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

Vol.:(0123456789) Annali di Matematica Pura ed Applicata (1923 -) (2022) 201:2691–2706 https://doi.org/10.1007/s10231-022-01215-9 1 3 Can trans‑S‑manifolds be defined fromtheGray‑Hervella classification foralmost Hermitian manifolds? PabloAlegre1 · LuisM.Fernández1 · ManuelSoriano‑Trigueros2 Received: 23 May 2021 / Accepted: 1 April 2022 / Published online: 30 April 2022 © The Author(s) 2022 Abstract 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. Keywords Almost trans-S-manifold· Trans-S-manifold· Gray-Hervella’s classification Mathematics Subject Classification 53C15· 53C25· 53C99 1 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 * Luis M. Fernández lmf[email protected] Pablo Alegre [email protected] Manuel Soriano-Trigueros [email protected] 1 Departamento de Geometría y Topología, c/Tarfia s/n, Universidad de Sevilla, Seville41012, Spain 2 Departamento de Matemática Aplicada I, Universidad de Sevilla, Av. Reina Mercedes s/n, Seville41012, Spain 2692 P.Alegre et al. 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 J2X=−X and an almost contact structure is a triple formed by a (1,1) tensor field 𝜙 , a vector field 𝜉 and its dual 1-form 𝜂 such that 𝜙2X=−X+𝜂(X)𝜉 . 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 f3+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 𝜂 i⊗𝜉 i where 𝜉i are global vector fields and 𝜂i 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 W1,W2,W3,W4 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 W1 3∇𝛺=d𝛺 W2 d𝛺=0 W3 𝛿𝛺 =NJ=0 (or (∇X𝛺)(Y,Z) − (∇JX 𝛺)(JY,Z)=0 ) W4 (∇ X𝛺)(Y,Z)= −1 2(n−1){G(X,Y)𝛿𝛺(Z)−G(X,Z)𝛿𝛺(Y ) −G(X,JY)𝛿𝛺(JZ)+G(X,JZ)𝛿𝛺(JY)} where 𝛺 is the fundamental 2-form 𝛺(X,Y)=G(X,JY) and NJ 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 W4 , containing locally conformal Kaehler manifolds. In addition, he proved that being trans-Sasakian is equivalent to being normal and satisfying NJ(X,Y)=−[X,Y]−J[JX,Y]−J[X,JY]+[JX,JY], (1) (∇X𝜙)Y=𝛼{g(X , Y)𝜉−𝜂(Y)X}+𝛽{g(𝜙X , Y)𝜉−𝜂(Y)𝜙X} 2693 Can trans‑S‑manifolds be defined fromtheGray‑Hervella… 1 3 for some differentiable functions 𝛼,𝛽 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 W2⊕W4 . 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 𝛷(X,Y)=g(X,𝜙Y) is the fundamental 2-form. Kenmotsu: d𝛷=2𝜂∧𝛷, d𝜂=0, normal Cosymplectic: Quasi-Sasakian: Trans-Sasakian: d𝛷=0, d𝜂=0, d𝛷=0, d𝛷=2𝛽(𝛷∧𝜂), normal normal d𝜂=𝛼𝛷, 𝜙∗(𝛿𝛷)=0, normal Sasakian: 𝛷=d𝜂 , normal Analogously, P. Alegre, L. M. Fernández and A. Prieto introduced in [1] a new class of metric f-manifold called tran-S-manifolds. They are defined as normal metric f-manifolds satisfying where 𝛼i,𝛽i 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: dF =2∑ s i=1 𝜂 i ∧F , d𝜂i=0 for any i, normal C-manifold [3]: K-manifold [3]: Trans-S-manifold [1]: Definition: Definition: Necessary Condition: dF =0, dF =0, dF =2F∧∑ s i=1 𝛽 i 𝜂 i, d𝜂i=0 for any i, normal d𝜂i=𝛼iFfor any i, normal f∗(𝛿F)=0, normal S-manifold [3]: Definition: F=d𝜂ifor any i, normal 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,d𝜂i and f∗(𝛿F) . On the other hand f-Kenmotsu, S-manifolds (2) ( ∇Xf ) Y= s ∑ i=1 [ 𝛼i{g(fX,fY)𝜉i+𝜂i(Y)f2X}+𝛽i{g(fX,Y)𝜉i−𝜂i(Y)fX} ], 2694 P.Alegre et al. 1 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 ℝs equipped with the Euclidean metric in order to classify it in the Gray-Hervella classification. Section4 is devoted to prove the main result: Theorem If (M,f,𝜂i,𝜉i,g) is a trans-S-manifold of dimension 2n+s and ℝs is equipped with the Euclidean metric, then the product manifold  M=M×ℝs is an Hermitian manifold and 1. If s=1 ,  M lies in W4 . 2. If s>1 ,  M lies in W3⊕W4 . 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 W3⊕W4 of the form M×ℝs , where M is a metric f-manifold but not a trans-S-manifold. Finally, in Sect.6, we give an example of a manifold M×ℝs in only the most general class W1⊕W2⊕W3⊕W4 , where M is an almost trans-S-manifold. 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. 2 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 f3+f=0 [9]) is said to be a metric f-man- ifold if there exist s global vector fields 𝜉1,…,𝜉s on M (called structure vector fields) and s 1-foms 𝜂1,…,𝜂s 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=−f2 and m=f2+I , it is clear that: (3) 𝜂i(𝜉j)=𝛿 j i,f𝜉i=0; 𝜂i◦f=0; f2=−I+ s ∑ i=1 𝜂i⊗𝜉 i; g (fX,fY)=g(X,Y)− s ∑ i=1 𝜂i(X)𝜂i(Y) , 2695 Can trans‑S‑manifolds be defined fromtheGray‑Hervella… 1 3 In other words, both the operators are projection onto two mutually orthogonal distributions L= Im(f) , M= span ( 𝜉 1 , … ,𝜉 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 , {X1,…,Xn,fX1,…,fXn} which together with {𝜉1,…,𝜉s} forms a orthonormal local basis for X(M) . Let F be the 2-form on M defined by F(X,Y)=g(X,fY) , forany X,Y∈X(M) . Since f is of rank 2n, then 𝜂1∧ ⋯ ∧𝜂s∧Fn≠0 and, inparticular, M is orientable. The f-structure f is said to be normal if where [f,f] denotes the Nijenhuis tensor of f given by for any X,Y∈X(M) . A metric f-manifold is said to be a K-manifold [3] if it is normal and dF =0 . In a K-man- ifold M, the structure vector fields are Killing vector fields [3]. A K-manifold is called an S-manifold if F=d𝜂i , for any i and a C-manifold if d𝜂i=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, 3 Almost Complex Manifolds induced byatrans‑S‑manifold A ( 2n+s )-dimensional metric f-manifold (M,f,𝜉1,…,𝜉s,𝜂1,…,𝜂s,g) is said to be an almost trans-S-manifold [1] if it satisfies l +m=I,l 2 =l,m 2 =m, fl=lf =f,mf =fm =0. [ f,f]+2 s ∑ i=1 𝜉i⊗d𝜂i= 0, [f , f](X , Y)=f2[X , Y]−f[fX , Y]−f[X , fY]+[fX , fY], (4) ∇X𝜉i=−fX,X∈X(M),i=1, …,s, (5) ∇X𝜉i=0, X∈X(M),i=1, …,s. (6) (∇ Xf)Y= s ∑ i=1 { g(fX,fY)𝜉i+𝜂i(Y)f2X }, (7) ∇f=0. 2696 P.Alegre et al. 1 3 where 𝛼i,𝛽i are smooth functions on M called structure functions. If M is normal, it is called a trans-S-manifold. When s=1 , trans-S-manifolds are actually trans-Sasakian manifolds. Some more relations studied in [1] are as follows: 1. M is a K-manifold if 𝛽i=0 for all i. 2. M is a S-manifold if and only if it is a K-manifold and 𝛼i=1 for all i. 3. M is a C-manifolds if and only if 𝛼i=𝛽i=0 for all i. 4. Generalized Kenmotsu manifolds are trans-S with 𝛼i=0 and 𝛽i=1 for all i. Our goal is to prove that this type of manifold can be embedded canonically into an almost complex manifold ( M,  J,  G ) which is in the Gray-Hervella class W3⊕W4 [5]. Consider a metric f-manifold (M,f,𝜂i,𝜉i,g) of dimension 2n+s , i=1, …,s . Let  M be the product manifold  M=M×ℝs of dimension 2n+2s , where ℝs is the Euclidean space of dimension s, equipped with the product metric  G= 𝜋 ∗( g )+ 𝜎 ∗( g e) , being 𝜋 and 𝜎 the projections of  M onto M and ℝs , respectively, and ge is the Euclidean metric of ℝs . From now on,  X,  Y and  Z will be tensor fields of  M . We extend the tensors in M to  M as follows for any  X and i=1, …,s . Then, if {x1,…,xs} denote the Euclidean coordinates of ℝs , from the definitions, we deduce for all i,j. Next, if ∇1 and ∇2 denote the Levi-Civita connections of M and ℝs , respectively, then, the Levi-Civita of  M is defined by the sum of ∇1 and ∇2 : This implies that we can extend by linearity the formulas involving derivatives from M and ℝs to  M . Therefore,  M is also a metric f-manifolds with structure tensors: Moreover, if M is a trans-S-manifold with structure functions (𝛼i,𝛽i) , i=1, …,s , from (8), we obtain that  M is also a trans-S-manifold with functions (8) (∇ Xf)Y= s ∑ i=1 [ 𝛼i{g(fX,fY)𝜉i+𝜂i(Y)f2X}+𝛽i{g(fX,Y)𝜉i−𝜂i(Y)fX} ], (9)  f X= ( f𝜋∗ X )∗ , 𝜂 i(  X ) =𝜂 i( 𝜋 ∗  X ) , (10) 𝜂 i ( 𝜕 𝜕xj ) =0,  f ( 𝜕 𝜕xj ) = 0, dxi(𝜉j)=0, dxi◦ f=0, 𝜂 i ◦ f=0. (11) ∇  Y  X= ( ∇1 𝜋 ∗  Y𝜋∗ X )∗ + ( ∇2 𝜎 ∗  Y𝜎∗ X )∗ . (  f,𝜂 i,dxi,𝜉i,𝜕 𝜕x i , G ) ,i=1, …,s . ( 𝛼 1 ,…,𝛼 s , 0, s) …, 0, 𝛽 1 ,…,𝛽 s , 0, s) …,0) , 2697 Can trans‑S‑manifolds be defined fromtheGray‑Hervella… 1 3 where, of course, we are also denoting 𝛼i=𝛼i◦𝜋 and 𝛽i=𝛽i ◦ 𝜋 , 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 ℝs 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 ℝs vanishes. A direct expansion using the definitions shows that  𝛺 can be expressed as the sum of more simple 2-forms where dxi are the coordinate 1-forms in ℝs . 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. Proposition 1 If M is a trans-S-manifold of dimension 2n+s , with structure functions 𝛼i,𝛽i and ℝs is equipped with the Euclidean metric, then the 2-form  𝛺 in the product manifold  M=M× ℝ s satisfies Proof Firstly, since ( ∇ X  F )( Y,  Z ) =  G ( Y, ( ∇ X  f ) Z ) , for any  X,  Y,  Z , and since  M is also a trans-S-manifold with structure functions taking into account (3) and using g( Y,𝜉 i) =𝜂 i (Y ) , for any i, we get (12)  J  X= f X− ∑ i dxi (  X ) 𝜉i+ ∑ j 𝜂 j (  X )𝜕 𝜕xj . (13)  𝛺( X,  Y ) =  G ( X,  J  Y ),  F (  X,  Y ) = G (  X,  f Y ) . (14)  𝛺(  X, Y ) = F (  X, Y ) + s ∑ i=1 ( dxi∧𝜂 i )(  X, Y ) (15) ( ∇ X 𝛺 )(  Y, Z ) = ( ∇ X F )(  Y, Z ) + ∑ i ∇ X ( dxi∧𝜂 i )(  Y, Z ). (16) ( ∇ X 𝛺 )(  Y, Z ) = ∑ i [ 𝛼i{𝜂 i (  Y )  G (  f X, f Z ) −𝜂 i (  Z )  G (  f Y, f X ) +dxi( Z) G( Y, f X)−dxi( Y) G( Z, f X)} +𝛽i{𝜂 i( Y) G( f X, Z)−𝜂 i( Z) G( Y, f X) +dxi (  Y )  G (  f Z, f X ) −dxi (  Z )  G (  f Y, f X ) } ] . ( 𝛼1,…,𝛼s, 0, s) …, 0, 𝛽1,…,𝛽s, 0, s) …,0 ), 2698 P.Alegre et al. 1 3 Moreover, we also have ( ∇ X dx i)( Y ) =∇  X( dx i( Y )) −dx i( ∇ X  Y ) = 0 and, consequently, But, we know that [1] and, so Putting together (15), (17) and (19), we complete the proof. ◻ 4 A Gray‑Hervella class for  M After computing the covariant derivative of  𝛺 , we can calculate the class of  M . This is done in the following proposition. Proposition 2 If M is a trans-S-manifold of dimension 2n+s and ℝs is equipped with the Euclidean metric, then the product manifold  M=M×ℝs lies in W3⊕W4 . Proof Remember from [5] that a manifold  M is in W3⊕W4 if and only if Therefore, we just have to compute ( ∇ J  X  𝛺 )( J  Y,  Z ) and see if it is equal to ( ∇ X  𝛺 )( Y,  Z ) . Using (12), we obtain Since, from the definition of  G , (17) ( ∇ X F )(  Y, Z ) = s ∑ i=1 [ 𝛼i{𝜂 i (  Y )  G (  f X, f Z ) −𝜂 i (  Z )  G (  f Y, f X )} +𝛽 i{ 𝜂 i(  Y )  G (  f X, Z ) −𝜂 i(  Z )  G (  Y, f X )}] . ∇  X ( dxi∧𝜂 i )( Y,  Z ) = (( ∇ Xdxi ) ∧𝜂 i )( Y,  Z ) + ( dxi∧ ( ∇ X𝜂 i ))( Y,  Z ) = ( dx i ∧ ( ∇ X 𝜂 i))(  Y, Z ) . (18) ( ∇ X 𝜂 i) Y=−𝛼 i G ( Y,  f  X ) +𝛽 i G ( f  Y,  f  X ), (19) ∇  X ( dxi∧𝜂 i )( 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)) +𝛽 i( dx i(  Y )  G (  f Z,f X ) −dx i(  Z )  G (  f Y, f X )). (20) ( ∇ X  𝛺 )( Y,  Z ) − ( ∇ J  X  𝛺 )( J  Y,  Z ) = 0. (21) 𝜂 i ( J  X ) = −dxi ( X ), d xi (  J X ) =𝜂 i (  X ) ,  J  f  X=  f 2 X . 2699 Can trans‑S‑manifolds be defined fromtheGray‑Hervella… 1 3 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 W3⊕W4 is NJ=0 [5]. Therefore, we can formulate the following corollary. Corollary 1 If M is a trans-S-manifold of dimension 2n+s and ℝs is equipped with the Euclidean metric, then the product manifold  M=M×ℝs is an Hermitian manifold. As  M lies in W3⊕W4 , we could ask if it is actually either in W3 or in W4 . The answer is, in general, negative. To verify this assertion, we need to compute 𝛿 𝛺 . Since the following lemma will be enough. Lemma 1 Let M be a trans-S-manifold of dimension 2n+s and ℝs equipped with the Euclidean metric. Then, for the product manifold  M=M×ℝs , 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 by  G( f  X,  Y ) =−  G ( X,  f  Y ) ,  G(𝜕 𝜕xi ,𝜕 𝜕xj)= G(𝜉i,𝜉j)=𝛿j i,  G (  f X,𝜉i ) = G (  f X,𝜕 𝜕x i) = G ( 𝜉i,𝜕 𝜕x j) = 0, (22) ( ∇ J X 𝛺 )(  J Y, Z ) = ∑ i [ 𝛼i { −dxi (  Y )  G (  f X, Z ) −𝜂 i (  Z )  G (  f Y, f X ) +dxi( Z) G( Y, f X)+𝜂 i( Y) G( f Z, f X)} +𝛽i{+dxi( Y) G( f X, f Z)−𝜂 i( Z) G( Y, f X) +𝜂 i (  Y )  G (  Z, f X ) −dxi (  Z )  G (  f Y, f X )} . 𝛿 𝛺=𝛿  F+ ∑ i 𝛿 ( dxi∧𝜂 i ), (23) 𝛿 F (  X ) =2n ∑ i 𝛼i𝜂 i (  X ) 𝛿( dx i ∧𝜂 i)(  X ) =2n𝛽 i dx i(  X ) . (24) {X1,…Xn,fX1,…,fXn,𝜉1,…,𝜉s} 2706 P.Alegre et al. 1 3 7 Conclusions Given a trans-S-manifold M, we have proved that the product manifold M×ℝs is a Hermitian manifold and lies in the class W3⊕W4 . This fact generalizes the trans-Sasakian case in which M×ℝ lies in the class W4 . However, there exist metric f-manifolds M such that M×ℝs lies in the class W3⊕W4 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×ℝs lies only in the most general class W1⊕W2⊕W3⊕W4 . Acknowledgements We would like to thank to the referee for his/her many valuable comments in order to improve the paper. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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