# Almostα-Cosymplectic Pseudo Metric Manifolds

Authors: Sermin Öztürk; Hakan Öztürk
Journal: Journal of Mathematics
 (2021)

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## Abstract

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


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## Body

## 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 almostα-cosymplectic manifolds [5]. The authors investigated canonical foliations of an almost α-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 studiedη-parallel contact metric spaces in [15]. They considered a milder condition that h is η-parallel, i.e.,(1)g∇XhY,Z=0,in contact metric manifolds for all X,Y,Z∈D.In [16], Ghosh et al. studied the η-parallelity of the torsion tensor τ for a contact metric manifold M2n+1. The torsion tensor field τ defined as(2)gτX,Y=LξgX,Y,for any vector fields X and Y on M2n+1 was firstly introduced by Hamilton and Chern [17].In this paper, we consider the almostα-cosymplectic pseudo metric manifold which is a wide subclass of almost contact pseudo metric manifolds. We first give the concept of almost α-cosymplectic pseudo metric manifolds and state general curvature properties. We derive several formulas on almost α-cosymplectic pseudo metric manifolds. These formulas would enable us to find the geometrical properties of almost α-cosymplectic pseudo metric manifolds with η-parallel tensor h and φh. We study the η-parallelity of the tensor fields h and φh. Next, we obtain some results related to the η-parallelity and η-cyclic parallelity of the torsion tensor τ. Moreover, we investigate the deformation of almost α-Kenmotsu pseudo metric structure. Finally, we give an illustrative example of almost α-cosymplectic pseudo metric manifolds.


## 2. Preliminaries

LetM2n+1 be a 2n+1-dimensional differentiable manifold equipped with a triple ϕ,ξ,η, whereϕ is a type of (1, 1) tensor field, ξ is a vector field, and η is a 1-form on M2n+1 such that(3)ηξ=1,ϕ2=−I+η⊗ξ,which implies(4)φξ=0,η∘φ=0,rankφ=2n.A pseudo Riemannian metricg on M2n+1 is said to be compatible with the almost contact structure φ,ξ,η if gφX,φY=gX,Y−εηXηY where ε=±1.A smooth manifoldM2n+1 furnished with an almost contact structure φ,ξ,ηand a compatible pseudo Riemannian metric g is called an almost contact pseudo metric manifold which is denoted by M2n+1,φ,ξ,η,g. It is clear that gφX,Y=−gX,φY, ηX=εgX,ξ, and gξ,ξ=ε.On such a manifold, the fundamental 2-formΦ of M2n+1 is defined by ΦX,Y=gX,φY for any vector fields X,Y on M2n+1 [18]. An almost contact pseudo metric manifold satisfying the conditions dη=0 and dΦ=2αη∧Φ is said to be an almost α-Kenmotsu pseudo metric manifold for α≠0 and α∈R. It is well known that the normality of almost contact structure is expressed by the vanishing of the tensor as follows:(5)NφX,Y=φX,φY−φφX,Y−φX,φY+φ2X,Y+2dηX,Yξ,where φ,φ is the Nijenhuis tensor of φ [19].An almost contact pseudo metric manifoldM2n+1,φ,ξ,η,g is said to be almost cosymplectic pseudo metric manifold if dη=0 and dΦ=0, where d is the exterior differential operator.If we join these two classes, we obtain the notion of an almostα-cosymplectic pseudo metric manifold, defined by dη=0and dΦ=2αη∧Φ, for any real number α [5]. When an almost α-cosymplectic pseudo metric manifold M2n+1 has a normal almost contact structure, we can say that M2n+1 is an α-cosymplectic pseudo metric manifold. In this paper, we shall denote by ΓTM and ∇ the Lie algebra of all tangent vector fields on M2n+1 and the Levi Civita connection of pseudo Riemannian metric g, respectively.


## 3. Certain Properties

In this section, we give the basic relations on almostα-cosymplectic pseudo metric manifolds.Proposition 1.
LetM2n+1 be an almost contact metric manifold and ∇ be the Riemannian connection. Then, the following equations are held [3]:(6)∇XΦY,Z=gY,∇XφZ,(7)∇XΦY,Z+∇XΦφY,φZ=ηZ∇XηφY−ηY∇XηφZ,(8)∇XηY=gY,∇Xξ=∇XΦξ,φY,(9)2dηX,Y=∇XηY−∇YηX,(10)3dΦX,Y,Z=⊕X,Y,Z∇XΦY,Z.
Here,⊕X,Y,Z denotes the cyclic sum over the vector fields X,Y, and Z [1].Lemma 1.
LetM2n+1 be an almost contact pseudo metric manifold. Then, the following equation is held:(11)2g∇XφY,Z=3dΦX,φY,φZ−3dΦX,Y,Z+gN0Y,Z,φX+εN1Y,ZηX+2εdηφY,XηZ−2εdηφZ,XηY,for any tangent vector fields X,Y,Z∈ΓTM where N0andN1 are defined by(12)N0X,Y=NφX,Y+2dηX,Yξ,(13)N1X,Y=LφXηY−LφYηX,respectively. Here, LX denotes the Lie derivative in the direction of X [20].Proposition 2.
LetM2n+1 be an almost α-cosymplectic pseudo metric manifold. Then, we have(14)hX=12LξφX,hξ=0,(15)∇Xξ=−αφ2X−φhX,(16)∇ξξ=0,∇ξφ=0,(17)φ∘hX+h∘φX=0,(18)∇XηY=αεgX,Y−ηXηY+εgφY,hX,(19)trh=0,h=0⟺∇ξ=−αφ2,for any tangent vector fields X,Y,Z∈ΓTM.Proof.
Considering the Koszul formula (11), we have(20)2g∇XφY,Z=3dΦX,φY,φZ−3dΦX,Y,Z+gN0Y,Z,φX+εN1Y,ZηX.
In view of (6), (7), and (10) for Y=ξ, we deduce(21)2g∇Xφξ,Z=2gφZ,X+gN0ξ,Z,φX+εN1ξ,ZηX.
Taking into account of (12) and (13), we get(22)N0ξ,Z=−ξ,Z+ηξ,Zξ,N1ξ,Z=ηφZ,ξ.
Then, making use of (22) in (21) and (20) reduces to(23)2g∇Xφξ,Z=2gφZ,X−gLξφZ,X+ηXεηξ,φZ+ηξ,φZ=−2gφX,Z−gLξφZ,X.
Here, if we choose the symmetric (1, 1)-type tensor fieldh as follows:(24)hZ=12LξφZ,for any vector fields Z, and Equation (24) takes the form(25)g∇Xφξ,Z=−gαφX,Z−ghZ,X,which completes the proof of (15). From (14) and (15), the first equation of (16) is obvious. Moreover, using (20) for X=ξ and putting X=ϕX and Y=ϕY in (13), we obtain the following:(26)2g∇ξφY,Z=−2gφY,Z−2gY,φZ+εN1Y,Z.
This means that∇ξφ=0 for any nonzero vector field Z. Now, considering the sum of φ∘h and h∘φ for any vector field X, we have(27)φhX+hφX=12φξ,φX−φ2ξ,X+ξ,φ2X−φξ,φX=12−η∇ξXξ+η∇Xξξ+ε∇ξgX,ξξ=η∇Xξξ.
Equation (27) shows that the sum of φ∘h and h∘φ vanishes identically. In addition, from (8), we get(28)∇XηY=εαgX,Y−ε2αηXηY−εgY,φhX,where ε2=1. Thus, we can complete the proof of (18). Also, from (14) and (15), we can easily obtain (19). Here, if h∗ is defined by h∗=−φh, then we have h∗φ=−φh∗,h2=h∗2, and hξ=trh=0.
Now, we investigate the curvature properties of almost contact pseudo metric manifolds. First, we have the following propositions.Proposition 3.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. Then, we have(29)RX,Yξ=α2ηXY−ηYX−αηXφhY−ηYφhX+∇YφhX−∇XφhY,for any tangent vector fields X,Y∈ΓTM.Proof.
Making use of the Riemannian curvature tensor and (15), we obtain (29) such that ηX=εgξ,X.Proposition 4.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. Then, the following relations are held:(30)RX,ξξ=α2φ2X+2αφhX−h2X+φ∇ξhX,(31)∇ξhX=−φRX,ξξ−α2φX−2αhX−φh2X,(32)RX,ξξ−φRφX,ξξ=2α2φ2X−h2X,(33)SX,ξ=−2nα2ηX−divφhX,(34)Sξ,ξ=−2nα2+trh2,(35)divξ=2αn,(36)divη=−2αnε,for any tangent vector fields X,Y∈ΓTM.Proof.
By the hypothesis, using (29) with Y=ξ and considering the following equations:(37)∇ξφhX=φ∇ξhX,∇Xφhξ=h2X−φhX,we obtain (30). Applying φ to (30) and remarking that g∇ξhX,ξ=0, we get (31). Also, with the help of (30) for φX, we have(38)RφX,ξξ=−α2φX−2αφ2hX−φh2X+φ∇ξhφX.
Then, we get(39)RX,ξξ−φRφX,ξξ=2α2φ2X−2h2X+φ∇ξhX+φ2∇ξhφX,which reduces to (32) where ∇ξh∘φ=−φ∘∇ξh.
Now, we may take a local orthonormalφ-basis as follows:(40)E1,…,E2n,ξ=e1,…,en,φe1,…,φen,ξ.
From (29) and the Ricci curvature tensor, we have(41)SX,ξ=∑i=12n+1εigREi,Xξ,Ei=∑i=1nεigRei,Xξ,ei+∑i=1nεi+ngRφei,Xξ,φei+ε2n+1gRξ,Xξ,ξ,where ε2n+1gRξ,Xξ,ξ=0. It follows that(42)SX,ξ=−2nα2ηX+αηX∑i=1nεigφhei,ei+εi+ngφhφei,φei+∑i=1nεig∇Xφhei,ei+εi+ng∇Xφhφei,φei−∑i=1nεig∇eiφhX,ei+εi+ng∇φeiφhX,φei.
Then, we have(43)SX,ξ=−2nα2ηX−∑i=12n+1εigREi,Xξ,Ei=−2nα2ηX−divφhX,such that(44)divφh=∑i=12n+1εig∇EiφhX,Ei.
Sincetrφh=0, we deduce(45)0=∑i=1nεigφhei,ei+εi+ngφhφei,φei.
Thus, the proof of (33) completes. Moreover, putting X=ξ in (33), we obtain (34) where divφhξ=trh2. This proof can also be given in another way. Consider the local orthonormal φ-basis on M2n+1. The sectional curvatures of nondegenerate planes spanned by ξ,ei and ξ,φei, respectively, are defined as(46)Kξ,ei=εεiRξ,ei,ξ,ei=εεiglei,ei,(47)Kξ,φei=εεiRξ,φei,ξ,φei=−εεigφlφei,ei,where εi=gei,ei=gφei,φei=±1 for all indices i=1,…,n and l is the Jacobi operator defined by l=R.,ξξ. Thus, we have(48)lX−φlφX=2α2φ2X−h2X,and from (46) and (47), it follows that(49)Sξ,ξ=∑i=1nRξ,ei,ξ,ei+∑i=1nRξ,φei,ξ,φei=ε∑i=1nεiKξ,ei+Kξ,φei=∑i=1nglei−φlφei,ei=−2nα2−trh2.
It is well known that(50)divη=−tr∇η=δη=−∑i=1n∇eiηei+∇φeiηφei.
From Equation (50), we obtain(51)divξ=α∑i=1nεigei−φhei,ei+gφei−φhφei,φei,divη=−tr∇η=−εdivξ,which completes the proof.


## 4. Main Results

In this section, we consider some certain parallel tensor conditions on almostα-cosymplectic pseudo metric manifolds. Also, we study the deformation of almost α-Kenmotsu pseudo metric manifolds with α>0. Firstly, we study the η-parallelity of the tensor fields h and φh on almost α-cosymplectic pseudo metric manifolds. As we know that we can take X=XT+ηXξ where XT is tangentially part of X and ηXξ is the normal part of X. So, the symmetric (1, 1)-type tensor field B on a Riemannian manifold M,g is said to be a η-parallel tensor if it holds the following:(52)g∇XTBYT,ZT=0,for all tangent vectors XT, YT, and ZTorthogonal to ξ [15].Proposition 5.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If h satisfies the η-parallelity condition, then we have(53)∇XhY=ηX∇ξhY−ηYαhX+φh2X−εgY,αhX+φh2Xξ,for any tangent vector fields X,Y∈ΓTM.Proof.
Assume thath is η-parallel. From (52), we have(54)0=g∇XhY,Z−ηXg∇ξhY,Z−ηYg∇Xhξ,Z−ηZg∇XhY,ξ+ηXηYg∇ξhξ,Z+ηYηZg∇Xhξ,ξ+ηZηXg∇ξhY,ξ−ηXηYηZg∇ξhξ,ξ,for any X,Y,Z∈ΓTM. It follows that(55)0=g∇XhY,−φ2Z−ηXg∇ξhY,Z−ηYg∇Xhξ,Z.
From Equation (55), we deduce(56)∇XhY=−ηXφlY+α2φY+2αhY+φh2Y−ηY−αφ2hX+φh2X+εg∇XhY,ξξ.
Thus, it completes the proof.Proposition 6.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If φh satisfies the η-parallelity condition, then we have(57)∇XφhY=ηXlY−α2φ2Y−2αφhY+h2Y−ηYαφhX−h2X−εgY,αφhX−h2Xξ,for any X,Y∈ΓTM.Proof.
By the hypothesis, we suppose thatφh is η-parallel. Then, we have(58)0=g∇XφhY,Z−ηXg∇ξφhY,Z−ηYg∇Xφhξ,Z−ηZg∇XφhY,ξ+ηXηYg∇ξφhξ,Z+ηYηZg∇Xφhξ,ξ+ηZηXg∇ξφhY,ξ−ηXηYηZg∇ξφhξ,ξ,for any X,Y,Z∈ΓTM. By a straightforward computation, we obtain(59)g∇XφhY,φ2Z=−ηXg∇ξφhY,Z−ηYg∇Xφhξ,Z.
With the help of (15) and (17), Equation (59) reduces to(60)∇XφhY=−ηYαφhX−h2X+ηX∇ξφhY+εghY,hXξ+αgφY,hXξ.
From Equation (60) and ∇ξφhY=φ∇ξhY, (57) is easily seen. Then, the proof is completed.Theorem 1.
An almostα-cosymplectic pseudo metric manifold with η-parallel tensor φh holds the following equation:(61)RX,Yξ=−ηXlY+ηYlX,for any X,Y∈ΓTM where l=R.,ξξ is the Jacobi operator with respect to ξ.Proof.
Making use of (29), we have(62)RX,Yξ=α2ηXY−α2ηYX−αηXφhY+αηYφhX−ηX∇ξφhY+ηYαφhX−h2X+εgY,αφhX−h2Xξ+ηY∇ξφhX−ηXαφhY−h2Y−εgX,αφhY−h2Yξ.
Then, simplifying Equation (62), we obtain(63)RX,Yξ=−ηXlY+ηYlX+ηXα2φ2Y−ηYα2φ2X+ηXα2Y−ηYα2X,which is desired result.Theorem 2.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold with η-parallel tensor φh. Then, ξ is the eigenvector of Ricci operator on M2n+1.Proof.
Lete1,…,e2n,ξ be an orthonormal basis of the tangent space at any point. Taking the inner product of both sides of (61) with respect to Z and contracting (61) for 1≤i≤2n+1 with X=W=ei, we have(64)∑i=12n+1εigRei,Yξ,ei=∑i=12n+1εiηYglei,ei−ηeiglY,ei,for any X,Y∈ΓTM. This means that(65)Sξ,ξ=∑i=12n+1εiglei,ei.
Thus, it completes the proof.Theorem 3.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If h satisfies the η-parallelity condition and ∇ξh=0, then the eigenvalues of h are constant.Proof.
LetZ∈D be an eigen unit vector field such that hZ=μZ where μ is an eigen function corresponding to the vector field Z. Then, (53) can be written as(66)g∇XhZ,Z=ηXg∇ξhZ,Z=ηXξμ,for Z∈D. Also, we have(67)g∇XhZ,Z=Xμ.
Taking into account of (66) and (67), we also get(68)Xμ=ηXξμ,dμ=η⊗ξμ.
Furthermore, since∇ξh=0, we obtain(69)g∇ξhZ,Z=0=ξμ.
Follows from (68) and (69), we have dμ=0. Thus, it completes the proof.Proposition 7.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. Then, the torsion tensor field τ holds the following:(70)τX=2∇Xξ,for any X∈ΓTM.Proof.
From the definition ofτ, we get(71)LξgX,Y=g−αφ2X−φhX,Y+gX,−αφ2Y−φhY,which completes the proof.Proposition 8.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If τ is η-parallel, then we have(72)∇XφhY=ηX∇ξφhY−ηYφh∇Xξ+εg∇Xφhξ,Yξ,for any X,Y∈ΓTM.Proof.
The hypothesis is essentially same as(73)g∇XTτYT,ZT=0,for all tangent vectors orthogonal to ξ. Putting XT=X−ηXξ and using the definition of τ, we obtain(74)∇XτY=ηX∇ξτY+ηY∇Xτξ+g∇XτY,ξξ.
It follows that(75)∇XτY=−2αηY∇Xξ−2αηYg∇Xξ,Yξ−2∇XφhY.
PuttingY=ξ in (75), we have(76)∇Xτξ=−2α∇Xξ+2φh∇Xξ.
Also, it is noted that(77)∇ξτY=−2∇ξφhY.
Finally, taking into account of (74)–(77), (72) holds. Then, we complete the proof.Theorem 4.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If τ is η-parallel, then ξ is the eigenvector of Ricci operator on M2n+1.Proof.
From (72), we have(78)∇YφhX−∇XφhY=ηY∇ξφhX−ηXφh∇Yξ+εg∇Yφhξ,Xξ−ηX∇ξφhY+ηYφh∇Xξ−εg∇Xφhξ,Yξ.
Simplifying Equation (78), we get(79)∇YφhX−∇XφhY=ηY∇ξφhX−ηXφh∇Yξ−ηX∇ξφhY+ηYφh∇Xξ.
With the help of (29), (31), and (79), we obtain(80)RX,Yξ=ηY−φ2lX−α2φ2X−2αφhX−φ2h2X−ηX−φ2lY−α2φ2Y−2αφhY−φ2h2Y−ηXφh−αφ2Y−φhY+α2ηXY+ηYφh−αφ2X−φhX−α2ηYX−αηXφhY+αηYφhX.
By a direct calculation, the desired result is achieved.Theorem 5.
LetM2n+1,φ,ξ,η,g be an almost α-cosymplectic pseudo metric manifold. If τ is cyclically η-parallel, then ξ is the eigenvector of Ricci operator on M2n+1.Proof.
According to the hypothesis, it means that(81)0=g∇XTτYT,ZT+g∇YTτZT,XT+g∇ZTτXT,YT,for all tangent vectors orthogonal to ξ. Simplifying Equation (81), we obtain(82)0=⊕X,Y,Zg∇XτY,Z−ηX⊕Y,Z,ξg∇YτZ,ξ−ηY⊕Z,X,ξg∇ZτX,ξ−ηZ⊕X,Y,ξg∇XτY,ξ,where ⊕X,Y,Z denotes the cyclic sum over X,Y,Z. It follows that(83)⊕X,Y,Zg∇XφhY,Z−ηXg∇ξφhY,Z+g∇YφhZ,ξ−gφh∇Zξ,Y−ηYg∇ξφhZ,X+g∇ZφhX,ξ−gφh∇Xξ,Z−ηZg∇XφhY,ξ+g∇ξφhX,Y−gφh∇Yξ,X=0.
Contracting (83) with respect to Y and Z, we have(84)ηX2nα2+3trh2+trl−2divφhX=0.
From (33), (84) reduces to(85)Qξ=−32trh2+13trl+2α2nξ.
From (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.
Letφ,ξ,η,g be an almost contact pseudo metric structure associated to a compatible pseudo Riemannian metric where gξ,ξ=ε on a smooth manifold M2n+1. With the help of [20], we have the following pseudo Riemannian metric formula:(86)g∗X,Y=αgX,Y−2εηXηY,for any X,Y∈ΓTM where g∗ξ,ξ=ε∗=εα−2, α>0. This means that (86) is still compatible pseudo metric with the same almost contact pseudo metric structure φ,ξ,η,g.
Thus, we give the following results. Here, we denote by∇∗ and R∗ as the semi Riemannian connection and the curvature tensor of g∗ on almost α-Kenmotsu pseudo metric manifold, respectively.Proposition 9.
LetM2n+1,φ,ξ,η,g be an almost α-Kenmotsu pseudo metric manifold. Then, M2n+1,φ,ξ,η,g∗ is also an almost α-Kenmotsu pseudo metric manifold in the sense of (86).Proof.
According to (86), we denote by Φ the fundamental 2-form with respect to M2n+1,φ,ξ,η,g∗ and then we have Φ∗X,Y=g∗X,φY=αΦX,Y with α>0. Furthermore, from Φ∗X,Y=αΦX,Y, we obtain dΦ∗X,Y=2αη∧Φ∗X,Y for any X,Y∈ΓTM. Thus, it completes the proof.Proposition 10.
LetM2n+1,φ,ξ,η,g be an almost α-Kenmotsu pseudo metric manifold and g∗ the pseudo Riemannian metric given by (86). Then, the following equations are held:(87)∇X∗Y=∇XY+2εαg∇Xξ,Yξ,(88)R∗X,YZ=RX,YZ−2εαg∇Xξ,Z∇Yξ−g∇Yξ,Z∇Xξ2εαg∇YφhX−∇XφhY,Zξ+2εg∇Yφ2X−∇Xφ2Y,Zξ,for any X,Y∈ΓTM.Proof.
According to the Koszul formula, we have(89)2g∗∇X∗Y,Z=Xg∗Y,Z+Yg∗X,Z−Zg∗X,Y+g∗X,Y,Z+g∗Z,X,Y+g∗Z,Y,X.
Using (86) and (89), we get(90)g∗∇X∗Y,Z=αg∇XY,Z−2εηZg∇Xξ,Y,where XηY=g∇Xξ,Y.
On the other hand, making use of (86) in (90), it follows that(91)αg∇X∗Y−∇XY,Z−2εηZη∇X∗Y−XηY=0.
Then, taking into account of (90) and (91), we obtain (87). Moreover, using the definition of Riemannian curvature tensor and (87), we have(92)R∗X,YZ=∇X∗∇YZ+2εαg∇Yξ,Zξ−∇Y∗∇XZ+2εαg∇Xξ,Zξ−∇X,Y∗Z−2εαg∇X,Yξ,Zξ.
From (92), we obtain(93)R∗X,YZ=RX,YZ+2εα∇Xg∇Yξ,Zξ+2εαg∇Xξ,∇YZ+4α2g∇Xξ,ξg∇Yξ,Zξ−2εα∇Yg∇Xξ,Zξ−2εαg∇Yξ,∇XZ−4α2g∇Yξ,ξg∇Xξ,Zξ−2εαg∇X,Yξ,Zξ.
Finally, substituting (15) into (92), it reduces to (88). Then, we complete the proof.


## 5. An Example

Consider theM⊂R13 manifold such that M=u,v,w∈R13:w≠0, where u,v,w are the standard coordinates in R13. The vector fields are(94)e1=e2w∂∂u,e2=e2w∂∂v,e3=∂∂w.Moreover, the following equations hold:(95)g=e−4wε1du2+ε2dv2+εdw2,η=dw,φξ=0,φe1=e2,φe2=−e1,φ2X=−X+ηXe3,ηX=εge3,X,ηe3=ge3,e3=ε=ε3,gφX,φY=gX,Y−εηXηY,εi=gei,ei,i=1,2,3.From Equation (95), there exists an almost contact pseudo metric structure φ,ξ,η,g on M. In order to check, whether it is almost α-cosymplectic pseudo metric or not, we verify the condition dΦ=2αη∧Φ. On the other hand, all Φijs vanish except for Φe1,e2=−εi. Hence, we have(96)Φ∂∂u,∂∂v=−εie−4w,Φ=−εie−4wdu∧dv.It follows that(97)dΦ=4εie−4wdu∧dv∧dw.Sinceη=dw, we have(98)dΦ=2−2εiη∧Φ.Here, it is noted thatNφ=0. Therefore, M is an α-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 almostα-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 α-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 α-cosymplectic pseudo metric manifolds.



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*Source: 4106025-2021-07-30.xml*