# ∗-Ricci Tensor on α-Cosymplectic Manifolds

Authors: M. R. Amruthalakshmi; D. G. Prakasha; Nasser Bin Turki; Inan Unal
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we studyα-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 tensor W∗ on M has been considered. Particularly, we show that a manifold M with vanishing ∗-Weyl curvature tensor is a weak ϕ-Einstein and a manifold M fulfilling the condition RE1,E2⋅W∗=0 is η-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold M 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.


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## Body

## 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 2n+1-dimension with the condition η∧dηn≠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 S1. 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:(1)S∗E1,E2=12TrE3⟶RE1,ϕE2ϕE3,for all E1,E2∈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 gQ∗E1,E2=λgE1,E2, 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]. 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 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.


## 2. Preliminaries

Here, we are going to recall some general facts onα-cosymplectic manifolds which are relevant to our work.An almost contact metric manifold of2n+1-dimension is a 5-tuple M,ϕ,ξ,η,g with the following resources [11].(2)ϕ2E1=−E1+ηE1ξ,ηξ=1,ϕξ=0,ηϕE1=0,gϕE1,ϕE2=gE1,E2−ηE1ηE2,E1,E2∈TM,for a 1,1− tensor field ϕ, a characteristic vector field ξ, a 1-form η is dual of ξ, and g is a Riemannian metric. It is easily seen that(3)gϕE1,E2=−gE1,ϕE2,gE1,ξ=ηE1,E1,E2∈TM.It is well known that the fundamental 2-formω is defined by ωE1,E2=gϕE1,E2 on M.For an almost contact metric manifoldM, we have the following classifications ([12, 13]):(1)
Ifdη=ω, then M is a contact metric manifold(2)
Ifdη=0 and dω=0, then M is an almost cosymplectic manifold [1](3)
Ifdη=0 and dω=2αη∧ω, then M is an almost α-Kenmotsu manifold for a nonzero scalar αIn 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 haveϕ,ϕ=0 which is the Nijenhuis tensor of the tensor field ϕ, is defined by(4)ϕ,ϕ=ϕ2E1,E2+ϕE1,ϕE2−ϕϕE1,E2−ϕE1,ϕE2,for all E1,E2∈TM. A normal almost cosymplectic manifold is a cosymplectic manifold.Almostα−cosymplectic manifolds have been defined by Kim and Pak [14] by combining an almost α-Kenmotsu and almost cosymplectic structures by the following formula:(5)dη=0,dω=2αη∧ω,for a constant α. On an α-cosymplectic manifold, we have(6)∇E1ϕE2=αgϕE1,E2ξ−ηE2ϕE1,where ∇ denotes the Riemannian connection. From (6), it is easy to see that(7)∇E1ξ=−αϕ2E1=αE1−ηE1ξ,and(8)∇E1ηE2=αgE1,E2−ηE1ηE2.On anα-cosymplectic manifold M of dimension 2n+1, the following relationships are valid:(9)Rξ,E1E2=α2ηE2E1−gE1,E2ξ,(10)RE1,E2ξ=α2ηE1E2−ηE2E1,(11)SE1,ξ=−2nα2ηE1,where R and S are the curvature and Ricci tensors, respectively.By the following lemma, we obtain some derivational features ofα-cosymplectic manifold.Lemma 1.
On anα− cosymplectic manifold of dimension 2n+1, we have(12)∇E1Qξ=−αQE1−2nα3E1,(13)∇ξQE1=−2αQE1−4nα3E1,(14)ξr=−2αr+2n2n+1α2.Proof.
Note that (11) implies Qξ=−2nα2ξ, for Q defined by SE1,E2=gQE1,E2. Differentiating this along E1 and using (7), we get (12). Next, differentiation of (10) with respect to T gives(15)∇TRE1,E2ξ=−αRE1,E2T+α3gE1,TE2−gE2,TE1.
Leteii=12n+1 be a local basis on M. Replacing E1=T=ei in the foregoing equation with summing over i gives(16)∑i=12n+1g∇eiRei,E2ξ,E3=αSE2,E3+2nα3gE2,E3.
Using second Bianchi’s identity leads to(17)∑i=12n+1g∇eiRE3,ξE2,ei=g∇E3Qξ,E2−g∇ξQE3,E2.
By considering (16) in (17) and then with the help of (12), we conclude(18)g∇ξQE1,E2=−2αSE1,E2−4nα3gE1,E2,which proves (13). Finally, contraction of (13) gives (14).From Riemannian geometry, the covariant derivative of a1,s-type of tensor field K is given by(19)divKE1,E1,…,E1s=∑i=12n+1g∇eiKE1,E1,…,E1,ei.for all E1,E1,…,E1∈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.(1)
Anα−cosymplectic manifold M is called by weak ϕ-Einstein if we have(20)SϕE1,E2=βgϕE1,E2,∀E1,E2∈TM,for some functionβ, where gϕE1,E2=gϕE1,ϕE2, and Sϕ is defined by(21)SϕE1,E2=12S∗E1,E2+Ric∗E2,E1,E1,E2∈TM,In other words,Sϕ denotes the symmetric part of S∗. If β is constant, then M is called ϕ-Einstein [16].(2)
M is called near quasi-Einstein manifold if the Ricci tensor is of the form(22)SE1,E2=agE1,E2+bEE1,E2,∀E1,E2∈TM,wherea and b are the nonzero scalars and E is a nonzero 0,2 tensor [17].(3)
M is called an η-Einstein manifold if we have(23)SE1,E2=αgE1,E2+γηE1ηE2,∀E1,E2∈TM,whereα and γ are the constants [18].By decomposition of Riemannian curvature tensorR, the Weyl conformal curvature tensor W has been obtained in this way:(24)WE1,E2E3=RE1,E2E3−12n−1SE2,E3E1−SE1,E3E2+gE2,E3QE1−gE1,E3QE2+r2n2n−1gE2,E3E1−gE1,E3E2.for all E1,E2,E3∈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.


## 3.∗-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:(25)S∗E1,E2=SE1,E2+2n−1α2gE1,E2+α2ηE1ηE2,for all E1,E2∈TM.Note thatS∗ is not symmetric. By contraction of (25), the ∗-scalar curvature is specified by(26)r∗=r+4n2α2.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.


### 3.1.∗-Ricci Semisymmetric α-Cosymplectic Manifolds

Anα-cosymplectic manifold M satisfying the condition RE1,E2⋅S=0 for all E1,E2∈TM is called Ricci semi symmetric, where RE1,E2 acts as a derivation on S. This 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 RE1,E2⋅S∗=0∀E1,E2∈TM.Theorem 1.
If a2n+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(27)SE1,E2=1−2nα2gE1,E2−α2ηE1ηE2,E1,E2∈TM.Proof.
Let us consider∗-Ricci semisymmetric α-cosymplectic manifold M. Then, condition RE1,E2.S∗=0 is equivalent to(28)S∗RE1,E2E3,E4+S∗E3,RE1,E2E4=0.
PuttingE1=ξ in (28) and then recalling (9), we have(29)α2S∗E2,E4ηE3−gE2,E3S∗ξ,E4+α2S∗E3,E2ηE4−gE2,E4S∗E3,ξ=0.
It is well known thatRic∗E1,ξ=0. Making use of this in (29), we find(30)α2S∗E2,E4ηE3+S∗E3,E2ηE4=0.
Again, pluggingE4 by ξ in (30) shows that M is ∗-Ricci flat, that is, S∗E3,E2=0,E3,E2∈TM. Moreover, in view of (25) and (30), we have the required result.


### 3.2.ϕ-Conformally Flat α-Cosymplectic Manifolds

Anα-cosymplectic manifold M is said to be ϕ-conformally flat if we have(31)ϕ2WϕE1,ϕE2ϕE3=0,for all E1,E2,E3∈TM. Sasakian manifolds which are ϕ-conformally flat have been studied in [21]. In the following, we study a ϕ-conformally flat α-cosymplectic manifold.Theorem 2.
Anφ-conformally flat α-cosymplectic manifold is ∗−η-Einstein manifold. Moreover, M is weak φ-Einstein.Proof.
Assume that anα-cosymplectic manifold is ϕ-conformally flat. So, it is easy to see that ϕ2CϕE1,ϕE2ϕE3=0 carry if and only if gCϕE1,ϕE2ϕE3,ϕE4=0∀E1,E2,E3,E4∈TM. Hence, ϕ-conformally flat means(32)gRϕE1,ϕE2ϕE3,ϕE4=12n−1SϕE2,ϕE3gϕE1,ϕE4−SϕE1,ϕE3gϕE2,ϕE4+gϕE2,ϕE3SϕE1,ϕE4−gϕE1,ϕE3SϕE2,ϕE4−r2n2n−1gϕE2,ϕE3gϕE1,ϕE4−gϕE1,ϕE3gϕE2,ϕE4.
For a local orthonormal basis ofTM with e1,…,e2n,ξ, if we put E1=E4=ei in (32) and sum up with respect to i, then we obtain(33)∑i=12ngRϕei,ϕE2ϕE3,ϕei=12n−1∑i=12ngϕei,ϕeiSϕE2,ϕE3−gϕE2,ϕeiSϕei,ϕE3+gϕE2,ϕE3Sϕei,ϕei−gϕei,ϕE3SϕE2,ϕei−r2n2n−1∑i=12ngϕei,ϕeigϕE2,ϕE3−gϕei,ϕE3gϕE2,ϕei,and therefore,(34)SϕE2,ϕE3−gRξ,E2E3,ξ=2n−12n−1SϕE2,ϕE3+12n−1r2n+α2gϕE2,ϕE3.
From (9), we obtain gRξ,E2E3,ξ=−α2gϕE2,ϕE3, and hence, from (34), we get(35)SϕE2,ϕE3=r2n+α2gϕE2,ϕE3.
Then, from (35), it follows from (32) that(36)gRϕE1,ϕE2ϕE3,ϕE4=r+4nα22n2n−1gϕE2,ϕE3gϕE1,ϕE4−gϕE1,ϕE3gϕE2,ϕE4.
In anα-cosymplectic manifold, in view of (3) and (9) for all E1,E2,E3,E4∈TM, we can verify that(37)Rφ2E1,φ2E2,φ2E3,φ2E4=RE1,E2,E3,E4+α2gE2,E3ηE1ηE2−gE1,E3ηE2ηE4+gE1,E4ηE2ηE3−gE2,E4ηE1ηE3.
TakingφE1,φE2,φE3,φE4 instead of E1,E2,E3,E4 in (37), respectively, and making use of (36), we obtain(38)RE1,E2,E3,E4=r+4nα22n2n−1gE1,E4gE2,E3−gE2,E4gE1,E3−r+2n2n+1α22n2n−1gE2,E3ηE1−gE1,E3ηE2ηE4+gE1,E4ηE2−gE2,E4ηE1ηE3.
By the definition ofS∗, direct computation yields(39)S∗E1,E2=∑i=12n+1RE1,ei,ϕei,ϕE2=βgE1,E2−βηE1ηE2,where β=r+4nα2/2n2n−1 reveals that M is ∗−η-Einstein. In view of (3), we have(40)S∗E1,E2=r+4nα22n2n−1gϕE1,E2,E1,E2∈TM.
Thus,S∗=Sϕ, and hence, it is weak ϕ-Einstein. This completes the proof.Next, for a constant scalar curvature ofM, in view of (40), we state the following.Corollary 1.
Aϕ-conformally flat α-cosymplectic manifold of constant scalar curvature is a ϕ--Einstein manifold.In anα-cosymplectic manifold, the ∗-Ricci tensor is given by (25), and so in view of (40), we state the following.Corollary 2.
Aϕ-conformally flat α-cosymplectic manifold is η-Einstein.Furthermore, anα-cosymplectic manifold M is called to have the η-parallel Ricci tensor if its Ricci tensor S satisfies the condition ∇E1SϕE2,ϕE3=0,E1,E2,E3∈TM. 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.
Anα-cosymplectic manifold M is said to have a η-parallel ∗-Ricci tensor if its ∗-Ricci tensor satisfies the condition ∇E1S∗ϕE2,ϕE3=0∀E1,E2,E3∈TM.ReplacingE1 by ϕE1 and E2 by ϕE2 in (39), we obtain S∗ϕE1,ϕE2=βgϕE1,ϕE2. Covariant derivative of the foregoing equation with respect to E4, we get ∇E4S∗ϕE1,ϕE2=drE4gϕE1,ϕE2. Therefore, from Definition 1, we have the following.Corollary 3.
LetM be a 2n+1-dimensional ϕ-conformally flat α-cosymplectic manifold. If M admits a η-parallel ∗-Ricci tensor, then M has a constant scalar curvature.


## 3.1.∗-Ricci Semisymmetric α-Cosymplectic Manifolds

Anα-cosymplectic manifold M satisfying the condition RE1,E2⋅S=0 for all E1,E2∈TM is called Ricci semi symmetric, where RE1,E2 acts as a derivation on S. This 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 RE1,E2⋅S∗=0∀E1,E2∈TM.Theorem 1.
If a2n+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(27)SE1,E2=1−2nα2gE1,E2−α2ηE1ηE2,E1,E2∈TM.Proof.
Let us consider∗-Ricci semisymmetric α-cosymplectic manifold M. Then, condition RE1,E2.S∗=0 is equivalent to(28)S∗RE1,E2E3,E4+S∗E3,RE1,E2E4=0.
PuttingE1=ξ in (28) and then recalling (9), we have(29)α2S∗E2,E4ηE3−gE2,E3S∗ξ,E4+α2S∗E3,E2ηE4−gE2,E4S∗E3,ξ=0.
It is well known thatRic∗E1,ξ=0. Making use of this in (29), we find(30)α2S∗E2,E4ηE3+S∗E3,E2ηE4=0.
Again, pluggingE4 by ξ in (30) shows that M is ∗-Ricci flat, that is, S∗E3,E2=0,E3,E2∈TM. Moreover, in view of (25) and (30), we have the required result.


## 3.2.ϕ-Conformally Flat α-Cosymplectic Manifolds

Anα-cosymplectic manifold M is said to be ϕ-conformally flat if we have(31)ϕ2WϕE1,ϕE2ϕE3=0,for all E1,E2,E3∈TM. Sasakian manifolds which are ϕ-conformally flat have been studied in [21]. In the following, we study a ϕ-conformally flat α-cosymplectic manifold.Theorem 2.
Anφ-conformally flat α-cosymplectic manifold is ∗−η-Einstein manifold. Moreover, M is weak φ-Einstein.Proof.
Assume that anα-cosymplectic manifold is ϕ-conformally flat. So, it is easy to see that ϕ2CϕE1,ϕE2ϕE3=0 carry if and only if gCϕE1,ϕE2ϕE3,ϕE4=0∀E1,E2,E3,E4∈TM. Hence, ϕ-conformally flat means(32)gRϕE1,ϕE2ϕE3,ϕE4=12n−1SϕE2,ϕE3gϕE1,ϕE4−SϕE1,ϕE3gϕE2,ϕE4+gϕE2,ϕE3SϕE1,ϕE4−gϕE1,ϕE3SϕE2,ϕE4−r2n2n−1gϕE2,ϕE3gϕE1,ϕE4−gϕE1,ϕE3gϕE2,ϕE4.
For a local orthonormal basis ofTM with e1,…,e2n,ξ, if we put E1=E4=ei in (32) and sum up with respect to i, then we obtain(33)∑i=12ngRϕei,ϕE2ϕE3,ϕei=12n−1∑i=12ngϕei,ϕeiSϕE2,ϕE3−gϕE2,ϕeiSϕei,ϕE3+gϕE2,ϕE3Sϕei,ϕei−gϕei,ϕE3SϕE2,ϕei−r2n2n−1∑i=12ngϕei,ϕeigϕE2,ϕE3−gϕei,ϕE3gϕE2,ϕei,and therefore,(34)SϕE2,ϕE3−gRξ,E2E3,ξ=2n−12n−1SϕE2,ϕE3+12n−1r2n+α2gϕE2,ϕE3.
From (9), we obtain gRξ,E2E3,ξ=−α2gϕE2,ϕE3, and hence, from (34), we get(35)SϕE2,ϕE3=r2n+α2gϕE2,ϕE3.
Then, from (35), it follows from (32) that(36)gRϕE1,ϕE2ϕE3,ϕE4=r+4nα22n2n−1gϕE2,ϕE3gϕE1,ϕE4−gϕE1,ϕE3gϕE2,ϕE4.
In anα-cosymplectic manifold, in view of (3) and (9) for all E1,E2,E3,E4∈TM, we can verify that(37)Rφ2E1,φ2E2,φ2E3,φ2E4=RE1,E2,E3,E4+α2gE2,E3ηE1ηE2−gE1,E3ηE2ηE4+gE1,E4ηE2ηE3−gE2,E4ηE1ηE3.
TakingφE1,φE2,φE3,φE4 instead of E1,E2,E3,E4 in (37), respectively, and making use of (36), we obtain(38)RE1,E2,E3,E4=r+4nα22n2n−1gE1,E4gE2,E3−gE2,E4gE1,E3−r+2n2n+1α22n2n−1gE2,E3ηE1−gE1,E3ηE2ηE4+gE1,E4ηE2−gE2,E4ηE1ηE3.
By the definition ofS∗, direct computation yields(39)S∗E1,E2=∑i=12n+1RE1,ei,ϕei,ϕE2=βgE1,E2−βηE1ηE2,where β=r+4nα2/2n2n−1 reveals that M is ∗−η-Einstein. In view of (3), we have(40)S∗E1,E2=r+4nα22n2n−1gϕE1,E2,E1,E2∈TM.
Thus,S∗=Sϕ, and hence, it is weak ϕ-Einstein. This completes the proof.Next, for a constant scalar curvature ofM, in view of (40), we state the following.Corollary 1.
Aϕ-conformally flat α-cosymplectic manifold of constant scalar curvature is a ϕ--Einstein manifold.In anα-cosymplectic manifold, the ∗-Ricci tensor is given by (25), and so in view of (40), we state the following.Corollary 2.
Aϕ-conformally flat α-cosymplectic manifold is η-Einstein.Furthermore, anα-cosymplectic manifold M is called to have the η-parallel Ricci tensor if its Ricci tensor S satisfies the condition ∇E1SϕE2,ϕE3=0,E1,E2,E3∈TM. 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.
Anα-cosymplectic manifold M is said to have a η-parallel ∗-Ricci tensor if its ∗-Ricci tensor satisfies the condition ∇E1S∗ϕE2,ϕE3=0∀E1,E2,E3∈TM.ReplacingE1 by ϕE1 and E2 by ϕE2 in (39), we obtain S∗ϕE1,ϕE2=βgϕE1,ϕE2. Covariant derivative of the foregoing equation with respect to E4, we get ∇E4S∗ϕE1,ϕE2=drE4gϕE1,ϕE2. Therefore, from Definition 1, we have the following.Corollary 3.
LetM be a 2n+1-dimensional ϕ-conformally flat α-cosymplectic manifold. If M admits a η-parallel ∗-Ricci tensor, then M has a constant scalar curvature.


## 4.∗-Weyl Curvature Tensor on α-Cosymplectic Manifolds

The 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:(41)W∗E1,E2E3=RE1,E2E3+r∗2n2n−1gE2,E3E1−gE1,E3E2−12n−1gQ∗E2,E3E1−gQ∗E1,E3E2+gE2,E3Q∗E1−gE1,E3Q∗E2,for all E1,E2,E3∈TM, where Q∗ is the ∗-Ricci operator and r∗ is the ∗-scalar curvature corresponding to Q∗.Using (25), we can write(42)Q∗E1=QE1+α22n−1E1+ηE1ξ.With the help of (41), (32), and (42), we obtain the expression for the ∗-Weyl curvature tensor on 2n+1-dimensional α-cosymplectic manifold M as(43)W∗E1,E2E3=WE1,E2E3−2n−1α22n−1gE2,E3E1−gE1,E3E2−α22n−1ηE2ηE3E1−ηE1ηE3E2+gE2,E3ηE1ξ−gE1,E3ηE2ξ.


### 4.1.α-Cosymplectic Manifold with Vanishing ∗-Weyl Curvature Tensor

Theorem 3.
Anα-cosymplectic manifold with vanishing ∗-Weyl curvature tensor is an η-Einstein manifold.Proof.
Let us consider anα-cosymplectic manifold M with vanishing ∗-Weyl curvature tensor, that is, W∗E1,E2E3=0. Thus, (43) infers that(44)WE1,E2E3=2n−1α22n−1gE2,E3E1−gE1,E3E2+α22n−1ηE2ηE3E1−ηE1ηE3E2+gE2,E3ηE1ξ−gE1,E3ηE2ξ.
Covariant differentiation of above relation alongE4 and then contracting the resultant equation over E4 yields(45)divWE1,E2E3=2n−1α3gE1,E3ηE2−gE2,E3ηE1,where “div” denotes the divergence. On the other side, differentiating W covariantly along E4 and then contracting with the aid of following well-known formulas,(46)divRE1,E2E3=g∇E1QE2,E3−g∇E2QE1,E3,divQE1=12E1r,we easily obtain(47)divWE1,E2E3=2n−12n−1g∇E1QE2,E3−g∇E2QE1,E3−n−12n2n−1gE2,E3E1r−gE1,E3E2r.
By virtue of (45) and (47), we have(48)2n−12n−1g∇E1QE2,E3−g∇E2QE1,E3=2n−1α3gE1,E3ηE2−gE2,E3ηE1+n−12n2n−1gE2,E3E1r−gE1,E3E2r.
ReplacingE2 by ξ in (48), we obtain(49)2n−12n−1g∇E1Qξ,E3−g∇ξQE1,E3=2n−1α3gE1,E3−ηE1ηE3+n−12n2n−1ηE3E1r−gE1,E3ξr.
Recalling Lemma1 to find(50)2n−12n−1αgQE1,E3+α2gE1,E3+2n−1α3ηE1ηE3=n−12n2n−1ηE3E1r−gE1,E3ξr.
Writingξ instead of E3 by ξ in the foregoing equation and by making use of (11), we derive(51)Xr=ξrηE1.
Making use of this equation in (50) yields(52)αgQE1,E3=−α3+14nξrgE1,E3+−2n−1α3+14nξrηE1ηE3.
Using (14) in (52), we get(53)SE1,E3=2nα2+r2ngE1,E3+−4nα2−r2nηE1ηE3.
This proves our result.Substituting (53) in (25), we have(54)S∗E1,E3=γgE1,E3−ηE1ηE3,where γ=1/2n+4n−1α2 shows that M is ∗−η-Einstein. In view of (3), we obtain(55)S∗E1,E3=r2n+4n−1α2gϕE1,E3,E1,E3∈TM.Thus,S∗=Sϕ, and hence, it is weak ϕ-Einstein. Thus, we state the following.Theorem 4.
Anα-cosymplectic manifold with vanishing ∗-Weyl curvature tensor is a weak ϕ-Einstein manifold.


### 4.2.α-Cosymplectic Manifold Satisfying the Condition RX,Y⋅W∗=0

Anα-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 RE1,E2⋅W∗=0. In this section, we make an attempt to study this condition in the framework of α-cosymplectic manifolds and prove the following.Theorem 5.
Anα-cosymplectic manifold satisfying the condition RE1,E2⋅W∗=0 is an η-Einstein manifold.Proof.
LetM be an 2n+1-dimensional α-cosymplectic manifold satisfying the condition RE1,E2⋅W∗U,VE4=0. This infers that(56)RE1,E2W∗U,VE4−W∗RE1,E2U,VE4−W∗U,RE1,E2VE4−W∗U,VRE1,E2E4=0.
Pluggingξ in place of E1 in the previous equation and then picking inner product with ξ for the resultant equation, we obtain(57)ηRξ,E2W∗U,VE4−ηW∗Rξ,E2U,VE4−ηW∗U,Rξ,E2VE4−ηW∗U,VRξ,E2E4=0.
In view of (9), it follows from (56) that(58)α2W∗U,V,E4,E2−gE2,UηW∗ξ,VE4−ηW∗U,VE4ηE2−gE2,VηW∗U,ξE4+ηUηW∗E2,VE4+ηVηW∗U,E2E4−gE2,E4ηW∗U,Vξ+ηE4ηW∗U,VE2=0.
ReplacingE2 by U in the above equation, we have(59)W∗U,V,E4,U−gU,UηW∗ξ,VE4−gU,VηW∗U,ξE4+ηE4ηW∗U,VU=0,provided α2≠0. By virtue of (43), one can easily see that(60)ηW∗E1,E2E3=−12n−1gQE2,E3ηE1−gQE1,E3ηE2+r2n2n−1−2n−12n−1α2gE2,E3ηE1−gE1,E3ηE2,ηW∗E1,E2ξ=0,(61)ηW∗E1,ξE3=12n−1gQE1,E3+12n−12n−1α2−r2ngE1,E3+12n−1r2n+2α2ηE1ηE3,(62)∑i=12n+1W∗ei,E2,E3,ei=−2n−1α2gE2,E3−α2ηE2ηE3,where eii=12n+1 is an orthonormal basis of the tangent space at any point of the manifold. Taking U=ei in (59) and summing over i and making use of (56)–(61), we get(63)SV,E4=12nr+α2gV,E4−12nr+4n2+1α2ηVηE4.
This completes the proof.


## 4.1.α-Cosymplectic Manifold with Vanishing ∗-Weyl Curvature Tensor

Theorem 3.
Anα-cosymplectic manifold with vanishing ∗-Weyl curvature tensor is an η-Einstein manifold.Proof.
Let us consider anα-cosymplectic manifold M with vanishing ∗-Weyl curvature tensor, that is, W∗E1,E2E3=0. Thus, (43) infers that(44)WE1,E2E3=2n−1α22n−1gE2,E3E1−gE1,E3E2+α22n−1ηE2ηE3E1−ηE1ηE3E2+gE2,E3ηE1ξ−gE1,E3ηE2ξ.
Covariant differentiation of above relation alongE4 and then contracting the resultant equation over E4 yields(45)divWE1,E2E3=2n−1α3gE1,E3ηE2−gE2,E3ηE1,where “div” denotes the divergence. On the other side, differentiating W covariantly along E4 and then contracting with the aid of following well-known formulas,(46)divRE1,E2E3=g∇E1QE2,E3−g∇E2QE1,E3,divQE1=12E1r,we easily obtain(47)divWE1,E2E3=2n−12n−1g∇E1QE2,E3−g∇E2QE1,E3−n−12n2n−1gE2,E3E1r−gE1,E3E2r.
By virtue of (45) and (47), we have(48)2n−12n−1g∇E1QE2,E3−g∇E2QE1,E3=2n−1α3gE1,E3ηE2−gE2,E3ηE1+n−12n2n−1gE2,E3E1r−gE1,E3E2r.
ReplacingE2 by ξ in (48), we obtain(49)2n−12n−1g∇E1Qξ,E3−g∇ξQE1,E3=2n−1α3gE1,E3−ηE1ηE3+n−12n2n−1ηE3E1r−gE1,E3ξr.
Recalling Lemma1 to find(50)2n−12n−1αgQE1,E3+α2gE1,E3+2n−1α3ηE1ηE3=n−12n2n−1ηE3E1r−gE1,E3ξr.
Writingξ instead of E3 by ξ in the foregoing equation and by making use of (11), we derive(51)Xr=ξrηE1.
Making use of this equation in (50) yields(52)αgQE1,E3=−α3+14nξrgE1,E3+−2n−1α3+14nξrηE1ηE3.
Using (14) in (52), we get(53)SE1,E3=2nα2+r2ngE1,E3+−4nα2−r2nηE1ηE3.
This proves our result.Substituting (53) in (25), we have(54)S∗E1,E3=γgE1,E3−ηE1ηE3,where γ=1/2n+4n−1α2 shows that M is ∗−η-Einstein. In view of (3), we obtain(55)S∗E1,E3=r2n+4n−1α2gϕE1,E3,E1,E3∈TM.Thus,S∗=Sϕ, and hence, it is weak ϕ-Einstein. Thus, we state the following.Theorem 4.
Anα-cosymplectic manifold with vanishing ∗-Weyl curvature tensor is a weak ϕ-Einstein manifold.


## 4.2.α-Cosymplectic Manifold Satisfying the Condition RX,Y⋅W∗=0

Anα-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 RE1,E2⋅W∗=0. In this section, we make an attempt to study this condition in the framework of α-cosymplectic manifolds and prove the following.Theorem 5.
Anα-cosymplectic manifold satisfying the condition RE1,E2⋅W∗=0 is an η-Einstein manifold.Proof.
LetM be an 2n+1-dimensional α-cosymplectic manifold satisfying the condition RE1,E2⋅W∗U,VE4=0. This infers that(56)RE1,E2W∗U,VE4−W∗RE1,E2U,VE4−W∗U,RE1,E2VE4−W∗U,VRE1,E2E4=0.
Pluggingξ in place of E1 in the previous equation and then picking inner product with ξ for the resultant equation, we obtain(57)ηRξ,E2W∗U,VE4−ηW∗Rξ,E2U,VE4−ηW∗U,Rξ,E2VE4−ηW∗U,VRξ,E2E4=0.
In view of (9), it follows from (56) that(58)α2W∗U,V,E4,E2−gE2,UηW∗ξ,VE4−ηW∗U,VE4ηE2−gE2,VηW∗U,ξE4+ηUηW∗E2,VE4+ηVηW∗U,E2E4−gE2,E4ηW∗U,Vξ+ηE4ηW∗U,VE2=0.
ReplacingE2 by U in the above equation, we have(59)W∗U,V,E4,U−gU,UηW∗ξ,VE4−gU,VηW∗U,ξE4+ηE4ηW∗U,VU=0,provided α2≠0. By virtue of (43), one can easily see that(60)ηW∗E1,E2E3=−12n−1gQE2,E3ηE1−gQE1,E3ηE2+r2n2n−1−2n−12n−1α2gE2,E3ηE1−gE1,E3ηE2,ηW∗E1,E2ξ=0,(61)ηW∗E1,ξE3=12n−1gQE1,E3+12n−12n−1α2−r2ngE1,E3+12n−1r2n+2α2ηE1ηE3,(62)∑i=12n+1W∗ei,E2,E3,ei=−2n−1α2gE2,E3−α2ηE2ηE3,where eii=12n+1 is an orthonormal basis of the tangent space at any point of the manifold. Taking U=ei in (59) and summing over i and making use of (56)–(61), we get(63)SV,E4=12nr+α2gV,E4−12nr+4n2+1α2ηVηE4.
This completes the proof.


## 5.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. 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 V is defined by(64)EVgE1,E2+2SE1,E2+2λgE1,E2=0,for some constant λ. The Ricci soliton is said to be shrinking, steady, and expanding accordingly as λ 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 g on M is called ∗-Ricci soliton, if the Lie derivative of a vector field V on M is given by(65)EVgE1,E2+2S∗E1,E2+2λgE1⋅E2=0,∀E1,E2∈TM.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 α-cosymplectic manifold. Now, we prove the following result.Theorem 6.
LetM be α-cosymplectic manifold admitting a ∗-Ricci soliton. If the potential vector field V is pointwise collinear with ξ, then M is a near quasi-Einstein manifold.Proof.
LetV be a pointwise collinear vector field with ξ. Then, we have V=bξ. From (7) and (65), we derive(66)EVgE1,E2=g∇E1V,E2+g∇E2V,E1=E1bηE2+E2bηE1+2bαgE1,E2−ηE1ηE2.
LetM be an α-cosymplectic manifold admitting a ∗-Ricci soliton. Then, from (61) and (62), we obtain(67)S∗E1,E2=−λ+bαgE1,E2+bαηE1ηE2−12E1bηE2+E2bηE1.
LetDb be a gradient of smooth function b on M, that is, E1b=gDb,E1 and E2b=gDb,E2. Then, by denoting the dual form of Db by v, we write(68)E1b=vE1,E2b=vE2.
By taking account of foregoing equations in (67), we get(69)S∗E1,E2=−λ+bαgE1,E2+bαηE1ηE2−12vE1ηE2+vE2ηE1.
Then, from (25), equation (65) reduces to(70)SE1,E2=−λ+bα+2n−1α2gE1,E2+bα−α2ηE1ηE2−12vE1ηE2+vE2ηE1.
Let us take a nonvanishing symmetric0,2 tensor E in (66), such that(71)SE1,E2=bα−α2ηE1ηE2−12vE1ηE2+vE2ηE1.
Then, equation (66) yields(72)SE1,E2=agE1,E2+EE1,E2,where a=−λ+bα+2n−1α2. So, M is a near quasi-Einstein.As an immediate outcome of Theorem6, we have the following corollary.Corollary 4.
Anα-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 (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 flatα-cosymplectic manifold admitting a ∗-Ricci soliton is a manifold of near quasi-constant 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-dimensionalα-cosymplectic manifold admitting ∗-Ricci soliton is a manifold of nearly quasiconstant curvature if V is a pointwise collinear with ξ.


## 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 isα-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, 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. 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 α-cosymplectic manifolds. By the way, important physical results have been stated in this study.



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*Source: 7939654-2022-02-21.xml*