# Certain Class of Almostα-Cosymplectic Manifolds

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

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## Abstract

This paper concerned with almostα-cosymplectic manifolds satisfying conformally flat condition. Firstly, we investigate Kaehler integral submanifolds of almost α-cosymplectic manifolds. Next, we study conformally flat almost α-cosymplectic manifolds of dim ≥ 5 whose integral submanifolds are Kaehler. Finally, an illustrative example is constructed to verify our result.


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## Body

## 1. Introduction

The notion of conformal flatness is one of the most primitive concepts in differential geometry. Notwithstanding this fact, most of the studies have been local character. However, Kulkarni classified conformally flat manifolds up to conformal equivalence [1].On a Riemannian manifoldM, Weyl established a tensor of type (1, 3) which vanishes whenever the metric is (locally) conformally equivalent to a flat metric. Therefore, this tensor is called the conformal curvature tensor of the metric and defined by(1)CX,YZ=RX,YZ−12n−1SY,ZX−SX,ZY+gY,ZQX−gX,ZQY+r2n2n−1gY,ZX−gX,ZY,for any vector fields X,Y,andZ on M. Here, we denote by R and r the Riemann curvature tensor and scalar curvature of M, respectively [2, 3]. A necessary condition being conformally flat for a Riemannian manifold is the vanishing of the Weyl curvature tensor. It is obvious that the Weyl tensor vanishes identically in 2 dimensions. In general, it is nonzero in dimensions ≥ 4. The metric is locally conformally flat provided that the Weyl tensor vanishes for 4 dimensions. In this case, the metric has a local coordinate system where it is proportional to a constant tensor. In dimension 3, we have(2)cX,Y=∇XQY−∇YQX−12n−2∇XrY−∇YrX.Equation (2) is a necessary and sufficient condition for three-dimensional Riemannian manifold being conformally flat. Here, c is the divergence operator of C [3].A (2n + 1)-dimensional Riemannian manifoldM2n+1 is conformally flat if and only if the Weyl conformal curvature tensor C vanishes for any vector fields X,Y,andZ when n>2 or the tensor L of type (1, 1) defined as(3)LX=12n−1QX−r4nX,for any vector field X, is of Codazzi type when n=1. Here, Q is the Ricci operator associated with the Ricci tensor S and r is the scalar curvature [4].In contact metric manifolds, Okumura obtained that a conformally flat Sasakian manifold of dimension> 3 is locally isometric to the unit sphere [5]. Later, this result was extended to the K-contact manifolds by Tanno for dimension ≥ 3. Gosh and Sharma showed that a conformally flat contact strongly pseudo-convex integrable CR manifold is locally isometric to a unit sphere if the characteristic vector field is an eigenvector of the Ricci tensor at each point [6]. Afterwards, Gosh et al. obtained that a conformally flat contact strongly pseudo-convex integrable CR manifold of dimension > 3 is of constant curvature 1 [7].Moreover, if an almost cosymplectic manifold with dimension (2n + 1) is conformally flatn≥2, then it is locally flat and cosymplectic [8]. Conversely to this, there exist conformally flat almost cosymplectic manifolds with Kaehler leaves which are not locally flat and not cosymplectic in dimension 3 [8].Recently, Blair et al. have focused on almost contact metric manifolds with conformally flat condition [9]. The authors construct an illustrative example of 3-dimensional conformally flat almost α-Kenmotsu manifold whose sectional curvature is nonconstant. Furthermore, they consider conformally flat almost contact metric manifolds which are ∗−η-Einstein manifold with dim ≥ 5. Moreover, Wang investigated conformally flat almost Kenmotsu manifold of k-dimension 3 [4]. After these studies, we also point out that Weyl conformal curvature tensor has been studied extensively by Venkatesha et al. [10].In this paper, we study the geometry of conformally flat almostα-cosymplectic manifolds. We aim at characterizing and classifying conformally flat almost α-cosymplectic manifolds. Then, we obtain the curvature properties of almost α-cosymplectic manifolds with Kaehler integral submanifolds and investigate almost α-cosymplectic manifolds of dim ≥ 5 which are Kaehler integral submanifolds. Finally, we give a concrete example of 3-dimensional almost α-Kenmotsu manifolds.


## 2. Preliminaries

LetM2n+1 be a 2n+1-dimensional smooth manifold endowed with a triple ϕ,ξ,η where ϕ is a type of (1, 1) tensor field, ξ is a vector field, and η is a 1-form on M2n+1 such that(4)ηξ=1,ϕ2=−I+η⊗ξ,ϕξ=0,η∘ϕ=0,rankϕ=2n.IfM2n+1 admits a Riemannian metric g, defined by(5)gϕX,ϕY=gX,Y−ηXηY,ηX=gX,ξ,then M2n+1 is called almost contact structure ϕ,ξ,η,g. Also, the fundamental 2-form Φ of M2n+1 is defined by ΦX,Y=gϕX,Y. If the Nijenhuis tensor vanishes, defined by(6)NϕX,Y=ϕX,ϕY−ϕϕX,Y−ϕX,ϕY+ϕ2X,Y+2dηX,Yξ,then M2n+1,ϕ,ξ,η is said to be normal [3]. It is obvious that a normal almost Kenmotsu manifold is said to be Kenmotsu manifold. In other words, an almost contact metric manifold is known as Kenmotsu if and only if ∇XϕY=gϕX,Yξ−ηYϕX [11]. An almost contact metric structure is cosymplectic if and only if ∇η and ∇Φ are closed [12].In the light of the above definitions, the generalization of almost Kenmotsu manifoldM2n+1 is called almost α-Kenmotsu manifold if dη=0 and dΦ=2αη∧Φ, where α is a nonzero real constant [13, 14]. If we combine almost α-Kenmotsu and almost cosymplectic manifolds, then we introduce a new notion of an almost α-cosymplectic manifold defined by dη=0 and dΦ=2αη∧Φ for any real number α [15]. A normal almost α-cosymplectic manifold is said to be α-cosymplectic manifold, and it is either cosymplectic or α-Kenmotsu under the condition α=0 or α≠0, respectively [16–19].LetM2n+1 be an almost α-cosymplectic manifold endowed with ϕ,ξ,η,g. D is the contact distribution of M2n+1 given by D=kerη. Since dη=0, D is integrable and the 2n-dimensional distribution is defined by ϕD=D. Moreover, it is clear that ξ is orthogonal to D. Assume that N is a maximal integral submanifold of D. Therefore, the ξ restricted to integral submanifold N is the normal vector of N. Thus, there exists a Hermitian structure and the tensor field ϕ induces an almost complex structure J defined by JX=ϕX for any vector field X tangent to N [12, 13, 20].Suppose thatG is the Riemannian metric induced on N defined by GX,Y=gX,Y. Then, J,G has an almost Hermitian structure on N given by GX,Y=GJX,JY for any vector field X and Y tangent to N. The fundamental 2-form Ω,ΩX,Y=GJX,Y of J,G induced on N. This means that ΩX,Y=ΦX,Y, i.e., Ω is the pull-back of the tensor field ϕ from M2n+1 to N. Since Ω is closed, we obtain dΩ=0. Thus, the pair J,G is an almost Kaehler structure on N of D. When the structure J is complex, J,G becomes a Kaehler structure on N. If the structure J,G is Kaehler on every integral submanifolds of the distribution D, this manifold is said to be an almost α-cosymplectic manifold with Kaehler integral submanifolds.Denote byA and h the (1, 1) tensor fields on M2n+1 defined by(7)A=−∇ξ,h=12Lξϕ,respectively. Here, L is the Lie derivative of g. Obviously, Aξ=0 and hξ=0. Thus, we the following relations for any vector fields X,Y on M2n+1 [16]:(8)∇Xξ=−αϕ2X−ϕhX,(9)ϕ∘hX+h∘ϕX=0,ϕ∘AX+A∘ϕX=−2αϕ,(10)trh=0,trA=−2αn,trϕA=0,Aϕ+ϕA=−2α,(11)∇XAξ=A2X,trA2=A2.


## 3. Curvature Properties

This section deals with the fundamental curvature equations of almostα-cosymplectic manifolds with Kaehler integral submanifolds. Let us give the basic propositions that we will use in later usage. The proof of some propositions are left to the reader for shortness.Proposition 1.
LetM2n+1 be an almost α-cosymplectic manifold. Then, we have(12)RX,Yξ=α2ηXY−ηYX−αηXϕhY−ηYϕhX+∇YϕhX−∇XϕhY,(13)RX,Yξ=−∇XAY+∇YAX,(14)RX,ξξ=α2ϕ2X+2αϕhX−h2X+ϕ∇ξhX,(15)∇ξhX=−ϕRX,ξξ−α2ϕX−2αhX−ϕh2X,(16)RX,ξξ−ϕRϕX,ξξ=2α2ϕ2X−h2X,(17)SX,ξ=−2nα2ηX−∑i=12n+1g∇eiϕhei,X,(18)trl=−2nα2+trh2,where l=R.,ξξ is the Jacobi operator with respect to ξ. By direct computations, we have the following proposition.Proposition 2.
An almostα-cosymplectic manifold M2n+1 with Kaehler integral submanifolds holds the following equation:(19)RX,YϕZ−ϕRX,YZ=gAX,ϕZAY−gAY,ϕZAX−gAX,ZϕAY+gAY,ZϕAX+ηZϕ∇XAY−∇YAX+g∇XAY−∇YAX,ϕZξ=gAX,ϕZAY−gAY,ϕZAX−gAX,ZϕAY+gAY,ZϕAX−ηZϕRX,Yξ−gRX,Yξ,ϕZξ.Remark 1.
The Ricci operatorQ does not have to commute with the basic collineation ϕ for a contact metric manifold. Now, we give this condition for almost α-cosymplectic manifold M2n+1 with Kaehler integral submanifolds.Proposition 3.
LetM2n+1 be an almost α-cosymplectic manifold with Kaehler integral submanifolds. The following identity is held:(20)Qϕ−ϕQ=lϕ−ϕl+4α1−nϕA+4α21−nϕX+η∘Qϕξ−η∘ϕQξ.Proof.
Using (5) and (19), one obtains(21)gϕRϕX,ϕYZ,ϕW=gϕRZ,WX,ϕY+gAZ,ϕXgAW,ϕY−gAW,ϕXgAZ,ϕY−gAZ,XgϕAW,ϕY+gAW,XgϕAZ,ϕY−ηXgϕRZ,Wξ,ϕY−ηRϕX,ϕYZηW.
PuttingX=ϕX and Y=ϕY in (19), we have(22)gRϕX,ϕYϕZ,ϕW−gϕRϕX,ϕYZ,ϕW=gAϕX,ϕZgAϕY,ϕW−gAϕY,ϕZgAϕX,ϕW−gAϕX,ZgϕAϕY,ϕW+gAϕY,ZgϕAϕX,ϕW−ηZgϕRϕX,ϕYξ,ϕW.
Substituting of (21) into (22) yields immediately(23)gRϕX,ϕYϕZ,ϕW=gϕRZ,WX,ϕY+gAZ,ϕXgAW,ϕY−gAW,ϕXgAZ,ϕY−gAZ,XgϕAW,ϕY+gAW,XgϕAZ,ϕY−ηXgϕRZ,Wξ,ϕY−ηRϕX,ϕYZηW+gAϕX,ϕZgAϕY,ϕW−gAϕY,ϕZgAϕX,ϕW−gAϕX,ZgϕAϕY,ϕW+gAϕY,ZgϕAϕX,ϕW−ηZgϕRϕX,ϕYξ,ϕW.
By the help of (5), (23) can be written as(24)gRϕX,ϕYϕZ,ϕW=gRZ,WX,Y−ηRZ,WXηY−gAZ,XgAW,Y+gAW,XgAZ,Y−ηXgRZ,Wξ,Y−ηRϕX,ϕYZηW+gAϕX,ϕZgAϕY,ϕW−gAϕY,ϕZgAϕX,ϕW−ηZgϕRϕX,ϕYξ,ϕW.
Contracting with respect toY and Z, we obtain(25)−ϕQϕX−QX=−ϕlϕX−lX+4α1−nA−4α21−nϕ2X−ηXQξ+∑i=12n+1ηRϕX,ϕeieiξ.
The rest of the proof follows actingϕ on the last equation.Definition 1.
Q and Q∗ are the Ricci and ∗-Ricci operators of M2n+1 defined by(26)gQX,Y=trX⟶RX,YZ,gQ∗X,Y=trX⟶ϕRX,YϕZ,and denote by r and r∗ the scalar and ∗-scalar curvatures of M2n+1, where(27)r=trQ,r=trQ∗,respectively [8].Proposition 4.
For the Ricci and∗-Ricci operators of almost α-cosymplectic manifold with Kaehler integral submanifolds M2n+1, we have(28)QY−Q∗Y=∑i=12n+1g∇eiAei,Yξ+∇ξAY−A2Y+2αn−1AY,where e1,e2,…,e2n,e2n+1=ξ is an orthonormal frame at any point of M2n+1.Proof.
Taking trace of (19) with respect to X yields(29)gQY,Z−gQ∗Y,Z=gϕAϕAY,Z+gA2Y,Z+2αngAY,Z−ηZ∑i=12n+1g∇eiAY−∇YAei,ei+g∇ξAY−∇YAξ,Z.
By using the properties ofA in (29), this ends the proof.Proposition 5.
For the scalar and∗-scalar curvatures of almost α-cosymplectic manifold with Kaehler integral submanifolds M2n+1, we obtain(30)r∗−r=2A2+4α2nn−1.Proof.
(30) is a direct consequence of (29) by means of (10) and the following equation:(31)∑i=12n+1g∇eiAξ,ei=trA2=A2.Proposition 6.
For the Ricci operator of almostα-cosymplectic manifold with Kaehler integral submanifolds M2n+1, we have(32)Qξ=−∑i=12n+1∇eiAei,gQξ,ξ=−A2.Proof.
By using the projection of (28) onto ξ and gQ∗Y,ξ=0, the proof is obvious.Proposition 7.
An almostα-cosymplectic manifold with Kaehler integral submanifolds M2n+1 satisfies the following equation:(33)ϕRξ,ϕYξ−Rξ,Yξ=−2A2Y−4αAY+4α2ϕ2Y.Proof.
From (13), we get(34)ϕRξ,ϕYξ−Rξ,Yξ=−ϕ∇ξAϕY+ϕ∇ϕYAξ+∇ξAY−∇YAξ.
The proof is clear from the right side of equation (34).Proposition 8.
LetM2n+1 be an almost α-Kenmotsu manifold n≥2. Then, it has Kaehler integral submanifolds of the distribution D if and only if it holds(35)∇XϕY=−gϕAX,Yξ+ηYϕAX,where AX=αϕ2X+ϕhX.Proof.
By using similar technique in [8], the proof is clear.Proposition 9.
LetM2n+1 be an almost α-Kenmotsu manifold n≥2. If M2n+1 is conformally flat, then M2n+1 is a space of constant negative curvature −α2.Proof.
By the help of [11, 15], we can complete the proof.


## 4. Conformal Flatness Condition

This section is devoted to study conformally flat almostα-cosymplectic manifolds whose integral submanifolds are Kaehler.Theorem 1.
IfM2n+1 is a conformally flat almost α-cosymplectic manifold with Kaehler integral submanifolds n≥2, then ξ is the eigenvector of Ricci operator on M2n+1Proof.
Substitutingξ for X in (19), we have(36)Rξ,YϕZ−ϕRξ,YZ=ηZϕlY−gϕlY,Zξ.
Letei be an orthonormal basis of vector fields on M2n+1. Taking Y=Z=ei in (36) for i=1,2,…,2n+1, then we get(37)∑i=12n+1Rξ,eiϕei=ϕQξ,where trϕl=0. Since M2n+1 is conformally flat and n≥2,C is identically zero. So, we can write(38)RX,YZ=12n−1SY,ZX−SX,ZY+gY,ZQX−gX,ZQY−r2n2n−1gY,ZX−gX,ZY.
PuttingX=ξ,Y=ei, and Z=ϕei in (38) and summing over i, we obtain(39)∑i=12n+1Rξ,eiϕei=12n−1trQϕξ+ϕQξ.
From (37) and (39), we have(40)ϕQξ=12n−1trQϕξ+ϕQξ,where trQϕ=0 and ϕQξ=0. Following from (40), we can complete the proof.Theorem 2.
IfM2n+1 is a conformally flat almost α-cosymplectic manifold with Kaehler integral submanifolds n≥2, then we have(41)trl=r2+α2n2n−1.Proof.
By the hypothesis, we have(42)gQ∗Y,Z=12n−1gQϕY,ϕZX+gQY,Z−gQξ,YηZ−gX,ZQY+r2n2n−1gY,Z−ηYηZ.
It follows that(43)r∗=12n−1r−2gQξ,ξ.
Then, making use of (32) and (43), we get(44)r∗=12n−1r+2A2.
Sincen≥2, from (30) and (44), we have(45)r∗=−2α2n,A2=−r2−α2n2n−1.
Taking into account (32) and (42), we deduce(46)gQ∗Y,Z−gQ∗Z,Y=12n−1∑i=12n+1g∇eiAei,YηZ−g∇eiAei,ZηY.
On the other hand, using (28) and (30) in (46), we have(47)gQ∗Y,Z−gQ∗Z,Y=∑i=12n+1g∇eiAei,YηZ−g∇eiAei,ZηY.
From (46) and (47), it follows that(48)∑i=12n+1∇eiAei=∑i=12n+1g∇eiAei,ξξ.
Taking into account (45) and (48), one obtains(49)∑i=12n+1∇eiAei=A2ξ=−r2ξ−α2n2n−1ξ.
Finally, making use of (32) and (49) and Theorem 1, the proof ends.Theorem 3.
A conformally flat almostα-cosymplectic manifold M2n+1n≥2 with Kaehler integral submanifolds satisfies the following equation:(50)Q=r4n+2n−14nrη⊗ξ+α222n−12η⊗ξ+α2n−1A+α222n−1.
Proof. From (42), we have(51)Q∗Y=12n−1−ϕQϕY+QY−12n2n−1rY−n−12n2n−1rηYξ−α2nηYξ.
PuttingX=ξ in (13), we get(52)Rξ,Yξ=−∇ξAY+A2Y.On the other hand, using (38) and (41), it follows that(53)Rξ,Yξ=−12n−1QY−n−12n2n−1rY+12nrηYξ+2α2nηYξ−α2nY.The last two equalities lead to(54)∇ξAY=A2Y+12n−1QY+n−12n2n−1rY−12nrηYξ−2α2nηYξ+α2nY.In view of (28), (49), (51), and (54), one finds(55)2n−3QY+ϕQϕY=n−22nrY+n−12nrηYξ+2α2nn−22n−1ηYξ+α2n2n−1Y+2αn−12n−1AY.SubstitutingϕY for Y in (55) and using (41), we have(56)2n−3ϕQϕY+QY=−n−22nrY+n−1nrηYξ+2α22−n2n−1ηYξ+α23n−42n−1Y+2αn−12n−1AY.Finally, this proof ends using (55) and (56).Theorem 4.
LetM2n+1 be a conformally flat almost α-cosymplectic manifold with Kaehler integral submanifolds n≥2. Then, the following relation is held:(57)Qϕ−ϕQ=lϕ−ϕl−4αn−1ϕA+4α2n−1ϕX.Proof.
The proof follows from (20) and Theorem 1 in [8].Theorem 5.
LetM2n+1 be an almost α-cosymplectic manifold with Kaehler integral submanifolds n≥2. If M2n+1 is conformally flat, then M2n+1 is either locally flat and cosymplectic or M2n+1 is an α-Kenmotsu manifold with constant negative curvature −α2.Proof.
LetM2n+1 be an almost α-cosymplectic manifold with Kaehler integral submanifolds. Assume additionally that M2n+1 is conformally flat and n≥2. At certain places of the main idea of the proof, we are inspired by the paper of Dacko and Olszak [8]. Also, the almost cosymplectic case α=0 is clear by means of Theorem 1 in [8].
Now, we shall prove the assertion of our theorem in case ofα≠0. Since n≥3, (50) holds. Thus, the auxiliary tensor L has the following shape:(58)LX=QX−14nrX.
From (50) and (58), L can be written as(59)LX=2n−14nrηXξ+α222n−12ηXξ+α2n−1AX+α222n−1X.
In view of (7), (10), and (59) with X=ξ, we obtain(60)2n−14nξrηYξ−Yrξ+α2n−1∇ξAY+2n−14nr+α222n−12−α2n−1AAY=0,where ξrηYξ=Yrξ. So, the last equation reduces to(61)0=α2n−1∇ξAY+2n−14nr+α222n−12−α2n−1AAY.
ReplacingX and Y by ξ in (38) and using (50), we find that(62)lX=−trl2nϕ2X+αϕhX.
Then, using (52) and (62), we have(63)∇ξAY=−trl2nϕ2X+αϕhX+A2Y.
Following from (61) and (63), we get(64)trl2n+α2ϕhX=0.
Also, using (18) and (64), one obtains(65)−trh22nϕhX=0,which reduces to h=0. Finally, we can use Proposition 9 to complete the proof.Remark 2.
It is noted that Theorem5 generalizes the result of Dacko and Olszak [8].Example.
ConsideringM3=x,y,z∈R3 such that x,y,z are the standard coordinates in R³, the vector fields are (66)E1=λ2e−αz∂∂x+λ1e−αz∂∂y,E2=−λ1e−αz∂∂x+λ2e−αz∂∂y,E3=∂∂z,where g1andg2 are given by g1z=λ2e−αzandg2z=λ1e−αz with λ12+λ22≠0,α≠0 for constants λ1,λ2, and α. Also, the set of E1,E2,E3 is linearly independent at each point of M3. Let g be the Riemannian tensor product given by(67)g=g12+g22−1dx⊗dx+dy⊗dy+dz⊗dz.
Letη be the 1-form defined by ηX=gX,E3 and ϕ be the (1, 1) tensor field defined by(68)ϕe1=e2,ϕe2=−e1,ϕe3=0.
Furthermore, we can calculate(69)E1,E3=αe1,E2,E3=αe2,E1,E2=0,∇E1E1=−αE3,∇E2E1=−E3,∇E3E1=0,∇E1E2=−E3,∇E2E2=−αE3,∇E3E2=0,∇E1E3=αE1,∇E2E3=αE2,∇E3E3=0.
Thus, we can check the only nonzero components ofΦ. Namely, we get(70)Φ∂∂x,∂∂y=−g12+g22−1=−λ12+λ22−1e2αz.
Sinceη=dz, it implies that dΦ=2αη∧Φ on M3. Moreover, Nijenhuis torsion tensor of ϕ vanishes.
For the curvature operatorR, the nonzero components are as follows:(71)RE1,E2E1=ααE2−E1,RE1,E3E2=αE3,RE1,E2E2=αE2−αE1,RE1,E3E1=α2E3,RE1,E3E3=−α2E1,RE2,E3E1=αE3,RE2,E3E3=−α2E2,RE2,E3E2=α2E3,RE3,E2E1=−αE3.
Clearly,g is not locally flat.
For the Ricci tensorS, assume Sij=SEi,Ej; then, we obtain Sii=−2α2 and r=−6α2. Consequently, the Ricci operator Q satisfies the equations(72)QE1=bE1+αE2,QE2=αE1+bE2,QE3=bE3,with b=−2α2. For the auxiliary operator L of M3, we have(73)LE1=aE1+αE2,LE2=αE1+aE2,LE3=aE3,with a=−2α2+r/4,r=4b−a where L is defined by L=Q−r/4I.
To obtain the conformal flatness ofg, it remains to verify the Codazzi condition for L. Namely,(74)0=−LEi,Ej−∇EjLEi+∇EiLEj,for 1≤i<j≤3. It is seen that the Codazzi condition does not hold. Thus, the manifold M3 is not conformally flat and has constant sectional curvature K=−α2.


## 5. Conclusion and Discussion

A Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation [2, 3]. There exist conformally flat contact metric manifolds which are not constant curvature [9]. However, this is an open problem in dimensions ≥ 5. In recent years, some authors have studied this area for almost contact metric manifolds [4, 8, 9].This paper deals with the conformally flat almostα-cosymplectic manifolds given by Kaehler integral submanifolds. Our main target is to make some generalizations and classifications on such manifolds, and certain results are proved in the last two sections. [20].



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