Kählerian Manifold on the Product of Two Trans-Sasakian Manifolds

I NTERNATIONAL E LECTRONIC J OURNAL OF G EOMETRY V OLUME 13 N O . 2 PAGE 135–143 (2020) DOI: HTTPS :// DOI . ORG /10.36890/ IEJG .756830 Kählerian Structure on the Product of Two Trans-Sasakian Manifolds Habib Bouzir and Gherici Beldjilali * (Communicated by Ramesh Sharma) A BSTRACT It’s shown that for some changes of metrics and structural tensors, the product of two transSasakian manifolds is a Kählerian manifold. This gives new positive answer and more generally to Blair-Oubiña’s open question. (See [7] and [17]). Concrete examples are given. Keywords: Trans-Sasakian manifolds; Kählerian manifolds; product manifolds. AMS Subject Classification (2020): Primary: 53C15 ; Secondary: 53C40. 1. Introduction On the product of two almost contact manifolds, A. Morimoto [11] defined a natural almost complex structure (see (4.2) in this paper) and proved that this almost complex structure is integrable if and only if the two factors are normal almost contact manifolds. Later, M. Capursi [8] investigated almost Hermitian geometry of the product of two almost contact metric manifolds with the product metric, with respect to the almost complex structure defined by Morimoto. He shows that this product is Hermitian, Kählerian, almost Kählerian or nearly Kählerian, if and only if, the two factors are normal, cosymplectic, almost cosymplectic or nearly cosymplectic respectively. Extending ideas from Capursi and Morimoto, Blair and Oubiña [7] considered conformal and related changes of the product metric with respect to a family of almost complex structures (see relation (3.1)) containing the one of Morimoto. Under the Kähler condition on the product manifold, Blair and Oubina proved that if one factor is Sasakian, the other is not, but that locally the second factor is of the type studied by Kenmotsu. The resuls are more general and given in terms of trans-Sasakian, α-Sasakian and β -Kenmotsu structures, finally they asked the open question: What kind of change of the product metric will make both factors Sasakian? In [18], Watanabe survey almost Hermitian, Kähler, almost quaternionic Hermitian and quaternionic Kähler structures, naturally constructed on products of manifolds with almost contact metric and Sasakian structures and open intervals, as an application of these constructions. Next, he investigated almost Hermitian structures, naturally defined on the product manifolds of two almost contact metric and Sasakian manifolds, and asked some problems related to these topics. In the same direction, Özdemir and al. [14], gave some properties that each factor should satisfy to make the almost Hermitian structure on the product manifold in a certain class of almost Hermitian manifolds. Recently, in [2], we introduced the notion of generalized doubly D-homothetic bi-warping. we gave an application to some questions of the characterization of certain geometric structures. Our work has supported the point of view of the Calabi-Eckmann manifolds that the almost Hermitian structures defined on the product of two Sasakian manifolds are never Kählerian. Here, again we based on the open question of Blair-Oubiña (see [7],[18]), but with a new techniques which requires a change in the two directions, metrics and structural tensors of the two Trans-Sasakian manifolds, Received : 23-June-2020, Accepted : 23-September-2020 * Corresponding author Kählerian Structure on the Product of Two Trans-Sasakian Manifolds which gave a positive response to the question see theorem (5.1)(main theorem). This paper is organized in the following way: Section 2, is devoted to the background of the structures which will be used in the sequel. In Section 3, we introduce a new deformation of almost contact metric structure and we give some geometric properties. Section 3 is devoted to the construction of a class of interesting examples in dimension 3. In the last section, we focus on our main goal where we construct Kählerian manifold using the product of two Trans-Sasakian manifolds with a concrete example. 2. Review Of Needed Notions An almost complex manifold with a Hermitian metric is called an almost Hermitian manifold. For an almost Hermitian manifold (M, J, g) we thus have J 2 = −1, g(JX, JY ) = g(X, Y ), (2.1) where X and Y denote arbitrary vector fields on M . An almost complex structure J is integrable, and hence the manifold is a complex manifold, if and only if its Nijenhuis tensor Nj vanishes, with NJ (X, Y ) = [JX, JY ] − [X, Y ] − J[X, JY ] − J[JX, Y ]. For an almost Hermitian manifold (M, J, g), we define the fundamental Kähler form Ω as Ω(X, Y ) = g(X, JY ). (M, J, g) is then called almost Kähler if Ω is closed i.e. dΩ = 0 where d denotes the exterior derivative. An almost Kähler manifold with integrable J is called a Kähler manifold, and thus is characterized by the conditions: dΩ = 0 and NJ = 0. One can prove that both these conditions combined are equivalent with the single condition ∇J = 0. An odd-dimensional Riemannian manifold (M 2n+1 , g) is said to be an almost contact metric manifold if there exist on M a (1, 1) tensor field ϕ, a vector field ξ (called the structure vector field) and a 1-form η such that η(ξ) = 1, ϕ2 (X) = −X + η(X)ξ and g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), (2.2) for any vector fields X , Y on M . In particular, in an almost contact metric manifold we also have ϕξ = 0 and η ◦ ϕ = 0. Such a manifold is said to be a contact metric manifold if dη = φ, where φ(X, Y ) = g(X, ϕY ) is called the fundamental 2-form of M . On the other hand, the almost contact metric structure of M is said to be normal if N (1) (X, Y ) = [ϕ, ϕ](X, Y ) + 2dη(X, Y )ξ = 0, (2.3) for any X and Y vector fields on M , where [ϕ, ϕ] denotes the Nijenhuis torsion of ϕ given by [ϕ, ϕ](X, Y ) = ϕ2 [X, Y ] + [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ]. An almost contact metric structures (ϕ, ξ, η, g) on M is said to be:   (a) : Sasaki ⇔ φ = dη and (ϕ, ξ, η) is normal, (b) : Cosymplectic ⇔ dφ = dη = 0 and (ϕ, ξ, η) is normal,  (c) : Kenmotsu ⇔ dη = 0, dφ = 2φ ∧ η and (ϕ, ξ, η) is normal. These manifolds can be characterized through their Levi-Civita connection, by requiring   (1) : Sasaki ⇔ (∇X ϕ)Y = g(X, Y )ξ − η(Y )X, (2) : Cosymplectic ⇔ ∇ϕ = 0,  (3) : Kenmotsu ⇔ (∇ ϕ)Y = g(ϕX, Y )ξ − η(Y )ϕX. X www.iejgeo.com (2.4) (2.5) 136 H. Bouzir and G. Beldjilali In [15], the author proves that (ϕ, ξ, η, g) is trans-Sasakian structure if and only if it is normal and dφ = 2βη ∧ φ, dη = αφ, (2.6) 1 1 where α = 2n δφ(ξ), β = 2n divξ and δ is the codifferential of g. It is well known that the trans-Sasakian condition may be expressed as an almost contact metric structure satisfying

(∇X ϕ)Y = α g(X, Y )ξ − η(Y )X + β g(ϕX, Y )ξ − η(Y )ϕX . (2.7) From this formula one easily obtains ∇X ξ = −αϕX − βϕ2 X, (2.8) (∇X η)Y = αg(X, ϕY ) + βg(ϕX, ϕY ). (2.9) It is clear that a trans-Sasakian manifold of type (1, 0) is a Sasakian manifold and a trans-Sasakian manifold of type (0, 1) is a Kenmotsu m (...truncated)