Certain Class of Almost α-Cosymplectic Manifolds

+e 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 manifold M, Weyl established a tensor of type (1, 3) which vanishes whenever the metric is (locally) conformally equivalent to a flat metric.+erefore, this tensor is called the conformal curvature tensor of the metric and defined by


Introduction
e 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 manifold M, Weyl established a tensor of type (1,3) which vanishes whenever the metric is (locally) conformally equivalent to a flat metric. erefore, this tensor is called the conformal curvature tensor of the metric and defined by for any vector fields X, Y, and Z 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. e 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 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 manifold M 2n+1 is conformally flat if and only if the Weyl conformal curvature tensor C vanishes for any vector fields X, Y, and Z when n > 2 or the tensor L of type (1, 1) defined as 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 flat (n ≥ 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]. e 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 almost α-cosymplectic manifolds. We aim at characterizing and classifying conformally flat almost α-cosymplectic manifolds. en, 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.

Preliminaries
Let M 2n+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 M 2n+1 such that η(ξ) � 1, If M 2n+1 admits a Riemannian metric g, defined by then M 2n+1 is called almost contact structure (ϕ, ξ, η, g). Also, the fundamental 2-form Φ of M 2n+1 is defined by Φ(X, Y) � g(ϕX, Y). If the Nijenhuis tensor vanishes, defined by then (M 2n+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 ( [11]. An almost contact metric structure is cosymplectic if and only if ∇η and ∇Φ are closed [12].
Let M 2n+1 be an almost α-cosymplectic manifold Moreover, it is clear that ξ is orthogonal to D. Assume that N is a maximal integral submanifold of D. erefore, the ξ restricted to integral submanifold N is the normal vector of N. us, 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 that G is the Riemannian metric induced on N defined by G(X, Y) � g(X, Y). en, (J, G) has an almost Hermitian structure on N given by G(X, Y) � G(JX, JY) for any vector field X and Y tangent to N. e fundamental 2- , Ω is the pull-back of the tensor field ϕ from M 2n+1 to N. Since Ω is closed, we obtain dΩ � 0. us, 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 by A and h the (1, 1) tensor fields on M 2n+1 defined by respectively. Here, L is the Lie derivative of g. Obviously, A(ξ) � 0 and h(ξ) � 0. us, we the following relations for any vector fields X, Y on M 2n+1 [16]:

Curvature Properties
is section deals with the fundamental curvature equations of almost α-cosymplectic manifolds with Kaehler integral submanifolds. Let us give the basic propositions that we will use in later usage. e proof of some propositions are left to the reader for shortness.
where l � R(., ξ)ξ is the Jacobi operator with respect to ξ. By direct computations, we have the following proposition.

Proposition 2. An almost α-cosymplectic manifold M 2n+1
with Kaehler integral submanifolds holds the following equation: Remark 1. e Ricci operator Q 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 M 2n+1 with Kaehler integral submanifolds. Proposition 3. Let M 2n+1 be an almost α-cosymplectic manifold with Kaehler integral submanifolds. e following identity is held: Proof. Using (5) and (19), one obtains
By the help of (5), (23) can be written as Contracting with respect to Y and Z, we obtain and denote by r and r * the scalar and * -scalar curvatures of M 2n+1 , where respectively [8].
Proof. Taking trace of (19) with respect to X yields By using the properties of A in (29), this ends the proof.
Proof. (30) is a direct consequence of (29) by means of (10) and the following equation: (32) Proof. By using the projection of (28) onto ξ and g(Q * Y, ξ) � 0, the proof is obvious.

Proposition 7. An almost α-cosymplectic manifold with
Kaehler integral submanifolds M 2n+1 satisfies the following equation: Proof. From (13), we get e proof is clear from the right side of equation (34).

Proposition 8. Let M 2n+1 be an almost α-Kenmotsu manifold (n ≥ 2). en, it has Kaehler integral submanifolds of the distribution D if and only if it holds
where AX � αϕ 2 X + ϕhX.
Proof. By using similar technique in [8], the proof is clear. Proof. By the help of [11,15], we can complete the proof.

Conformal Flatness Condition
is section is devoted to study conformally flat almost α-cosymplectic manifolds whose integral submanifolds are Kaehler.

Theorem 4.
Let M 2n+1 be a conformally flat almost α-cosymplectic manifold with Kaehler integral submanifolds (n ≥ 2). en, the following relation is held: Proof. e proof follows from (20) and eorem 1 in [8]. Proof. Let M 2n+1 be an almost α-cosymplectic manifold with Kaehler integral submanifolds. Assume additionally that M 2n+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 eorem 1 in [8].
Now, we shall prove the assertion of our theorem in case of α ≠ 0. Since n ≥ 3, (50) holds. us, the auxiliary tensor L has the following shape: From (50) and (58), L can be written as In view of (7), (10), and (59) with X � ξ, we obtain where ξ(r)η(Y)ξ � Y(r)ξ. So, the last equation reduces to Replacing X and Y by ξ in (38) and using (50), we find that en, using (52) and (62), we have Following from (61) and (63), we get Also, using (18) and (64), one obtains which reduces to h � 0. Finally, we can use Proposition 9 to complete the proof.
Example. Considering M 3 � (x, y, z) ∈ R 3 such that (x, y, z) are the standard coordinates in R, the vector fields are where g 1 and g 2 are given by g 1 (z) � λ 2 e − αz and g 2 (z) � λ 1 e − αz with λ 2 1 + λ 2 2 ≠ 0, α ≠ 0 for constants λ 1 , λ 2 , and α. Also, the set of E 1 , E 2 , E 3 is linearly independent at each point of M 3 . Let g be the Riemannian tensor product given by Let η be the 1-form defined by η(X) � g(X, E 3 ) and ϕ be the (1, 1) tensor field defined by Furthermore, we can calculate us, we can check the only nonzero components of Φ. Namely, we get Since η � dz, it implies that dΦ � 2α(η∧Φ) on M 3 . Moreover, Nijenhuis torsion tensor of ϕ vanishes.
For the curvature operator R, the nonzero components are as follows: Clearly, g is not locally flat. For the Ricci tensor S, assume S ij � S(E i , E j ); then, we obtain S ii � − 2α 2 and r � − 6α 2 . Consequently, the Ricci operator Q satisfies the equations with b � − 2α 2 . For the auxiliary operator L of M 3 , we have with a � − (2α 2 + (r/4)), r � 4(b − a) where L is defined by L � (Q − (r/4)I).
To obtain the conformal flatness of g, it remains to verify the Codazzi condition for L. Namely, for 1 ≤ i < j ≤ 3. It is seen that the Codazzi condition does not hold. us, the manifold M 3 is not conformally flat and has constant sectional curvature K � − α 2 .

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]. ere 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]. is paper deals with the conformally flat almost α-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].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this study.