Can trans-S-manifolds be defined from the Gray-Hervella classification for almost Hermitian manifolds?

Recently, trans-S-manifolds have been defined as a wide class of metric f-manifolds which includes, for instance, f-Kenmotsu manifolds, S-manifolds and C-manifolds and generalize well-studied trans-Sasakian manifolds. The definition of trans-S-manifolds is formulated using the covariant derivative of the tensor f and although this formulation coincides with the characterization of trans-Sasakian manifolds in such a particular case, this latter type of manifolds were not initially defined in this way but using the Gray-Hervella classification of almost Hermitian manifolds. The aim of this paper is to study how (almost) trans-S-manifolds relate with the Gray-Hervella classification and to establish both similarities and differences with the trans-Sasakian case.


Introduction
Since its beginning, manifolds and Riemannian geometry has been an intensive area of research. The discovery 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 were found, the study of some structures gained importance, for instance, complex and symplectic structures in even dimensions and contact structures in odd dimensions. In 1 3 particular, a Kaehler manifold is a Riemannian manifold with both complex and symplectic structure and a Sasakian manifold is a contact manifold M such that M × ℝ is conformal to a Kaehler manifold. Despite all these objects having desirable properties, it is a difficult task to find out when they can be defined on a particular manifold. This is one of the reasons why almost complex and almost contact structures were found so convenient at first. An almost complex structure is a (1, 1) tensor field J which satisfies J 2 X = −X and an almost contact structure is a triple formed by a (1, 1) tensor field , a vector field and its dual 1-form such that 2 X = −X + (X) . Both are much easier to define than a proper complex or contact structure and, if these tensors satisfy some properties, they actually induce not only complex and contact structures, but more rigid ones like Kaehler and Sasakian structures. Moreover, one can prove that in both cases, there exists a Riemannian metric compatible with the structure.
Moreover, 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, K. Yano introduced in [9] the f-structures, tensors fields of type (1,1) satisfying f 3 + f = 0 . This means that almost complex and almost contact structures are particular cases of f-structures. So, an f.pK-manifold, briefly named f-manifold, is a manifold endowed with an f-structure with parallelizable Kernel and satisfying f 2 = −I + ∑ s i=1 i ⊗ i where i are global vector fields and i their dual 1-forms. 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. Again, it is possible to find a compatible Riemannian metric with an f-structure, and then, we deal with metric f-manifolds.
In [5], Gray and Hervella studied how the covariant derivative of the associated 2-form can be used to create 16 classes of almost complex manifolds. The four basic classes are denoted by W 1 , W 2 , W 3 , W 4 and the rest are given by their direct sum. Kaehler manifolds are precisely the intersection of these four classes. In fact, if (N, J, G) is a 2n-dimensional almost Hermitian manifold, these four classes are shown in the following table [5]:

Class
Defining conditions where is the fundamental 2-form (X, Y) = G(X, JY) and N J is the Nijenhuis tensor field of J given by for any two vector fields X, Y tangent to N.
In 1985, Oubiña [7] defined trans-Sasakian manifolds as almost contact manifolds M such that M × ℝ is an almost complex manifold in the class W 4 , containing locally conformal Kaehler manifolds. In addition, he proved that being trans-Sasakian is equivalent to being normal and satisfying for some differentiable functions , on M. Almost Sasakian manifolds set up another class of almost contact metric manifolds [7]. If M is almost Sasakian, then it is not normal, because M × ℝ falls in the class W 2 ⊕ W 4 .
The objective of Oubiña was to generalize Kenmotsu, cosymplectic and Sasakian manifolds. The relation between these manifolds and trans-Sasakian manifolds is given by the following table, where (X, Y) = g(X, Y) is the fundamental 2-form. Kenmotsu: Quasi-Sasakian: Trans-Sasakian: where i , i are differentiable functions on the manifold. If the normality condition is removed, the metric f-manifold is called an almost trans-S-manifold. We can resume the situation showed in [1] by using the following table, f-Kenmotsu [2,8]: Definition: Trans-S-manifold [1]: Definition: Definition: Necessary Condition: where F(X, Y) = g(X, fY) is the fundamental 2-form. It should be noted that there exist K-manifolds which are not trans-S-manifolds (see [1]). In other words, trans-S-manifolds cannot be defined using dF, d i and f * ( F) . On the other hand f-Kenmotsu, S-manifolds

3
and C-manifolds are always trans-S-manifolds. Moreover, there are more types of metric f-manifolds which actually are trans-S-manifolds: for instance, the homothetic s-th Sasakian manifolds [6] and f-manifolds of Kenmotsu type introduced by Falcitelli and Pastore [4]. Consequently, the new class of trans-S-manifolds deserves to be studied.
Then, one question remains: Can we use the Gray-Hervella classification to define (almost) trans-S-manifolds?
This paper tries to clarify this question and compare the results with the trans-Sasakian case. In Sect. 2, we recall some results about f-structures for later use. In Sect. 3, we study the product manifold of a (2n + s)-dimensional trans-S-manifold and ℝ s equipped with the Euclidean metric in order to classify it in the Gray-Hervella classification. Section 4 is devoted to prove the main result: is a trans-S-manifold of dimension 2n + s and ℝ s is equipped with the Euclidean metric, then the product manifold M = M × ℝ s is an Hermitian manifold and We also show that the situation in the case s > 1 is the best possible. In Sect. 5, we present a manifold in the class Therefore, we will conclude that although the Gray-Hervella classification cannot be used to define (almost) trans-S-manifold, there is still an interesting relation given by the previous theorem.

f-Structures
A (2n + s)-dimensional Riemannian manifold (M, g) endowed with an f-structure f (that is, a tensor field of type (1,1) and rank 2n satisfying f 3 + f = 0 [9]) is said to be a metric f-manifold if there exist s global vector fields 1 , … , s on M (called structure vector fields) and s 1-foms 1 , … , s such that, for any X, Y ∈ X(M) and i, j = 1, … , s . If s = 0 , f is an almost complex structure and M an almost Hermitian manifold. When s = 1 , f is an almost contact structure and M an almost contact manifold.
If we fix l = −f 2 and m = f 2 + I , it is clear that: In other words, both the operators are projection onto two mutually orthogonal distributions L = Im(f ) , M = span( 1 , … , s ) and X(M) = L ⊕ M . Because of (3), these two distributions are complementary, and in any neighborhood, we can find an orthonormal local basis of L , .
A metric f-manifold is said to be a K-manifold [3] if it is normal and dF = 0 . In a K-manifold M, the structure vector fields are Killing vector fields [3]. A K-manifold is called an S-manifold if F = d i , for any i and a C-manifold if d i = 0 , for any i. Note that, for s = 0 , a K-manifold is a Kaehlerian manifold and, for s = 1 , a K-manifold is a quasi-Sasakian manifold, an S-manifold is a Sasakian manifold and a C-manifold is a cosymplectic manifold. When s ≥ 2 , non-trivial examples can be found in [3,6]. Moreover, a K-manifold M is an

S-manifold if and only if and it is a C-manifold if and only if
It is easy to show that in an S-manifold, for any X, Y ∈ X(M) and in a C-manifold,

Almost Complex Manifolds induced by a trans-S-manifold
When s = 1 , trans-S-manifolds are actually trans-Sasakian manifolds. Some more relations studied in [1] are as follows: Our goal is to prove that this type of manifold can be embedded canonically into an almost complex manifold M ,J,Ḡ which is in the Gray-Hervella class W 3 ⊕ W 4 [5].
Consider This implies that we can extend by linearity the formulas involving derivatives from M and ℝ s to M . Therefore, M is also a metric f-manifolds with structure tensors: Moreover, if M is a trans-S-manifold with structure functions ( i , i ) , i = 1, … , s , from (8), we obtain that M is also a trans-S-manifold with functions where, of course, we are also denoting i = i • and i = i • , for any i (see Corollary 4.7 of [1]). Next, we define the (1,1) tensor field J on M by Using (9), we can extend the formulas involving these tensors fields from M and ℝ s to M . Moreover, it is straightforward to show that J is an almost complex structure and M an almost Hermitian manifold. We define the 2-forms associated to J and f as follows: Obviously, F restricted to M coincides with the 2-form associated to f and its restriction to ℝ s vanishes. A direct expansion using the definitions shows that ̄ can be expressed as the sum of more simple 2-forms where dx i are the coordinate 1-forms in ℝ s .
Moreover, due to the linearity of the covariant derivative, (14) can be easily rewritten as follows: In this context, we can prove the following proposition. Proof Firstly, since ∇XF Ȳ ,Z =Ḡ Ȳ , ∇Xf Z , for any X ,Ȳ,Z , and since M is also a trans-S-manifold with structure functions taking into account (3) and using g Ȳ , i =̄i(Y) , for any i, we get

Proposition 1 If M is a trans-S-manifold of dimension
X ,Ȳ =Ḡ X ,JȲ ,

A Gray-Hervella class for M
After computing the covariant derivative of ̄ , we can calculate the class of M . This is done in the following proposition.

Proof Remember from [5] that a manifold M is in W 3 ⊕ W 4 if and only if
Therefore, we just have to compute ∇̄JX̄ JȲ ,Z and see if it is equal to ∇X̄ Ȳ ,Z . Using (12), we obtain Since, from the definition of Ḡ , from (16) together with (3), we deduce Sorting the addends, we see that (22) is equal to (16), and then, (20) is satisfied. ◻ We recall that an equivalent defining condition of the class W 3 ⊕ W 4 is N J = 0 [5]. Therefore, we can formulate the following corollary.

Corollary 1 If M is a trans-S-manifold of dimension 2n + s and ℝ s is equipped with the Euclidean metric, then the product manifold M = M × ℝ s is an Hermitian manifold.
As M lies in W 3 ⊕ W 4 , we could ask if it is actually either in W 3 or in W 4 . The answer is, in general, negative. To verify this assertion, we need to compute ̄ . Since the following lemma will be enough.

Lemma 1 Let M be a trans-S-manifold of dimension
2n + s and ℝ s equipped with the Euclidean metric. Then, for the product manifold M = M × ℝ s , the following formulas are satisfied Proof The first equality can be found in [1]. For the second equality, consider a local orthonormal basis of tangent vector fields in M given bȳ This allows us to establish that if M is in W 3 , then it is a C-manifold, because in [5] is proved that in case of being M in W 3 , then ̄= 0 . Furthermore, we also can assert that if M is a trans-S-manifold of dimension 2n + s with s > 1 , the product manifold M = M × ℝ s is not, in general, in the class W 4 . In fact, since the real dimension of M is 2(n + s) , the condition for it to be in W 4 is [5]: By using the definition of Ḡ , (3) for f , (16) and (25), a straightforward computation gives that 5 Trans-S manifolds cannot be defined using W 3 ⊕ W 4 In this section, we will prove there are manifolds M × ℝ s in W 3 ⊕ W 4 with M a metric f-manifold which are not trans-S-manifolds. Firstly, since from (15), we have using (5) and (7), we can easily state

Proposition 4 The almost complex manifold induced by a C-manifold is a Kaelher manifold.
Next, we prove the following theorem.

Almost trans-S-manifolds and the Gray-Hervella classification
By Proposition 1, it follows that if M is a trans-S-manifold, then M × ℝ s lies neither in the class W 1 ⊕ W 2 ⊕ W 3 nor in the class W 1 ⊕ W 2 ⊕ W 4 . In this section, starting by a suitable (2n + s + s 1 )-dimensional almost trans-S-manifold M, we prove that M = M × ℝ s+s 1 falls neither in W 2 ⊕ W 3 ⊕ W 4 nor in W 1 ⊕ W 2 ⊕ W 4 . First, we remind the following expression for almost trans-S-manifolds which appears in [1].
We also have that Consequently, where A X ,Ȳ,Z is the second part of the equality in Eq. (22). Remember we have proved in Proposition 2, without using the normality condition, for any X ,Ȳ,Z. From [5], the condition that an almost Hermitian manifold needs to satisfy for being in W 2 ⊕ W 3 ⊕ W 4 is as follows: where ∑ cyc is the cyclic sum of X ,Ȳ,Z . Then, from (29), (30) and taking into account the definition of J , it follows that

Conclusions
Given a trans-S-manifold M, we have proved that the product manifold M × ℝ s is a Hermitian manifold and lies in the class W 3 ⊕ W 4 . This fact generalizes the trans-Sasakian case in which M × ℝ lies in the class W 4 . However, there exist metric f-manifolds M such that M × ℝ s lies in the class W 3 ⊕ W 4 without M being trans-S-manifolds. Consequently, this class cannot be used to define trans-S manifolds. Finally, we have presented an example of an almost trans-S-manifold M such that the product manifold M × ℝ s lies only in the most general class W 1 ⊕ W 2 ⊕ W 3 ⊕ W 4 .