Almost contact metric manifolds with certain condition

The object of this article is to study a new class of almost contact metric structures which are integrable but non normal. Secondly, we explain a method of construction for normal manifold starting from a non-normal but integrable manifold. Illustrative examples are given.

The fundamental 2-form φ is defined by φ(X , Y ) = g(X , ϕY ). It is known that the almost contact structure (ϕ, ξ, η) is said to be normal if and only if N (1) (1.2) for any X , Y on M, where N ϕ denotes the Nijenhuis torsion of ϕ, given by Given an almost contact structure, one can associate in a natural manner an almost CRstructure (D, ϕ| D ), where D := K er(η) = I m(ϕ) is the distribution of rank 2n transversal to the characteristic vector field ξ . If this almost CR-structure is integrable (i.e., N ϕ = 0), the manifold M 2n+1 is said to be CR-integrable. It is known that normal almost contact manifolds are CR-manifolds.
In [8], Olszak completely characterized the local nature of the normal almost contact metric manifold of dimension three. He proved that, for an arbitrary 3-dimensional almost contact metric manifold (M, ϕ, ξ, η, g), we have where 2α = tr g (ϕ∇ξ) , 2β = divξ and ∇ denotes the Levi-Civita connection of g. Any almost contact metric manifold of dimension three is normal if and only if This means that any normal almost contact metric manifold of dimension three is a trans-Sasakian manifold (C 5 ⊕ C 6 -manifold), according to the concept introduced by J. A. Oubiña [9]. Recently, in [3], the authors studied the 3-dimensional C 12 -structures. They are integrable, non-normal (i.e. N ϕ = 0 and N (1) = 0) and are characterized by (1.6) In this paper, we introduce the concept of generalized C 12 -structure, proving that manifolds endowed with a generalized C 12 -structure set up the Chinea-Gonzales class C 5 ⊕ C 12 . In particular, for this manifolds one has N ϕ = 0, namely N (1) Recently, in [7], The authors have developed a systematic study of the curvature of this class and obtain some classification theorems for those manifolds that satisfy suitable curvature conditions.
Here, we show that such manifolds in dimension 3 are completely controllable and we will also present some techniques to extract such structures from others. As an application, we present a method to obtain a normal structure starting by an integrable one.
We are going to prove that generalized C 12 -manifolds set up the class C 5 ⊕ C 12 . In this paper, we assume that ∇ ξ we consider generalized C 12 -manifolds that are not normal.
where ψ denotes the vector field ∇ ξ ξ , which represents the mean curvature vector field of any integral curve of D ⊥ , i.e.
Proof Let (M, ϕ, ξ, η, g) be an almost contact metric manifold. It is well known that for any almost contact metric manifold of dimension 2n + 1, we have (see [1], p.82) and also, from ( [1], p.81) we have where L indicates the Lie derivative. Assume that (ϕ, ξ, η, g) is a generalized C 12 -structure. By (2.1) one has and and hence Moreover, using hypothesis and Finally, we use (2.2) and the relation This completes the proof.
We construct examples of generalized C 12 -manifolds that are closely related to the conformal deformation.
The second type of examples is closely related to the doubly twisted products [10]. Let (N 1 , g 1 ) and (N 2 , g 2 ) be Riemannian manifolds and σ 1 and σ 2 are positive smooth functions defined on N 1 and N 2 , respectively. Consider the product manifold N 1 × N 2 with its projection π 1 : N 1 × N 2 → N 1 and π 2 : N 1 × N 2 → N 2 . Then the doubly twisted product of (N 1 , g 1 ) and (N 2 , g 2 ) is the product manifold M = N 1 ×N 2 equipped with the metric g = σ 2 2 g 1 +σ 2 1 g 2 given by where π * i (g i ) is the pullback of g i via π i for i =∈ {1, 2}. In this construction, we consider the case where N 1 = N is a 2n-dimensional Riemannian manifold equipped with the Käkler structure (J , h) and N 2 = R. We define a Riemannian metric tensor g, a vector field ξ , a 1-form η and a (1, 1)-tensor field ϕ on the product space M = N × R as follows: where ∂ t denotes the unit tangent field to R and ρ, τ are smooth functions on N and R, respectively, and X a vector field on N . By a direct calculation using (1.1), one can check that (ϕ, ξ, η, g) is an almost contact metric structure. In addition, we have dη = dρ ∧ η. Also, the fundamental 2-form φ of (ϕ, ξ, η, g) is we can check that is very simply. It follows where is the Kähler form of (J , h). Hence Given the definition of ϕ, we can show that By (2.12), one has an almost contact metric structure (ϕ, ξ, η, g) on M that fulfills the following conditions

3-dimensional generalized C 12 -manifold
Through the rest of this paper, M always denotes a 3-dimensional differential manifold unless stated otherwise. Firstly, based on theorem 2.2, we give a more efficient main theorem: where X is a vector field on M and ψ = ∇ ξ ξ .
The following proposition provides another characterization:

2)
where X is a vector field on M.
Using the fundamental frame, one can get the following: Theorem 3.5 For any generalized C 12 -manifold, we have Proof (1) The first three relations come from formula (3.1).

A class of examples
We denote the Cartesian coordinates in a 3-dimensional Euclidean space R 3 by (x 1 , x 2 , x 3 ) and define a symmetric tensor field g by where f = f (x 1 , x 2 ), τ = τ (x 1 ) and σ = 0 are smooth functions. Further, we define an almost contact metric structure (ϕ, ξ, η) on R 3 by The fundamental 1-form η and the 2-form φ have the forms, ξ, η, g) is a generalized C 12 -structure.
Therefore, to continue studying this example, it suffices to take f 1 = 0 or f 2 = 0 to ensure that the structure is ϕ-integrable not normal.
Thus , (ϕ, ξ, η, g) is a generalized C 12 -manifold with Now, using the orthonormal basis the components ∇ e i e 3 are given by: Then, one can easily check that for all i, j ∈ {1, 2, 3}, In fact, by Theorem 3.1, these formulas imply that the structure is a generalized C 12structure.

From integrability to normality
In this section, we present the sufficient and necessary techniques to construct normal almost contact metric structures starting from integrable ones. According to the previous section, the 3-dimensional generalized C 12 -manifold is completely controllable.

The twin structures of generalized C 12 -structure
On an arbitrary oriented Riemannian 3-manifold, one can canonically define a cross-product × of two vector fields X and Y on M as follows: for any vector fields Z on M. where dv g denotes the volume form defined by g. When  (M, ϕ, ξ, η, g) is an almost contact metric 3-manifold, the cross-product is given by the formula [4,5]: for all X and Y vector fields on M. Easily, we can notice that ϕ X = ξ × X .
For our first application, let's define a (1, 1)-tensor field ϕ on M by where X any vector field on M. That is, (M, ϕ, ξ, η, g) be a generalized C 12 -manifold and ϕ the tensor field defined in (5.3). Then, (M, ϕ, V , θ 1 , g) is an almost contact metric manifold.

Proposition 5.2 Let
Proof Easily we can see that ϕV = 0 and for all vector field X on M, we have For the condition of compatibility, we have which completes the proof.