided α 2 ≠ 0. By virtue of (43), one can easily see that 8 Advances in Mathematical Physics where e i 2n+1 i�1 is an orthonormal basis of the tangent space at any point of the manifold. Taking U � e i in (59) and summing over i and making use of (56)-(61), we get is completes the proof.

α-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. ese 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 V is defined by for some constant λ. e Ricci soliton is said to be shrinking, steady, and expanding accordingly as λ is negative, zero, and positive, respectively. e 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 g on M is called * -Ricci soliton, if the Lie derivative of a vector field V on M 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][30][31][32][33][34] and drawn several interesting results. In this section, we intended to * -Ricci soliton on a α-cosymplectic manifold. Now, we prove the following result. Proof. Let V be a pointwise collinear vector field with ξ. en, we have V � bξ. From (7) and (65), we derive Let M be an α-cosymplectic manifold admitting a * -Ricci soliton. en, from (61) and (62), we obtain

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Let Db be a gradient of smooth function b on M, that is, en, by denoting the dual form of Db by v, we write (68) By taking account of foregoing equations in (67), we get en, from (25), equation (65) reduces to Let us take a nonvanishing symmetric (0, 2) tensor E in (66), such that en, equation (66) yields where a � − λ + bα + (2n − 1)α 2 . So, M is a near quasi-Einstein.

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

Corollary 4. An α-cosymplectic manifold admitting a * -Ricci soliton is an η-Einstein manifold if V � ξ.
A near quasi-Einstein manifold is not a manifold of nearly quasiconstant curvature. But, it is noted ( eorem 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 α-cosymplectic manifold admitting a * -Ricci soliton is a manifold of near quasiconstant curvature if V is a pointwise collinear with ξ.
However, since a 3-dimensional Riemannian manifold is conformally flat, we have following.

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

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 α-cosymplectic manifold. is 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.
is notion carries significant curvature features, and this feature gives valuable information about the geometry of the manifold. In this study, α-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. e 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 α-cosymplectic manifolds. By the way, important physical results have been stated in this study.

Data Availability
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Conflicts of Interest
e authors declare that they have no conflicts of interest.