Las trans-S-variedades y la clasificación de Gray-Hervella.

Las Trans-𝑆-Variedades y la Clasificación de Gray-Hervella. Manuel Soriano Trigueros Memoria presentada como parte de los requisitos para la obtención del título de Máster Universitario en Matemáticas por la Universidad de Sevilla. Tutelado por Prof. Luis M. Fernández Fernández Contents Abstract 1 Introduction 3 1. Preliminaries 9 1.1. 𝑓-structures ................................ 9 1.2. TensorCalculus .............................. 11 2. Almost Complex Manifolds induced by a trans-𝑆13 2.1. Computing the associated 2-form. . . . . . . . . . . . . . . . . . . . . 15 2.2. A Gray-Hervella’s Class for  𝑀...................... 19 2.3. Trans-𝑆cannot be defined using 𝑊3⊕ 𝑊4............... 24 3. Almost Trans-𝑆-manifolds and the Gray-Hervella’s Classification 27 Conclusion 31 Bibliography 33 abstract 1 Abstract Recently, trans-𝑆manifolds have been defined as a natural generalization of 𝑓- Kenmotsu, 𝑆-manifolds and 𝐶-manifolds. This uses 𝑓-structure techniques to extend results for almost contact manifolds, where trans-Sasakian are a generalization of Kenmotsu, Sasakian and cosymplectic manifolds. The defintion of trans-𝑆is formulated using the covariant derivative of the tensor 𝑓. Although this formulation coincides with the characterization of trans-Sasakian when 𝑠= 1, the latter type of manifolds were not initially defined in this way. Trans-Sasakian manifolds were defined in 1985 using the Gray-Hervella’s classification of almost Hermitian manifolds. Therefore, one could ask whether trans-𝑆 manifolds could be defined using this classification and how they relate with almost Hermitian manifolds. The objective of this project is to study how trans-𝑆-manifolds and almost trans- 𝑆-manifolds relates with the Gray-Hervella’s classification. We will find both similarities and differences with the trans-Sasakian case (𝑠= 1). 2las trans-𝑠-variedades y la clasificación de gray-hervella. Resumen Las trans-𝑆-variedades han sido definidas recientemente como una generalización natural de las 𝑓-Kenmotsu, 𝑆-variedades y 𝐶-variedades. Estos resultados representan una versión para 𝑓-structuras de algunos resulados ya conocidos en variedades casi contacto, donde las trans-Sasakianas generalizan variedades como las Kenmotsu, las Sasakianas y las cosimpléticas. La definición de variedad trans-𝑆ha sido formulada en términos de la derivada covariante del tensor 𝑓. Aunque esta definición coincide con la caracterización de las trans-Sasakianas cuando 𝑠= 1, este último tipo de variedad no fue definida de esta manera en un principio. Las variedades trans-Sasakianas fueron definidas en 1985 usando la clasificación de Gray y Hervella para variedades casi Hermitianas. Por tanto, una pregunta interesante sobre variedades trans-𝑆es si podrían definirse de la misma manera y cómo se relacionan con la clase de variedades Hermitianas. El objetivo de este proyecto es estudiar como se relacionan las variedades trans-𝑆 y casi trans-𝑆con la clasificación de Gray y Hervella. Descubriremos algunas similitudes con el caso trans-Sasakiano (𝑠= 1) pero también algunas diferencias. Introduction Since its beginning at the end of nineteenth century, manifolds and Riemannian geometry has been an intensive area of research. The discover of general relativity, which would have been impossible without this theory, was the greatest proof of its power to explain phenomena which could not be treated before. While relations of this new geometry with other fields of mathematics and science was found, some structures of study gained importance. For example, complex and symplectic manifolds in even dimensions, and contact manifolds in odd dimensions. In particular a Kähler manifold is a Riemannian manifold with both complex and symplectic structure. A Sasakian manifold is a contact manifold 𝑀such that 𝑀×ℝ is conformal to a Kähler manifold. Despite all these objects have desirable properties, it is a difficult task to find out when they can be defined in a particular manifold. This is the reason why almost complex and almost contact structures were found so convenient at first. An almost complex structure is a (1,1) tensor 𝐽which satisfies 𝐽2𝑋= −𝑋and an almost contact structure is a (1,1) tensor 𝜙, a vector field 𝜉, and a 1-form 𝜂such that 𝜂(𝜉) = 1 and 𝜙2𝑋= −𝑋+𝜂(𝑋)𝜉. Both are much easier to define than a proper complex or contact structure and, if these tensors satisfies some properties, they actually induce not only complex and contact structures, but more rigid ones like Kähler and Sasakian. Nevertheless, their interest does not finish there. The existence of these two weaker structures provides some properties to the manifolds which make them of interest by their own right. Keeping this idea in mind, Yano introduced in [10] 𝑓structures: tensors of type (1,1) satisfying 𝑓3+𝑓= 0. In particular, metric 𝑓manifolds satisfy 𝑓2= −𝐼+∑𝑠 𝑖=1 𝜂𝑖⊗ 𝜉𝑖where 𝜉𝑖are global vector fields and 𝜂𝑖there 4las trans-𝑠-variedades y la clasificación de gray-hervella. associates 1-forms. Both, almost contact and almost complex are particular cases of 𝑓-structures. The study of these tensors and how they relate with the already known results involving almost contact and almost complex structures is an active area of research. In [5] Gray and Hervella studied how the covariant derivative of the associated 2-form can be used to create 16 classes for almost complex manifolds. The four basic classes are 𝑊1, 𝑊2, 𝑊3, 𝑊4, the rest are given by their direct sum. Kähler manifolds are precisely the intersection of these four classes. At the end of this introduction we show a chart from the original article where all classes are summarized. In 1985 Oubiña defined trans-Sasakian manifolds as almost contact manifolds 𝑀 such that 𝑀×ℝwas an almost complex manifold in the class 𝑊4, containing locally conformal Kähler manifolds (see [8]). In addition it was shown that being trans- Sasakian was equivalent to being normal and satisfying (∇𝑋𝑓)𝑌=𝛼{𝑔(𝑋, 𝑌 )𝜉−𝜂(𝑌)𝑋} + 𝛽{𝑔(𝑓𝑋, 𝑌 )𝜉−𝜂(𝑌)𝑓𝑋}(1) for some differentiable functions 𝛼, 𝛽 on 𝑀. Another almost contact manifold defined in the article was almost Sasakian-manifolds. This type was not normal and the product with ℝgives an almost complex manifold in the class 𝑊2⊕ 𝑊4. The objective of Oubiña was to generalize Kenmotsu, cosymplectic, Sasakian and Quasi-Sasakian manifolds. The relation between these manifolds and trans-Sasakian manifolds is given by the following chart where Φ(𝑋, 𝑌 ) = 𝑔(𝑋, 𝜙𝑌 )is the fundamental 2-form. Kenmotsu: 𝑑𝜂 = 0,normal Cosymplectic: 𝑑Φ=0, 𝑑𝜂 = 0, normal Sasakian: Φ = 𝑑𝜂, normal Quasi-Sasakian: 𝑑Φ=0, normal Trans-Sasakian: 𝑑Φ=2𝛽(Φ ∧ 𝜂), 𝑑𝜂 =𝛼Φ, 𝜙∗(𝛿Φ) = 0, normal introduction 5 Analogously, P. Alegre, L. M. Fernández and A. Prieto defined in [1] a new class of metric 𝑓-manifold called tran-𝑆-manifolds. They are characterized as normal manifolds satisfying (∇𝑋𝑓)𝑌= 𝑠 ∑ 𝑖=1 (𝛼𝑖{𝑔(𝑓𝑋, 𝑓𝑌 )𝜉𝑖+𝜂𝑖(𝑌)𝑓2𝑋} + 𝛽𝑖{𝑔(𝑓𝑋, 𝑌 )𝜉𝑖−𝜂𝑖(𝑌)𝑓𝑋}),(2) where 𝛼𝑖, 𝛽𝑖are differentiable functions on the manifold. If the normality condition is removed, the manifold is called almost trans-𝑆. We can resume the situation showed in [1] using the following chart. 𝑓-Kenmotsu: definition 𝑑𝐹 = 2 ∑𝑠 𝑖=1 𝜂𝑖∧𝐹 , ∀𝑖 𝑑𝜂𝑖= 0, normal 𝐶-manifold: definition 𝑑𝐹 = 0,∀𝑖 𝑑𝜂𝑖= 0, normal 𝑆-manifold: definition ∀𝑖 𝐹 =𝑑𝜂𝑖,normal 𝐾-manifold: definition 𝑑𝐹 = 0, normal Trans-𝑆-manifold: necessary condition 𝑑𝐹 = 2𝐹∧∑𝑠 𝑖=1 𝛽𝑖𝜂𝑖, ∀𝑖 𝑑𝜂𝑖=𝛼𝑖𝐹 , 𝑓∗(𝛿𝐹 )=0, normal Where 𝐹(𝑋, 𝑌 ) = 𝑔(𝑋, 𝑓 𝑌 )is the fundamental 2-form. It should be noted that it is necessary, but not sufficient, for a trans-𝑆-manifold to satisfy the equations given in the right cell. In fact, there exist 𝐾-manifolds which are not trans-𝑆. In other words, trans-𝑆manifolds cannot be defined using 𝑑𝐹 , 𝑑𝜂𝑖and 𝑓∗(𝛿𝐹 ). On the other hand; 𝑓-Kenmotsu, 𝑆-manifolds and 𝐶-manifolds are always trans-𝑆. One question remains: Can we use the Gray-Hervella’s classification for defining (almost) trans-𝑆-manifolds? 12 las trans-𝑠-variedades y la clasificación de gray-hervella. where 𝜄1, 𝜄2are the inclusion of 𝑀1and 𝑀2into 𝑀1×𝑀2,𝜄∗ 1, 𝜄∗ 2their pull-back and 𝜄1 ∗, 𝜄2 ∗their push-forward. 2 Almost Complex Manifolds induced by a trans-𝑆 The type of metric 𝑓-manifolds we are defining now will be the central object of study in this chapter. Definition 2.1. A (2𝑛+𝑠)-dimensional metric 𝑓-manifold 𝑀is said to be an almost trans-𝑆-manifold if it satisfies (∇𝑋𝑓)𝑌= 𝑠 ∑ 𝑖=1 (𝛼𝑖{𝑔(𝑓𝑋, 𝑓𝑌 )𝜉𝑖+𝜂𝑖(𝑌)𝑓2𝑋} + 𝛽𝑖{𝑔(𝑓𝑋, 𝑌 )𝜉𝑖−𝜂𝑖(𝑌)𝑓𝑋}), (2.1) where 𝛼𝑖, 𝛽𝑖are smooth functions on 𝑀. If 𝑀is normal, it is called a trans-𝑆-manifold. When 𝑠= 1 trans-𝑆-manifolds are called trans-Sasakian. Some more relations studied in [1] are 1. 𝑀is a 𝐾-manifold if 𝛽𝑖= 0 for all 𝑖. When 𝑠= 1 it is called quasi-Sasakian. 2. 𝑀is a 𝑆-manifold if and only if it is a 𝐾-manifold and 𝛼𝑖= 1 for all 𝑖. When 𝑠= 1 it is called Sasakian. 3. We have 𝛼𝑖=𝛽𝑖= 0 for all 𝑖if and only if 𝑀is a 𝐶-manifold. 4. Generalized Kenmotsu manifolds are trans-𝑆with 𝛼𝑖= 0 and 𝛽𝑖= 1 for all 𝑖. Our goal is to prove this type of manifold can be embedded canonically into an almost complex manifold ( 𝑀, 𝐽, 𝑔)which is in the Gray-Hervella’s Class 𝑊3⊕ 𝑊4. In order to achieve it we need to compute the Levi-Civita connection (∇) of  𝑀and check ∇ 𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍) = 0 where Ω(𝑋, 𝑌 ) = 𝑔(𝑋, 𝐽𝑌 ). This will be done in the following sections. 14 las trans-𝑠-variedades y la clasificación de gray-hervella. Consider a trans-𝑆-manifold (𝑀, 𝑓, 𝜂𝑖, 𝜉𝑖, 𝑔)of dimension 2𝑟+𝑠. We define  𝑀as the product manifold  𝑀=𝑀×ℝ𝑠of dimension 2𝑟+ 2𝑠equipped with the product metric 𝑔( 𝑋,  𝑌) = 𝑔(𝜄∗ 1 𝑋, 𝜄∗ 1 𝑌) + 𝑔𝑒(𝜄∗ 2 𝑋, 𝜄∗ 2 𝑌)where 𝜄1, 𝜄2are the projection, 𝜄∗ 1, 𝜄∗ 2their pull-back and 𝑔𝑒is the Euclidean metric. From now on,  𝑋,  𝑌and  𝑍will be tensor fields of  𝑀. Definition 2.2. We extend the tensors in 𝑀to  𝑀as follows: 𝜂𝑖=𝜂𝑖◦𝜄∗ 1,(2.2)  𝑓=𝜄1 ∗◦𝑓◦𝜄∗ 1,(2.3) where 𝜄1is the inclusion of 𝑀into  𝑀and 𝜄1 ∗, 𝜄∗ 1its push-forward and pull-back respectively. Definition 2.3. if {𝜕 𝜕𝑥𝑗 }is a basis of ℝ𝑠and 𝑑𝑥𝑗their dual, then 𝐽is the following (1,1) tensor of  𝑀 𝐽( 𝑋) =  𝑓 𝑋−∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝜉𝑖+∑ 𝑗 𝜂𝑗( 𝑋)𝜕 𝜕𝑥𝑗 .(2.4) The next lemma follows directly from the definitions Lemma 2.1.for all 𝑖, 𝑗 we have 𝜂𝑖(𝜕 𝜕𝑥𝑗 )=0, 𝑓(𝜕 𝜕𝑥𝑗 )=0, 𝑑𝑥𝑖(𝜉𝑗)=0, 𝑑𝑥𝑖◦ 𝑓= 0, 𝜂𝑖◦ 𝑓= 0. (2.5) Using definition 2.2 we can extend the formulas involving these tensors from 𝑀 and ℝ𝑠to  𝑀. The following is another simple result. Proposition 2.1.𝐽is an almost complex structure and  𝑀an almost Hermitian Manifold. 2. almost complex manifolds induced by a trans-𝑠15 Proof. Evaluating 𝐽twice we obtain, 𝐽2( 𝑋) = 𝐽( 𝑓 𝑋−∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝜉𝑖+∑ 𝑗 𝜂𝑗( 𝑋)𝜕 𝜕𝑥𝑗) =𝐽( 𝑓 𝑋) − ∑ 𝑖 𝑑𝑥𝑖𝐽(𝜉𝑖) + ∑ 𝑗 𝜂𝑗( 𝑋)𝐽(𝜕 𝜕𝑥𝑗 ) = 𝑓2 𝑋−∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝜕 𝜕𝑥𝑖 −∑ 𝑗 𝜂𝑗( 𝑋)𝜉𝑗 = −𝑋+∑ 𝑗 𝜂𝑗( 𝑋)𝜉𝑗−∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝜕 𝜕𝑥𝑖 −∑ 𝑗 𝜂𝑗( 𝑋)𝜉𝑗 = −  𝑋. (2.6) Now, using the bilinearity of 𝑔, the orthogonality of ,and ℝ𝑠, and that {𝜉𝑖}and {𝜕 𝜕𝑥𝑖 }are orthonormal basis of and ℝ𝑠: 𝑔(𝐽 𝑋, 𝐽  𝑌) = =𝑔 ( 𝑓 𝑋−∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝜉𝑖+∑ 𝑖 𝜂𝑖( 𝑋)𝜕 𝜕𝑥𝑖 , 𝑓 𝑌−∑ 𝑗 𝑑𝑥𝑗( 𝑌)𝜉𝑗+∑ 𝑗 𝜂𝑗( 𝑌)𝜕 𝜕𝑥𝑗) =𝑔 ( 𝑓 𝑋,  𝑓 𝑌)+∑ 𝑖𝑗 𝑑𝑥𝑖( 𝑋)𝑑𝑥𝑗( 𝑌)𝑔(𝜉𝑖, 𝜉𝑗) + ∑ 𝑖𝑗 𝜂𝑖( 𝑋)𝜂𝑗( 𝑌)𝑔 (𝜕 𝜕𝑥𝑖 ,𝜕 𝜕𝑥𝑗) =𝑔 ( 𝑓 𝑋,  𝑓 𝑌)+∑ 𝑖 𝑑𝑥𝑖( 𝑋)𝑑𝑥𝑖( 𝑌) + ∑ 𝑖 𝜂𝑖( 𝑋)𝜂𝑖( 𝑌).(2.7) Because of (1.2) we have 𝑔 ( 𝑓 𝑋,  𝑓 𝑌)+∑𝑖𝜂𝑖( 𝑋)𝜂𝑖( 𝑌)is equal to 𝑔(𝜄∗ 1 𝑋, 𝜄∗ 1 𝑌). Then, 𝑔(𝐽 𝑋, 𝐽  𝑌) = 𝑔(𝜄∗ 1 𝑋, 𝜄∗ 1 𝑌) + 𝑔𝑒(𝜄∗ 2 𝑋, 𝜄∗ 2 𝑌) = 𝑔( 𝑋,  𝑌).(2.8) 2.1 Computing the associated 2-form. Our next step is to extend the Levi-Civita connections ∇1of 𝑀and ∇2of ℝ𝑠to  𝑀. We can use Lemma 1.2 to conclude the Levi-Civita connection must be defined as in (1.11), ∇ 𝑌 𝑋=𝜄1 ∗(∇1 𝜄∗ 1 𝑌𝜄∗ 1 𝑋)+𝜄2 ∗(∇2 𝜄∗ 2 𝑌𝜄∗ 2 𝑋).(2.9) 16 las trans-𝑠-variedades y la clasificación de gray-hervella. In other words, the connection of the product is just the sum of the connections ∇1and ∇2. This implies we can extend by linearity the formulas involving derivatives from 𝑀and ℝ𝑠to  𝑀. For example, remember that a trans-𝑆-manifold satisfies (∇1 𝑋𝑓)(  𝑌) = 𝑠 ∑ 𝑖=1 [𝛼𝑖{𝑔(𝑓𝑋, 𝑓𝑌 )𝜉𝑖+𝜂𝑖(𝑌)𝑓2𝑋} + 𝛽𝑖{𝑔(𝑓𝑋, 𝑌 )𝜉𝑖−𝜂𝑖(𝑌)𝑓𝑋}] (2.10) Then, we obtain Lemma 2.2.The covariant derivative of the 𝑓-strucuture in  𝑀is given by (∇  𝑋 𝑓)(  𝑌) = 𝑠 ∑ 𝑖=1 [𝛼𝑖{𝑔( 𝑓 𝑋,  𝑓 𝑌)𝜉𝑖+𝜂𝑖( 𝑌) 𝑓2 𝑋} + 𝛽𝑖{𝑔( 𝑓 𝑋,  𝑌)𝜉𝑖−𝜂𝑖( 𝑌) 𝑓 𝑋}]. (2.11) As we said in the beginning, our objective is to check in which Gray-Hervella’s Class lies  𝑀. So as to find it out, we need to study the covariant derivative of the associated 2-form. Definition 2.4. We define the 2-forms associated to 𝐽and  𝑓as Ω(  𝑋,  𝑌) = 𝑔( 𝑋, 𝐽  𝑌)(2.12) 𝐹( 𝑋,  𝑌) = 𝑔( 𝑋,  𝑓 𝑌).(2.13) Obviously, 𝐹restricted to 𝑀coincide with the 2-form associated to 𝑓and when it is restricted to ℝ𝑠it is equivalent to zero. As it is shown in the following proposition, Ωcan be expressed as the sum of more simple 2-forms where 𝑑𝑥𝑖are the coordinate 1-forms in ℝ𝑠. Proposition 2.2. Ω(  𝑋,  𝑌) = 𝐹( 𝑋,  𝑌) + 𝑠 ∑ 𝑖=1 𝑑𝑥𝑖∧𝜂𝑖( 𝑋,  𝑌)(2.14) 2. almost complex manifolds induced by a trans-𝑠17 Proof. Using the bilinearity of 𝑔 and the definition of 𝐽we deduce the result. Ω(  𝑋,  𝑌) = 𝑔( 𝑋, 𝐽  𝑌) = 𝑔 ( 𝑋,  𝑓 𝑌−∑ 𝑗 𝑑𝑥𝑗( 𝑌)𝜉𝑗+∑ 𝑗 𝜂𝑗( 𝑌)𝜕 𝜕𝑥𝑗) =𝑔( 𝑋,  𝑓 𝑌) − ∑ 𝑗 𝑑𝑥𝑗( 𝑌)𝑔( 𝑋, 𝜉𝑗) + ∑ 𝑗 𝜂𝑗( 𝑌)𝑔( 𝑋, 𝜕 𝜕𝑥𝑗 ) =𝑔( 𝑋,  𝑓 𝑌) − ∑ 𝑗 𝑑𝑥𝑗( 𝑌)𝜂𝑗( 𝑋) + ∑ 𝑗 𝜂𝑗( 𝑌)𝑑𝑥𝑗( 𝑋) =𝐹( 𝑋,  𝑌) + 𝑠 ∑ 𝑖=1 𝑑𝑥𝑖∧𝜂𝑖( 𝑋,  𝑌).(2.15) The next lemma is just a note which will simplify the computations. Lemma 2.3. (∇  𝑋Ω)(  𝑌 ,  𝑍) = (∇  𝑋𝐹)(  𝑌 ,  𝑍) + ∑ 𝑖 ∇ 𝑋(𝑑𝑥𝑖∧𝜂𝑖)(  𝑌 ,  𝑍).(2.16) Proof. Just use the linearity of the covariant derivative, see definition 1.4. Then, if we find an explicit formula of ∇ 𝑋𝐹and ∇ 𝑋𝑑𝑥𝑖∧𝜂𝑖in terms of the metric, we will have one for ∇ 𝑋Ω. On one hand, using metric compatibility of the connection and the formula (1.8), (∇  𝑋𝐹)(  𝑌 ,  𝑍) =∇  𝑋𝑔( 𝑌 ,  𝑓 𝑍) − 𝑔(∇  𝑋 𝑌 ,  𝑓 𝑍) − 𝑔( 𝑌 ,  𝑓∇ 𝑋 𝑍) =𝑔(∇  𝑋 𝑌 ,  𝑓 𝑍) + 𝑔( 𝑌 , ∇ 𝑋( 𝑓 𝑍)) − 𝑔(∇  𝑋 𝑌 ,  𝑓 𝑍) + 𝑔( 𝑌 , (∇  𝑋 𝑓) 𝑍) −𝑔( 𝑌 , ∇ 𝑋( 𝑓 𝑍)) = 𝑔( 𝑌 , (∇  𝑋 𝑓)).(2.17) Now, from (2.11) and using 𝑔( 𝑌 , 𝜉𝑖) = 𝜂𝑖(𝑌). (∇  𝑋𝐹)(  𝑌 ,  𝑍) = 𝑔( 𝑌 , (∇  𝑋 𝑓)) =𝑔 ( 𝑌 , 𝑠 ∑ 𝑖=1 [𝛼𝑖{𝑔( 𝑓 𝑋,  𝑓 𝑍)𝜉𝑖+𝜂𝑖( 𝑍) 𝑓2 𝑋} + 𝛽𝑖{𝑔( 𝑓 𝑋,  𝑍)𝜉𝑖−𝜂𝑖( 𝑍) 𝑓 𝑋}]) = 𝑠 ∑ 𝑖=1 𝛼𝑖{𝜂𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑓 𝑍) + 𝜂𝑖( 𝑍)𝑔( 𝑌 ,  𝑓2 𝑋)} +𝛽𝑖{𝜂( 𝑌)𝑔( 𝑓 𝑋,  𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋)}.(2.18) 18 las trans-𝑠-variedades y la clasificación de gray-hervella. Finally, using the Lemma 1.1. (∇  𝑋𝐹)(  𝑌 ,  𝑍) = 𝑠 ∑ 𝑖=1 𝛼𝑖{𝜂𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑓 𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋)} +𝛽𝑖{𝜂( 𝑌)𝑔( 𝑓 𝑋,  𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋)}.(2.19) We also have (∇  𝑋𝑑𝑥𝑖)(  𝑌)=∇ 𝑋(𝑑𝑥𝑖( 𝑌)) − 𝑑𝑥𝑖(∇  𝑋 𝑌) = 𝜕 𝑌𝑖 𝜕 𝑋−𝜕 𝑌𝑖 𝜕 𝑋= 0 where  𝑌𝑖 is the coordinate 𝜕 𝜕𝑥𝑖 . Using this equality and the Leibniz’s rule, we obtain ∇ 𝑋(𝑑𝑥𝑖∧𝜂𝑖)(  𝑌 ,  𝑍) =(∇  𝑋𝑑𝑥𝑖) ∧ 𝜂𝑖( 𝑌 ,  𝑍) + 𝑑𝑥𝑖∧ (∇  𝑋𝜂𝑖)(  𝑌 ,  𝑍) =𝑑𝑥𝑖∧ (∇  𝑋𝜂𝑖)(  𝑌 ,  𝑍).(2.20) From [1] we know that (∇  𝑋𝜂𝑖) 𝑌=𝑔(∇  𝑋𝜉𝑖, 𝑌) = 𝑔(−𝛼𝑖 𝑓 𝑋−𝛽𝑖 𝑓2 𝑋,  𝑌)=−𝛼𝑖𝑔( 𝑌 ,  𝑓 𝑋) + 𝛽𝑖𝑔( 𝑓 𝑌 ,  𝑓 𝑋). (2.21) Then, using (2.21) ∇ 𝑋(𝑑𝑥𝑖∧𝜂𝑖)(  𝑌 ,  𝑍) =𝑑𝑥𝑖∧ (∇  𝑋𝜂𝑖)(  𝑌 ,  𝑍) =𝑑𝑥𝑖( 𝑌)(−𝛼𝑖𝑔( 𝑍,  𝑓 𝑋) + 𝛽𝑖𝑔( 𝑓 𝑍,  𝑓 𝑋)) −𝑑𝑥𝑖( 𝑍)(−𝛼𝑖𝑔( 𝑌 ,  𝑓 𝑋) + 𝛽𝑖𝑔( 𝑓 𝑌 , 𝑓  𝑋)) =𝛼𝑖(𝑑𝑥𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋) − 𝑑𝑥𝑖( 𝑌)𝑔( 𝑍,  𝑓 𝑋)) +𝛽𝑖(𝑑𝑥𝑖( 𝑌)𝑔( 𝑓 𝑍, 𝑓  𝑋) − 𝑑𝑥𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋)).(2.22) Finally we have found an explicit formula for ∇ 𝑋Ωdepending on the metric. Putting together (2.16), (2.17) and (2.22) we obtain Proposition 2.3.(∇  𝑋Ω)(  𝑌 ,  𝑍)is equal to ∑ 𝑖 𝛼𝑖{𝜂𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑓 𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋) + 𝑑𝑥𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋) − 𝑑𝑥𝑖( 𝑌)𝑔( 𝑍,  𝑓 𝑋)} 𝛽𝑖{𝜂𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋) + 𝑑𝑥𝑖( 𝑌)𝑔( 𝑓 𝑍,  𝑓 𝑋) − 𝑑𝑥𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋)}. (2.23) Now we can figure out in which class lies  𝑀. This is the aim of the following section. 2. almost complex manifolds induced by a trans-𝑠19 2.2 A Gray-Hervella’s Class for  𝑀 After computing the covariant derivative of Ωwe can calculate the class of  𝑀. This is done in the following proposition. Proposition 2.4.If (𝑀, 𝑓, 𝜂𝑖, 𝜉𝑖, 𝑔)is a trans-𝑆-manifold of dimension 2𝑟+𝑠and ℝ𝑠 is equipped with the Euclidean metric, then the product manifold  𝑀=𝑀×ℝ𝑠has an almost complex structure and lies in 𝑊3⊕ 𝑊4. Proof. We have already shown in Proposition 2.1 it is almost complex. Remember from [5] that a manifold  𝑀is in 𝑊3⊕ 𝑊4if and only if ∇ 𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)=0.(2.24) Therefore, we just have to compute ∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)and see if it is equal to ∇ 𝑋Ω(  𝑌 ,  𝑍). Using (2.4) we obtain 𝜂𝑖(𝐽 𝑋)=−𝑑𝑥𝑖( 𝑋), 𝑑𝑥𝑖(𝐽 𝑋) = 𝜂𝑖( 𝑋),(2.25) 𝐽 𝑓 𝑋= 𝑓2 𝑋. (2.26) From (2.23) we have ∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)is equal to ∑ 𝑖 𝛼𝑖{−𝑑𝑥𝑖( 𝑌)𝑔( 𝑓2 𝑋,  𝑓 𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑓2 𝑌 ,  𝑓2 𝑋) +𝑑𝑥𝑖( 𝑍)𝑔(𝐽 𝑌 ,  𝑓2 𝑋) − 𝜂𝑖( 𝑌)𝑔( 𝑍,  𝑓2 𝑋)} +𝛽𝑖{−𝑑𝑥𝑖( 𝑌)𝑔( 𝑓2 𝑋,  𝑍) − 𝜂𝑖( 𝑍)𝑔(𝐽 𝑌 ,  𝑓2 𝑋) +𝜂𝑖( 𝑌)𝑔( 𝑓 𝑍,  𝑓2 𝑋) − 𝑑𝑥𝑖( 𝑍)𝑔( 𝑓2 𝑌 ,  𝑓2 𝑋)}.(2.27) Now, using 𝑔( 𝑓 𝑋,  𝑌)=−𝑔( 𝑋,  𝑓 𝑌),(2.28) 𝑔(𝜕 𝜕𝑥𝑖 ,𝜕 𝜕𝑥𝑗 ) = 𝑔(𝜉𝑖, 𝜉𝑗) = 𝛿𝑗 𝑖,(2.29) 𝑔( 𝑓 𝑋, 𝜉𝑖) = 𝑔( 𝑓 𝑋, 𝜕 𝜕𝑥𝑖 ) = 𝑔(𝜉𝑖,𝜕 𝜕𝑥𝑗 )=0,(2.30) (2.31) 20 las trans-𝑠-variedades y la clasificación de gray-hervella. together with (1.2) we obtain 𝑔( 𝑓2 𝑋,  𝑓2 𝑌) = 𝑔(𝑓2𝑋, 𝑓2𝑌) = 𝑔(𝑓𝑋, 𝑓𝑌 ) − 𝜂(𝑌)𝜂(𝑋) =𝑔(𝑓𝑋, 𝑓𝑌 ) = 𝑔( 𝑓 𝑋,  𝑓 𝑌),(2.32) 𝑔(𝐽 𝑌 ,  𝑓2 𝑋) = 𝑔( 𝑓𝐽  𝑌 ,  𝑓 𝑋)=−𝑔( 𝑓2𝑌 ,  𝑓 𝑋) = −(  𝑓𝑌 ,  𝑋) +∑ 𝑖 𝜂𝑖( 𝑓𝑌 )𝜂𝑖( 𝑋) = 𝑔( 𝑌 ,  𝑓 𝑋).(2.33) Then, we can reduce (2.27) to ∑ 𝑖 𝛼𝑖{−𝑑𝑥𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋) + 𝑑𝑥𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋) + 𝜂𝑖( 𝑌)𝑔( 𝑓 𝑍,  𝑓 𝑋)} 𝛽𝑖{+𝑑𝑥𝑖( 𝑌)𝑔( 𝑓 𝑋,  𝑓 𝑍) − 𝜂𝑖( 𝑍)𝑔( 𝑌 ,  𝑓 𝑋) + 𝜂𝑖( 𝑌)𝑔( 𝑍,  𝑓 𝑋) − 𝑑𝑥𝑖( 𝑍)𝑔( 𝑓 𝑌 ,  𝑓 𝑋)}. (2.34) Sorting the addends we see it is equal to (2.23) and then (2.24) is satisfied. Note that in Figure 1 there are two equivalent definitions of 𝑊3⊕𝑊4. The second one 𝑆= 0 where 𝑆is the Nijenhuis tensor 𝑁𝐽, see equation (1.6). Therefore, we can formulate the following corollary. Corollary 2.1.If (𝑀, 𝑓, 𝜂𝑖, 𝜉𝑖, 𝑔)is a trans-𝑆-manifold of dimension 2𝑟+𝑠and ℝ𝑠is equipped with the Euclidean metric, then the product manifold  𝑀=𝑀×ℝ𝑠is an Hermitian manifold. Proof. Just remember Definition 1.2. As  𝑀lies in 𝑊3⊕ 𝑊4we could ask if it is actually in 𝑊3or 𝑊4, the answer is: in general it does not. To verify this assertion we need to compute 𝛿Ω. As 𝛿Ω = 𝛿𝐹 +∑𝑖𝛿(𝑑𝑥1∧𝜂𝑖). The following lemma will be enough. Lemma 2.4. 𝛿𝐹 ( 𝑋)=2𝑛∑ 𝑖 𝛼𝑖𝜂𝑖( 𝑋)(2.35) 𝛿(𝑑𝑥1∧𝜂𝑖)(  𝑋)=2𝑛𝛽𝑖𝑑𝑥𝑖( 𝑋)(2.36) Proof. The first equality can be found in the article [1]. 2. almost complex manifolds induced by a trans-𝑠21 For the second, remember when we defined metric 𝑓-manifolds we pointed out there exist in each neighbourhood vector fields such that {𝑋1…𝑋𝑛, 𝑓𝑋1,…, 𝑓𝑋𝑛, 𝜉1,…, 𝜉𝑠}(2.37) was an orthonormal basis for 𝑀. Then, together with {𝜕 𝜕𝑥1 ,…,𝜕 𝜕𝑥𝑠 }we can form an orthonormal basis for  𝑀. Using (1.10) it follows 𝛿(𝑑𝑥𝑖∧𝜂𝑖)(  𝑋) = − ∑ 𝑗[𝑑𝑥𝑖∧ (∇𝑋𝑗𝜂𝑖)(𝑋𝑗, 𝑋) + 𝑑𝑥𝑖∧ (∇  𝑓𝑋𝑗𝜂𝑖)(  𝑓𝑋𝑗, 𝑋) +𝑑𝑥𝑖∧ (∇𝜉𝑗𝜂𝑖)(𝜉𝑗, 𝑋) + 𝑑𝑥𝑖∧ (∇ 𝜕 𝜕𝑥𝑗 𝜂𝑖)( 𝜕 𝜕𝑥𝑗 , 𝑋)].(2.38) Now, using the formula (2.21) and 𝑑𝑥𝑖(𝑋𝑗) = 𝑑𝑥𝑖( 𝑓𝑋𝑗) = 𝑑𝑥𝑖(𝜉𝑗)=0.(2.39) we have 𝛿(𝑑𝑥𝑖∧𝜂𝑖)(  𝑋) = − ∑ 𝑗 −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔(𝑋𝑗, 𝑓𝑋𝑗) + 𝛽𝑖𝑔( 𝑓𝑋𝑗, 𝑓𝑋𝑗)) −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔( 𝑓𝑋𝑗, 𝑓2𝑋𝑗) + 𝛽𝑖𝑔( 𝑓2𝑋𝑗, 𝑓2𝑋𝑗)) −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔(𝜉𝑖, 𝑓𝜉𝑖) + 𝛽𝑖𝑔( 𝑓𝜉𝑖, 𝑓𝜉𝑖)) +𝑑𝑥𝑖(𝜕 𝜕𝑥𝑖 )(−𝛼𝑖𝑔( 𝑋,  𝑓𝜕 𝜕𝑥𝑖 ) + 𝛽𝑖𝑔( 𝑓 𝑋,  𝑓𝜕 𝜕𝑥𝑖 )) −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔(𝜕 𝜕𝑥𝑖 , 𝑓𝜕 𝜕𝑥𝑖 ) + 𝛽𝑖𝑔( 𝑓𝜕 𝜕𝑥𝑖 , 𝑓𝜕 𝜕𝑥𝑖 )).(2.40) Using the orthonormality of the basis and  𝑓𝜉𝑖= 𝑓𝜕 𝜕𝑥𝑖 = 0, 𝑓2 𝑋𝑗= −  𝑋𝑗.(2.41) We obtain 𝛿(𝑑𝑥𝑖∧𝜂𝑖)(  𝑋) = − ∑ 𝑗 −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔(𝑋𝑗, 𝑓𝑋𝑗) + 𝛽𝑖𝑔( 𝑓𝑋𝑗, 𝑓𝑋𝑗)) −𝑑𝑥𝑖( 𝑋)(−𝛼𝑖𝑔( 𝑓𝑋𝑗, 𝑓2𝑋𝑗) + 𝛽𝑖𝑔( 𝑓2𝑋𝑗, 𝑓2𝑋𝑗)).(2.42) Finally, using (1.2) and the orthonormality of the basis 𝑔( 𝑓2𝑋𝑗, 𝑓2𝑋𝑗) = 𝑔( 𝑓𝑋𝑗, 𝑓𝑋𝑗) = 𝑔(𝑋𝑗, 𝑋𝑗)=1 𝑔( 𝑓𝑋𝑗, 𝑓2𝑋𝑗) = 𝑔(𝑋𝑗, 𝑓𝑋𝑗)=0.(2.43) 28 las trans-𝑠-variedades y la clasificación de gray-hervella. From [5] the condition an almost Hermitian manifold needed to satisfy for being in 𝑊2⊕ 𝑊3⊕ 𝑊4is: ∑ 𝑐𝑦𝑐 (∇  𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)) = 0,(3.5) Where ∑ 𝑐𝑦𝑐 is the cyclic sum of (𝑋, 𝑌 , 𝑍). For 𝑊1⊕ 𝑊3⊕ 𝑊4the defining condition is ∇ 𝑋Ω(  𝑋,  𝑌)−∇𝐽 𝑋Ω(𝐽 𝑋,  𝑌)=0.(3.6) We will compute first (3.5). From (3.3) it follows ∑ 𝑐𝑦𝑐 [∇ 𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)]=∑ 𝑐𝑦𝑐 [𝐴( 𝑋,  𝑌 ,  𝑍) − 𝐴(𝐽 𝑋, 𝐽  𝑌 ,  𝑍) + 𝑠 ∑ 𝑗,𝑖=1 𝑑𝑥𝑖( 𝑋)𝜂𝑗(∇𝑋𝜉𝑖)𝜂𝑗( 𝑍) − 𝑑𝑥𝑖( 𝑍)𝜂𝑗(∇𝑋𝜉𝑖)𝜂𝑗( 𝑌) + 𝑠 ∑ 𝑗,𝑖=1 −𝑑𝑥𝑖(𝐽 𝑌)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗( 𝑍) + 𝑑𝑥𝑖( 𝑍)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗(𝐽 𝑌)].(3.7) Then, using (3.4) and the definition of 𝐽: ∑ 𝑐𝑦𝑐 (∇  𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)) = ∑ 𝑐𝑦𝑐 𝑠 ∑ 𝑗,𝑖=1 {𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑍) −𝑑𝑥𝑖( 𝑍)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑌) − 𝜂𝑖( 𝑌)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗( 𝑍) − 𝑑𝑥𝑖( 𝑍)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝑑𝑥𝑗( 𝑌)}, (3.8) which is not zero. Analogously, the equation (3.6) is equal to ∇ 𝑋Ω(  𝑋,  𝑌)−∇𝐽 𝑋Ω(𝐽 𝑋,  𝑌) = 𝐴( 𝑋,  𝑋,  𝑌) − 𝐴(𝐽 𝑋, 𝐽  𝑋,  𝑌) + 𝑠 ∑ 𝑗,𝑖=1 𝑑𝑥𝑖( 𝑋)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑌) − 𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑋) + 𝑠 ∑ 𝑗,𝑖=1 −𝑑𝑥𝑖(𝐽 𝑋)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗( 𝑌) + 𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗(𝐽 𝑋).(3.9) 3. almost trans-𝑠-manifolds and the gray-hervella’s classification 29 Again, using (3.3), (3.4) and the definition of 𝐽. ∇ 𝑋Ω(  𝑋,  𝑌)−∇𝐽 𝑋Ω(𝐽 𝑋,  𝑌) = 𝑠 ∑ 𝑗,𝑖=1 {𝑑𝑥𝑖( 𝑋)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑌) −𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑋) − 𝜂𝑖( 𝑋)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗( 𝑌) − 𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝑑𝑥𝑗( 𝑋)}. (3.10) Actually, we can find an explicit example where these formulas are not zero. Theorem 5 from [1] claims: Let 𝑁be a (2𝑛+𝑠1)-dimensional trans-𝑆-manifold with functions 𝛼𝑖, 𝛽𝑖. Then, if 𝑓,𝜉𝑖come from the 𝑓-structure of 𝑁and { 𝜕∕𝜕𝑡𝑖} is a basis for ℝ𝑠,𝜄the inclusion of 𝑁into the warped product 𝑀=ℝ𝑠×ℎ𝑁and 𝑓∗the following metric 𝑓-structure in 𝑀. 𝑓∗(𝑋) = 𝜄∗(𝑓(𝜄∗𝑋)),(3.11) 𝜉∗ 𝑖=⎧ ⎪ ⎨ ⎪ ⎩ 𝜕 𝜕𝑡𝑖 1≤𝑖≤𝑠 1 ℎ𝜉𝑖−𝑠𝑠+ 1 ≤𝑖≤𝑠+𝑠1, (3.12) Then, we have that 𝑀is a (2𝑛+𝑠+𝑠1) almost trans-𝑆-manifold with functions 𝛼∗ 𝑖=⎧ ⎪ ⎨ ⎪ ⎩ 0 1 ≤𝑖≤𝑠 𝛼𝑖−𝑠 ℎ𝑠+ 1 ≤𝑖≤𝑠+𝑠1, (3.13) 𝛽∗ 𝑖=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ℎ𝑖) ℎ1≤𝑖≤𝑠 𝛽𝑖−𝑠 ℎ𝑠+ 1 ≤𝑖≤𝑠+𝑠1, (3.14) and ∇𝑋𝜉∗ 𝑖=ℎ𝑖) ℎ𝜄∗𝑋. (3.15) Now, consider the induced manifold  𝑀=𝑀×ℝ𝑠+𝑠1=ℝ𝑠×ℎ𝑁×ℝ𝑠+𝑠−1. Then 30 las trans-𝑠-variedades y la clasificación de gray-hervella. if we fix  𝑋= 𝑌= − 𝜕 𝜕𝑥𝑠+1 ⇒𝐽 𝑌=𝐽 𝑋=𝜉∗ 𝑠+1 = (1∕ℎ)𝜉1,(3.16)  𝑍= − 𝜕 𝜕𝑥1 ⇒𝐽 𝑌=𝜉∗ 1=𝜕 𝜕𝑡1 ,(3.17) we have ∇ 𝑋𝜉∗ 𝑖= ∇  𝑌𝜉∗ 𝑖= ∇  𝑍𝜉∗ 𝑖= 0,(3.18) ∇𝐽 𝑋𝜉∗ 𝑖= ∇𝐽 𝑌𝜉∗ 𝑖= ∇𝜉∗ 𝑠+1 𝜉∗ 𝑖=ℎ𝑖) ℎ𝜄∗𝜉∗ 𝑠+1 =ℎ𝑖) ℎ𝜉∗ 𝑠+1,(3.19) ∇𝐽 𝑍𝜉∗ 𝑖=ℎ𝑖) ℎ𝜄∗𝜉∗ 1= 0,(3.20) 𝜂𝑖( 𝑋) = 𝜂𝑖( 𝑌) = 𝜂𝑖( 𝑍)=0,(3.21) 𝑑𝑥𝑖( 𝑋) = 𝑑𝑥𝑖( 𝑌) = 𝛿𝑖 𝑠+1, 𝑑𝑥𝑖( 𝑍) = 𝛿𝑖 1.(3.22) Finally, evaluating this vector fields at the formula (3.8) we obtain ∑ 𝑐𝑦𝑐 (∇  𝑋Ω(  𝑌 ,  𝑍)−∇𝐽 𝑋Ω(𝐽 𝑌 ,  𝑍)) = −𝑑𝑥1( 𝑍)𝜂𝑠+1(∇𝐽 𝑋𝜉1)𝑑𝑥𝑠+1( 𝑌) −𝑑𝑥1( 𝑍)𝜂𝑠+1(∇𝐽 𝑌𝜉1)𝑑𝑥𝑠+1( 𝑋) = −2ℎ1) ℎ≠0,(3.23) and in (3.10) ∇ 𝑋Ω(  𝑋,  𝑌)−∇𝐽 𝑋Ω(𝐽 𝑋,  𝑌) = = 𝑠 ∑ 𝑗,𝑖=1 {𝑑𝑥𝑖( 𝑋)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑌) − 𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇  𝑋𝜉𝑖)𝜂𝑗( 𝑋) −𝜂𝑖( 𝑋)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝜂𝑗( 𝑌) − 𝑑𝑥𝑖( 𝑌)𝜂𝑗(∇𝐽 𝑋𝜉𝑖)𝑑𝑥𝑗( 𝑋)} = −𝑑𝑥𝑠+1( 𝑌)𝜂𝑠+1(∇𝐽 𝑋𝜉𝑖)𝑑𝑥𝑠+1( 𝑋)=−ℎ𝑠+1) ℎ≠0,(3.24) as expected. conclusion 31 Conclusion We have proved that if 𝑀is a trans-𝑆-manifold, then the product manifold 𝑀×ℝ𝑠 is Hermitian and lies in 𝑊3⊕ 𝑊4. 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