Research Article Almost α -Cosymplectic Pseudo Metric Manifolds

The main purpose of this paper is to study almost α -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 , φ h , 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.


Introduction
Manifolds known as almost contact metric manifolds have been studied in [1][2][3]. e class of almost contact metric manifolds which are called almost Kenmotsu manifolds is firstly introduced by Kenmotsu. ese 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 almost α-cosymplectic manifolds [5]. e authors investigated canonical foliations of an almost α-cosymplectic manifold. Later, most of the research is devoted to this topic [6][7][8][9][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][12][13][14].
On that account, Wang and Liu introduced the geometry of almost Kenmotsu pseudo metric manifolds [12]. ey 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 studied η-parallel contact metric spaces in [15]. ey considered a milder condition that h is η-parallel, i.e., 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 M 2n+1 . e torsion tensor field τ defined as for any vector fields X and Y on M 2n+1 was firstly introduced by Hamilton and Chern [17].
In this paper, we consider the almost α-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. ese 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.

Preliminaries
Let M 2n+1 be a (2n + 1)-dimensional differentiable manifold equipped 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 which implies A pseudo Riemannian metric g on M 2n+1 is said to be compatible with the almost contact structure A smooth manifold M 2n+1 furnished with an almost contact structure (φ, ξ, η)and a compatible pseudo Riemannian metric g is called an almost contact pseudo metric manifold which is denoted by On such a manifold, the fundamental 2-form Φ of M 2n+1 is defined by Φ(X, Y) � g(X, φY) for any vector fields X, Y on M 2n+1 [18]. An almost contact pseudo metric manifold satisfying the conditions dη � 0 and dΦ � 2α(η ∧ 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: where [φ, φ] is the Nijenhuis tensor of φ [19].
If we join these two classes, we obtain the notion of an almost α-cosymplectic pseudo metric manifold, defined by dη � 0 and dΦ � 2α(η ∧ Φ), for any real number α [5]. When an almost α-cosymplectic pseudo metric manifold M 2n+1 has a normal almost contact structure, we can say that M 2n+1 is an α-cosymplectic pseudo metric manifold. In this paper, we shall denote by Γ(TM) and ∇ the Lie algebra of all tangent vector fields on M 2n+1 and the Levi Civita connection of pseudo Riemannian metric g, respectively.

Certain Properties
In this section, we give the basic relations on almost α-cosymplectic pseudo metric manifolds.

Proposition 1.
Let M 2n+1 be an almost contact metric manifold and ∇ be the Riemannian connection. en, the following equations are held [3]: Here, a X,Y,Z denotes the cyclic sum over the vector fields X, Y, and Z [1]. Lemma 1. Let M 2n+1 be an almost contact pseudo metric manifold. en, the following equation is held: Journal of Mathematics for any tangent vector fields X, Y, Z ∈ Γ(TM) where N (0) and N (1) are defined by respectively. Here, L X denotes the Lie derivative in the direction of X [20].
Now, we investigate the curvature properties of almost contact pseudo metric manifolds. First, we have the following propositions. □ Proposition 3. Let (M 2n+1 , φ, ξ, η, g) be an almost α-cosymplectic pseudo metric manifold. en, we have for any tangent vector fields X, Y ∈ Γ(TM).
for any tangent vector fields X, Y ∈ Γ(TM).
is proof can also be given in another way. Consider the local orthonormal φ-basis on M 2n+1 . e sectional curvatures of nondegenerate planes spanned by ξ, e i and ξ, φe i , respectively, are defined as K ξ, e i � εε i R ξ, e i , ξ, e i � εε i g l e i , e i , where ε i � g(e i , e i ) � g(φe i , φe i ) � ± 1 for all indices i � 1, . . . , n and l is the Jacobi operator defined by l � R(., ξ)ξ. us, we have and from (46) and (47), it follows that It is well known that which completes the proof.

Main Results
In this section, we consider some certain parallel tensor conditions on almost α-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 � X T + η(X)ξ where X T is tangentially part of X and η(X)ξ 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: for all tangent vectors X T , Y T , and Z T orthogonal to ξ [15]. (M 2n+1 , φ, ξ, η, g) be an almost α-cosymplectic pseudo metric manifold. If h satisfies the η-parallelity condition, then we have

Journal of Mathematics
for any X, Y ∈ Γ(TM) where l � R(., ξ)ξ is the Jacobi operator with respect to ξ.
Proof. Let e 1 , . . . , e 2n , 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 for any X, Y ∈ Γ(TM). is means that us, it completes the proof. Proof. Let Z ∈ D be an eigen unit vector field such that h(Z) � μZ where μ is an eigen function corresponding to the vector field Z. en, (53) can be written as for Z ∈ D. Also, we have Taking into account of (66) and (67), we also get Furthermore, since ∇ ξ h � 0, we obtain Follows from (68) and (69), we have dμ � 0. us, it completes the proof. □ Proposition 7. Let (M 2n+1 , φ, ξ, η, g) be an almost α-cosymplectic pseudo metric manifold. en, the torsion tensor field τ holds the following: for any X ∈ Γ(TM).
Proof. e hypothesis is essentially same as for all tangent vectors orthogonal to ξ. Putting X T � X − η(X)ξ and using the definition of τ, we obtain It follows that Putting Y � ξ in (75), we have Also, it is noted that Finally, taking into account of (74)-(77), (72) holds. en, we complete the proof. □ Theorem 4. Let (M 2n+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 M 2n+1 .
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.
us, 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.