On Generalized D-Conformal Deformations of Certain Almost Contact Metric Manifolds

In this work, we consider almost contact metric manifolds. We investigate the generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds. The new Levi–Civita covariant derivative of the new metric corresponding to deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Under some restrictions, deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Then, the scalar curvature of these three classes of deformed manifolds are analyzed.


Introduction
There exist several type of deformations of almost contact metric structures. For example conformal deformations, D-homothetic deformations etc. These deformations were studied by several authors [1][2][3][4]. In [4], generalized D-conformal deformations are applied to trans-Sasakian manifolds where the covariant derivatives of the deformed metric is evaluated under the condition that the functions used in deformation depend only on the direction of the characteristic vector field of the trans-Sasakian structure. In this study, in order to simplify tedious calculations, we obtain the new covariant derivatives of deformed almost contact metric structures seperately for the cases where the characteristic vector field is parallel, Killing and the one form dual to the characteristic vector field is closed. Then, we study generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and β-Kenmotsu manifolds. We analyse how the class of almost contact metric structures change. Under some restrictions, deformed nearly K-cosymplectic, quasi-Sasakian and C = α ∈ ⊗ 0 3 T p M : α(X, Y, Z) = −α(X, Z, Y) = −α (X, ϕ(Y), ϕ(Z)) +η(Y)α(X, ξ, Z) + η(Z)α(X, Y, ξ)} for all p ∈ M. The space C was decomposed into 12 U(n) × 1 irreducible components, denoted by C 1 , . . . , C 12 as shown in Table 1. There exist 2 12 invariant subspaces, each corresponding to a class of almost contact metric manifolds. For example, the trivial class corresponds to the class of cosymplectic (called co-Kähler by some authors) manifolds, C 1 is the class of nearly K-cosymplectic manifolds, C 6 ⊗ C 7 is the class of quasi-sasakian manifolds, C 5 is the class of β-Kenmotsu manifolds etc [5]. Also, a similar classification was made by [6]. In this work, we will use the definitions of some other classes in the context by using the notation in [5,6]. According to this classification; some special classes of almost contact metric manifolds coincide with a suitable sum of some classes of C i . When the dimension of the manifold is 3, then C = C 5 ⊕ C 6 ⊕ C 9 ⊕ C 12 [5]. Table 1. Defining relations for each of the twelve classes [5].

Classes Defining Relations
Let (M, ϕ, ξ, η, g) be an almost contact metric manifold. If we takẽ ϕ := ϕ,ξ := 1 a ξ,η := aη,g := bg + ( where a and b are positive functions on M, one can easily check that M,φ,ξ,η,g is an almost contact metric manifold too. This deformation is called a generalized D-conformal deformation [4]. After this deformation, the derivation of the new fundamental 2-formΦ is

Generalized D-Conformal Deformations of Nearly K-Cosymplectic Manifolds
Let (M, ϕ, ξ, η, g) be a nearly K-cosymplectic manifold (that is, belongs to class C 1 ). Defining relations of this class are To calculate the new Levi-Civita covariant derivative ∇ of a nearly K-cosymplectic manifold after applying a generalized D-conformal deformation, we need only to consider the property that ξ is parallel. Hence, we state the following lemma. Lemma 1. Let (M, ϕ, ξ, η, g) be an almost contact metric manifold such that characteristic vector field ξ is parallel. If a generalized D-conformal deformation is applied, then the new covariant derivative of the new metric g is obtained as Proof. Using Kozsul's formula and that ξ is parallel, one can get Take Z = ξ in the Equation (2), to obtain Hence, we write the new Levi-Civita covariant derivative of the new metricg as in (1).
Now we show that under some restrictions, it is possible to obtain a nearly K-cosymplectic structure from an old one by a generalized D-conformal deformation. Theorem 1. Let (M, ϕ, ξ, η, g) be a nearly K-cosymplectic manifold and consider a generalized D-conformal deformation on M with positive functions a and b. M,φ,ξ,η,g is a nearly K-cosymplectic manifold if and only if grad a = ξ[a]ξ and b is a constant.
Proof. If we take Y =ξ in (1), since ξ is parallel, We give the following example on nearly K-cosymplectic structures.
where f and g are C ∞ functions on M × R, X, Y are any vector fields on M. Then, M × R is nearly K-cosymplectic if and only if M is nearly Kaehlerian [5]. Consider a generalized D-conformal deformation with a C ∞ function a : M × R −→ R, a(x, t) = F(t) > 0 and a positive constant b. Since the function a depends only on t, grada = ξ[a]. Then, by Theorem 1, the new almost contact metric structure on M × R is also nearly K-cosymplectic.

Generalized D-Conformal Deformations of Quasi-Sasakian Manifolds
In this case, we consider generalized D-conformal deformations of quasi-Sasakian manifolds. An almost contact metric manifold is called quasi-Sasakian if it is normal and its fundamental 2-form Φ is closed, that is, [ϕ, ϕ](X, Y) + dη(X, Y)ξ = 0 and dΦ = 0, for all vector field X and Y. Quasi-Sasakian manifolds are the class C 6 ⊕ C 7 . The most important feature of a quasi-Sasakian manifold is that the fundamental vector field ξ is a Killing vector field [7]. Let (M, ϕ, ξ, η, g) be an almost contact metric manifold such that ξ is Killing. The Levi-Civita covariant derivative of g is calculated using Kozsul's formula, only by considering that ξ is Killing. Lemma 2. Let (M, ϕ, ξ, η, g) be an almost contact metric manifold such that characteristic vector field ξ is Killing. If a generalized D-conformal deformation is applied, then the new covariant derivative of the new metric g is obtained as Now our aim is to obtain a quasi-Sasakian manifold after applying a generalized D-conformal deformation to a quasi-Sasakian manifold. First we give the condition for ξ to be Killing with respect tog.

Lemma 3. ξ is Killing vector field if and only if ξ[b] = 0 and grada = ξ[a]ξ.
Proof. Let ξ be a Killing vector field. Then if we take X = ξ in (9), we obtain In addition, if we take grada = ξ[a]ξ in (9), we get ξ[b] = 0. Converse of the lemma is trivial.
We can obtain quasi-Sasakian manifolds by deforming the old ones as follows.
Conversely, let b be a constant and grad a = ξ[a]ξ. By Lemma 3,ξ is Killig. In addition, since a quasi-Sasakian manifold is normal, we have Since grad a = ξ[a]ξ, the equation is satisfied. Also, for a quasi-Sasakian manifold dΦ = 0, thus we get Since the function b is a constant, we obtain dΦ(X, Y, Z) = 0. As a result M,φ,ξ,η,g is a quasi-Sasakian manifold.
In addition, one can obtain the following corollary: If (M, ϕ, ξ, η, g) is a quasi-Sasakian manifold and a and b are positive functions such that ξ[b] = 0 (b need not be constant) and grada = ξ[a]ξ, then in new almost contact metric manifold M,φ,ξ,η,g , one can compute directly coderivation of δη and δΦ as follows: and δΦ = δΦ(X) + 1 − n b g (gradb, ϕ(X)) Let (M, ϕ, ξ, η, g) be a quasi-Sasakian manifold, b be a positive constant and a be a function such that grada = ξ[a]ξ. After a generalized D-conformal deformation, the new covariant derivative is Moreover, by direct calculation, one can get Hence, we obtain new Ricci operatorQ and scalar curvatureS as Example 2. Let M be a seven dimensional 3-Sasakian manifold. Since this manifold is Sasakian (C 6 ), it is in particular quasi-Sasakian (C 6 ⊕ C 7 ). It is known that its scalar curvature is 42 and also ∇ X ξ = −ϕ(X) for all vector fields X. For definition and properties of 3-Sasakian manifolds, see [8]. Let (M, ϕ, ξ, η, g) be one of the three Sasakian structures on a seven dimensional 3-Sasakian manifold and assume that a generalized D-conformal deformation is applied to this structure. Note that by Theorem [3], the deformed structure is also quasi-Sasakian. Now we calculate the new scalar curvatureS of the deformed manifold. Since g(∇ e i ξ, ∇ e i ξ) = g(ϕ(e i ), ϕ(e i )) = g(e i , e i ) − η(e i )η(e i ), from the Equation (18),S Let a and b be positive constants satisfying Then, the Equation (19) impliesS = 1 orS = −1. Thus the new quasi-Sasakian manifold is locally isometric to the sphere or the hyperbolic space, respectively.
On the other hand, if positive constants a and b are chosen as a 2 = 8b, thenS = 0 and the new quasi-Sasakian manifold has zero scalar curvature.

Generalized D-Conformal Deformations of β-Kenmotsu Manifolds
An almost contact metric manifold (M, ϕ, ξ, η, g) is called β-Kenmotsu manifold, if the relation is satisfied, where β is a smooth function on M. It is known that if (M, ϕ, ξ, η, g) is a β-Kenmotsu manifold, then the equation is satisfied.
Then, we obtain the following lemma: Proof. First by taking Y = ξ in (21), we get and Then, by the definition of the generalized D-conformal deformation and the Equations (22) and (23), we havẽ The Equation (24) implies Take Y = ξ in (25) to obtain that is grad a = g(grad a, ξ)ξ.
For any β-Kenmotsu manifold, we know that dη = 0. After deformation, derivation of η is obtained as: The new Levi-Civita covariant derivative ofΦ is Note that the following theorem can be deduced from Lemma 4.1 of [4]. In [4], first the new Levi-Civita covariant derivative is calculated under the restriction that a and b are positive functions depending on the direction of ξ and then Lemma 4.1 in [4] is stated for trans-Sasakian manifolds by using this covariant derivative. In our study, however, we obtain the new Levi-Civita covariant derivative only by assuming that dη = 0 (equivalently g(∇ X ξ, Y) = g(∇ Y ξ, X)) in Lemma 5 and then we state the following theorem. (M, ϕ, ξ, η, g) be a β-Kenmotsu manifold, and consider a generalized D-conformal deformation with a and b positive functions. If grad a = g(grad a, ξ)ξ and grad b = g(grad b, ξ)ξ, then (M,φ,ξ,η,g) is aβ-Kenmotsu manifold, whereβ
Take Y = ξ in (27), then we obtainβ = 1 a β + 1 2ab ξ [b]. We have been unable to find any restriction on the function b.
In addition, by long direct calculation, the new scalar curvatureS is