A new class of metric f-manifolds

arXiv:1612.07063v2 [math.DG] 2 Aug 2017 A NEW CLASS OF METRIC f-MANIFOLDS PABLO ALEGRE, LUIS M. FERN´ ANDEZ, AND ALICIA PRIETO-MART´ IN Abstract. We introduce a new general class of metric f-man- ifolds which we call (almost) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifolds studied previously. We prove their main properties and we present many examples which justify their study. 1. Introduction In complex geometry, the relationships between the different classes of manifolds can be summarize in the well known diagram by Blair [3]: Complex metric //Hermitian dΩ=0 //Kaehler Almost Complex [J,J]=0 OO metric // Almost Hermitian [J,J]=0 OO dΩ=0 // ∇J=0 88 r r r r r r r r r r r Almost Kaehler [J,J]=0 OO And the same for contact geometry: Normal Almost Contact metric // Normal Almost Contact Metric Φ=dη //Sasakian Almost Contact normal OO metric // Almost Contact Metric normal OO Φ=dη // (1) 66 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Contact Metric normal OO 2010 Mathematics Subject Classification. 53C15,53C25,53C99. Key words and phrases. Almost trans-S-manifold, trans-S-manifold, generalized D-conformal deformation, warped product. The first and the second authors are partially supported by the project MTM2014-52197-P (MINECO, Spain). 1 2 PABLO ALEGRE, LUIS M. FERN ´ ANDEZ, AND ALICIA PRIETO-MART´ IN In the above diagram the almost contact structure (φ, η, ξ) is said to be normal if [φ, φ] + 2dη ⊗ξ= 0 and condition (1) is (∇Xφ)Y=g(X, Y )ξ−η(Y)X, for any tangent vector fields Xand Y. Moreover, an almost contact metric manifold (M, φ, ξ, η, g) is said to have an (α, β) trans-Sasakian structure if (see [11] for more details) (1.1) (∇Xφ)Y=α{g(X, Y )ξ−η(Y)X}+β{g(φX, Y )ξ−η(Y)φX}, where α, β are differentiable functions (called characteristic functions) on M. Particular cases of trans-Sasakian manifolds are Sasakian (α= 1, β = 0), cosymplectic (α=β= 0) or Kenmotsu (α= 0, β = 1) manifolds. In fact, we can extend the above diagram to Normal Almost Contact Metric (1.3) //Trans −Sasakian Almost Contact Metric normal OO (1.2) // (1.1) 77 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Almost Trans −Sasakian normal OO where (1.2) dΦ = Φ ∧(φ∗(δΦ) −(δη)η), dη =1 2n{δΦ(ξ)Φ −2η∧φ∗(δΦ)}. and: (1.3) dΦ = −1 nδη(Φ ∧η), dη =1 2nδΦ(ξ)Φ, φ∗(δΦ) = 0. More in general, K. Yano [15] introduced the notion of f-structure on a (2n+s)-dimensional manifold as a tensor field fof type (1,1) and rank 2nsatisfying f3+f= 0. Almost complex (s= 0) and almost contact (s= 1) structures are well-known examples of f-structures. In this context, D.E. Blair [2] defined K-manifolds (and particular cases of S-manifolds and C-manifolds). Then, K-manifolds are the analogue of Kaehlerian manifolds in the almost complex geometry and S-manifolds (resp., C-manifolds) of Sasakian manifolds (resp., cosymplectic manifolds) in the almost contact geometry. Consequently, one can obtain a similar diagram for metric f-manifolds, that is, manifolds endowed with an f-structure and a compatible metric. The purpose of the present paper is to introduce a new class of metric f-manifolds which generalizes that one of trans-Sasakian manifolds. In this context, we notice that there has been a previous generalization of A NEW CLASS OF METRIC f-MANIFOLDS 3 (α, 0)-trans-Sasakian manifolds for metric f-manifolds. It was due to I. Hasegawa, Y. Okuyama and T. Abe who introduced the so-called homothetic s-contact Riemannian manifolds in [8] as metric f-manifolds such that 2cig(fX, Y ) = dηi(X, Y ) for certain nonzero constants ci, i= 1,...,s (actually, they use pinstead of s). In particular, if the structure vector fields ξiare Killing vector fields and the f-structure is also normal, the manifold is called a homothetic s-th Sasakian manifold. They proved that a homothetic s-contact Riemannian manifold is a homothetic s-th Sasakian manifold if and only if (∇Xf)Y=− s X i=1 ci{g(fX, fY )ξi+ηi(Y)f2X}, and ∇Xξi=cifX, for any tangent vector fields Xand Yand any i= 1,...,s. More recently, M. Falcitelli and A.M. Pastore have introduced fstructures of Kenmotsu type as those normal f-manifolds with dF = 2η1∧Fand dηi= 0 for i= 1,...,s [5]. In this context, L. Bhatt and K.K. Dube [1] and A. Turgut Vanli and R. Sari [14] have studied a more general type of Kenmotsu f-manifolds for which all the structure 1-forms ηiare closed and: dF = 2 s X i=1 ηi∧F. These examples justify the idea of introducing the mentioned new more general class of metric f-manifolds, including the above ones, which we shall call trans-S-manifolds because trans-Sasakian manifolds become to be a particular case of them. The paper is organized as follows: after a preliminaries section concerning metric f-manifolds, in Section 3 we define almost trans-S- manifolds and trans-S-manifolds in terms of the derivative of the fstructure and some characteristic functions and study their main properties. Specially, we prove a characterization theorem which gives a necessary and sufficient condition for an almost trans-S-manifold to be a trans-S-manifold, that is, for the normality of the structure, concerning the derivative of the structure vector fields in any direction. Moreover, we observe that S-manifolds, C-manifolds and Kenmotsu f-manifolds actually are trans-S-manifolds. On the other hand, we get some desirable conditions to be satisfied for trans-S-manifolds in order to generalize those ones of trans-Sasakian manifolds. By using them, we characterize what trans-S-manifolds are K-manifolds and we 4 PABLO ALEGRE, LUIS M. FERN ´ ANDEZ, AND ALICIA PRIETO-MART´ IN justify that these two classes of metric f-manifolds are not related by inclusion. Finally, in the last section, we present many non-trivial examples, that is, with non-constant characteristic functions, of (almost) trans-S- manifolds. To this end, we use generalized D-conformal deformations and warped products as tools. Acknowledgement: The first and the second authors are partially supported by the project MTM2014-52197-P (MINECO, Spain). 2. Metric f-manifolds 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 2nsatisfying f3+f= 0 [15]) is said to be a metric f-manifold if, moreover, there exist sglobal vector fields ξ1,...,ξson M(called structure vector fields) such that, if η1,...,ηsare the dual 1-forms of ξ1,...,ξs, then fξα= 0; ηα◦f= 0; f2=−I+ s X α=1 ηα⊗ξα; (2.1) g(X, Y ) = g(fX, fY ) + s X i=1 ηi(X)ηi(Y), for any X, Y ∈ X (M) and i= 1,...,s. The distribution on Mspanned by the structure vector fields is denoted by Mand its complementary orthogonal distribution is denoted by L. Consequently, TM =L⊕M. Moreover, if X∈ L, then ηα(X) = 0, for any α= 1,...,s and if X∈ M, then fX = 0. For a metric f-manifold Mwe can construct very useful local orthonormal basis of tangent vector fields. To this end, let Ube a coordinate neighborhood on Mand X1any unit vector field on U, orthogonal to the structure vector fields. Then, fX1is another unit vector field orthogonal to X1and to the structure vector fields too. Now, if it is possible, we choose a unit vector field X2orthogonal to the structure vector fields, to X1and to fX1. Then, fX2is also a unit vector field orthogonal to the structure vector fields, to X1, to fX1 and to X2. Proceeding in this way, we obtain a local orthonormal basis {Xi, fXi, ξj},i= 1,...,n and j= 1,...,s, called an f-basis. Let Fbe the 2-form on Mdefined by F(X, Y ) = g(X, fY ), for any X, Y ∈ X (M). Since fis of rank 2n, then η1∧ · · · ∧ ηs∧Fn6= 0 A NEW CLASS OF METRIC f-MANIFOLDS 5 and, in particular, Mis orientable. A metric f-manifold is said to be ametric f-contact manifold if F= dηi, for any i= 1,...,s. The f-structure fis said to be normal if [f, f] + 2 s X i=1 ξi⊗dηi= 0, where [f, f] denotes the Nijenhuis tensor of f. If fis normal, then [7] (2.2) [ξi, ξj] = 0, for any i, j = 1,...,s. A metric f-manifold is said to be a K-manifold [2] if it is normal and dF= 0. In a K-manifold M, the structure vector fields are Killing vector fields [2]. A K-manifold is called an S-manifold if F= dηi, for any iand 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 aC-manifold is a cosymplectic manifold. When s≥2, non-trivial examples can be found in [2, 8]. Moreover, a K-manifold Mis an S-manifold if and only if (2.3) ∇Xξi=−fX, X ∈ X (M), i = 1,...,s, and it is a C-manifold if and only if (2.4) ∇Xξi= 0, X ∈ X (M), i = 1,...,s. It is easy to show that in an S-manifold, (2.5) (∇Xf)Y= s X i=1 g(fX, fY )ξi+ηi(Y)f2X, for any X, Y ∈ X (M) and in a C-manifold, (2.6) ∇f= 0. 3. Definition of trans-S-manifolds and main properties The original idea to define (α, β) trans-Sasakian manifolds is to generalize cosymplectic, Kenmotsu and Sasakian manifolds. 6 PABLO ALEGRE, LUIS M. FERN ´ ANDEZ, AND ALICIA PRIETO-MART´ IN 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 In the same way, our idea is to define trans-S-manifolds generalizing C-manifolds, f-Kenmotsu and S-manifolds. As we said in the Introduction, an almost contact manifold is trans- Sasakian if and only if it (1.1) holds. This property aims us to introduce trans-S-manifolds. Definition 3.1. A(2n+s)-dimensional metric f-manifold Mis said to be a almost trans-S-manifold if it satisfies (∇Xf)Y= s X i=1 αi{g(fX, fY )ξi+ηi(Y)f2X} +βi{g(fX, Y )ξi−ηi(Y)fX}], (3.1) for certain smooth functions (called the characteristic functions) αi, βi,i= 1....s, on Mand any X, Y ∈ X (M). If, moreover, Mis normal, then it is said to be a trans-S-manifold. So, if s= 1, a trans-S-manifold is actually a trans-Sasakian manifold. Furthermore, in this case, condition (3.1) implies normality. However, for s≥2, this does not hold. In fact, it is straightforward to prove that, for any X, Y ∈ X (M), [f, f](X, Y )+2 s X i=1 dηi(X, Y )ξi = s X i,j=1 [ηj(∇Xξi)ηj(Y)−ηj(∇Yξi)ηj(X)] ξi, (3.2) which is not zero in general. But, in a trans-S-manifold, (3.2) implies that, for any X∈ X (M) and any i= 1,...,s: s X j=1 ηj(∇Xξi)ηj(Y)− s X j=1 ηj(∇Yξi)ηj(X) = 0. A NEW CLASS OF METRIC f-MANIFOLDS 7 If we put Y=ξk, from (2.2) we get that (3.3) ηk(∇Xξi) = 0, for any i, k = 1,...,s. Using this fact, from (3.1), we deduce that (3.4) ∇Xξi=−αifX −βif2X, for any X∈ X (M) and any i= 1,...,s. Now, we can prove: Theorem 3.1. A almost trans-S-manifold Mis a trans-S-manifold if and only if (3.4) holds for any X∈ X (M)and any i= 1,...,s. Proof. From (3.1) we have that, for any X∈ X (M) and any i= 1,...,s: ∇Xξi=−αifX −βif2X+ s X j=1 ηj(∇Xξi)ξj. Comparing this equality and (3.4) we have that ηj(∇Xξi) = 0, for any i, j = 1, . . . , s. So, from (3.2), the metric f-manifold Mis normal and, consequently, a trans-S-manifold. The converse is obvious.  Observe that (3.1) can be re-written as (∇XF)(Y, Z) = s X i=1 [αi{g(fX, fZ)ηi(Y)−g(fX, fY )ηi(Z)} +βi{g(X, fY )ηi(Z)−g(X, fZ)ηi(Y)}], for any X, Y, Z ∈ X (M). Then, if X∈ L is a unit vector field, we have: (∇XF)(X, ξi) = −αi,(∇XF)(ξi, fX) = βi, i = 1,...,s. Moreover, from (3.4), we deduce (∇Xηi)Y=αig(X, fY ) + βig(fX, fY ), for any X, Y ∈ X (M) and any i= 1,...,s. Again, if X∈ L is a unit vector field, we get: (∇Xηi)fX =−αi,(∇Xηi)X=βi, i = 1,...,s. For trans-S-manifolds, we can prove the following theorem. Theorem 3.2. Let Mbe a trans-S-manifold. Then, (δF)ξi= 2nαi and δηi=−2nβi, for any i= 1,...,s. 8 PABLO ALEGRE, LUIS M. FERN ´ ANDEZ, AND ALICIA PRIETO-MART´ IN Proof. Taking a f-basis {X1,...,Xn, fX1,...,fXn, ξ1,...,ξs}, since (δF)X=− n X k=1 {(∇XkF)(Xk, X) + (∇fXkF)(fXk, X)} − s X j=1 (∇ξjF)(ξj, X) = n X k=1 {g(Xk,(∇Xkφ)X) + g(fXk,(∇fXkφ)X)}, for any X∈ X (M), by using (3.1) it is straightforward to obtain (3.5) (δF)X= 2n s X j=1 αjηj(X) and, putting X=ξi, it follows that (δF)ξi= 2nαi. Moreover, δηi=− n X k=1 {(∇Xkηi)Xk+ (∇fXkηi)fXk} − s X j=1 (∇ξjηi)ξj, for any i= 1, . . . , s. But, from (3.3) we get that (∇ξjηi)ξj= 0, for any j= 1,...,s. Consequently, by using (3.4) δηi=− n X k=1 {g(Xk,∇Xkξi) + g(fXk,∇fXkξi)} =− n X k=1 βi{g(Xk, Xk) + g(fXk, fXk)}=−2nβi. which concludes the proof.  The above theorem generalizes the result given by D.E. Blair and J.A. Oubi˜na in [4] for trans-Sasakian manifolds. Moreover, trans-S- manifolds verify certain desirable conditions. Proposition 3.1. Let Mbe a trans-S-manifold. The following equations are verified: (i) dF = 2F∧ s X i=1 βiηi; (ii) dηi=αiF,i= 1,...,s; (iii) f∗(δF ) = 0. A NEW CLASS OF METRIC f-MANIFOLDS 9 Proof. From (3.1), a direct computation gives, for any X, Y, Z ∈ X (M): dF(X, Y, Z) = −g((∇Xf)Y, Z) + g((∇Yf)X, Z)−g((∇Zf)X, Y ) =2 s X i=1 {−βiηi(Z)g(fX, Y ) + βiηi(Y)g(fX, Z) −βiηi(X)g(fY, Z)} =2(F∧ s X i=1 βiηi)(X, Y, Z). Next, from (3.4) it is obtained the second statement. Finally, from (3.5) we get (iii).  From (ii) of the above proposition we observe that if one of the functions αiis a non-zero constant function, then the 2-form Fis closed and the trans-S-manifold Mis a K-manifold. Moreover we can prove: Theorem 3.3. A trans-S-manifold Mis a K-manifold if and only if β1=···=βs= 0. Proof. Firstly, if all the functions βiare equal to zero, from (i) of Proposition 3.1 we get dF = 0 and Mis a K-manifold. Conversely, it is known (see [6]) that, for K-manifolds, the following formula holds, for any X, Y, Z ∈ X (M): g((∇Xf)Y, Z) = s X i=1 {dηi(fY, X)ηi(Z)−dηi(fZ, X)ηi(Y)}. Consequently, from (ii) of Proposition 3.1 and (3.1) we conclude that βi= 0, for any i= 1,...,s. From Theorem 3.2 we deduce: Corollary 3.1. A trans-S-manifold Mis a K-manifold if and only if δηi= 0, for any i= 1,...,s. Furthermore, taking into account (2.3) and (2.4), we have: Corollary 3.2. Any trans-S-manifold is an S-manifold if and only if αi= 1,βi= 0 and it is a C-manifold if and only if αi=βi= 0, in both cases for any i= 1,...,s. In next section, we shall present some examples of trans-S-manifolds which are not K-manifolds due to not all their characteristic functions βiare zero. Now, the natural question is if any K-manifold is a trans- S-manifold. In general, the answer in negative and to this end, we can consider the following example. 16 PABLO ALEGRE, LUIS M. FERN ´ ANDEZ, AND ALICIA PRIETO-MART´ IN Joining the addends appropriately, it takes the form of (3.1) with the desired functions. Therefore, f Mis a almost trans-S-manifold.  Observe that, in the above conditions, (3.4) is not verified in general. In fact, consider e ξiwith 1 ≤i≤s. Then, for any e X∈ X (f M), e ∇e Xe ξi=hi) hU=hi) h(e X− s X j=1 eηj(e X)ξj) and so, if his not a constant function, from Theorem 3.1, we get that f Mis not a trans-S-manifold. Corollary 4.5. The warped product Rs×hM, being Ma trans-S- manifold, is a trans-S-manifold if and only if his constant. In particular, the Riemannian product Rs×Mis a trans-S-manifold with functions (0,s) . . ., 0, α1,...,αs1,0,s) . . ., 0, β1,...,βs1), where (αi, βi),i= 1,...,s1, denote the characteristic functions of M. Corollary 4.6. Let Mbe a Sasakian manifold. Then, the warped product R×hMis a almost trans-S-manifold with functions: α1= 0, α2=1 h, β1=h′ hand β2= 0. Corollary 4.7. Let Mbe a three dimensional trans-Sasakian, with non-constant characteristic functions αand β. Then, the warped product R×hMis a four dimensional almost trans-S-manifold not trans- S-manifold with functions: α1= 0, α2=α h, β1=h′ hand β2=β h. References [1] L. Bhatt and K.K. Dube, Semi-invariant submanifolds of r-Kenmotsu manifolds, Acta Cienc. Indica Math. 29(1) (2003), 167-172. [2] D.E. Blair, Geometry of manifolds with structural group U(n)× O(s), J. Diff. Geom. 4(1970), 155-167. [3] D.E. Blair, “Riemannian Geometry of Contact and Symplectic Manifolds”, 2nd. Ed., Progress in Mathematics, Vol. 209, Birkh¨auser, New York, 2010. [4] D.E. Blair and J.A. Oubi˜na, Conformal and related changes of metric on the product of two almost contact metric manifolds, Pub. Mat. 34 (1990), 199-207. [5] M. Falcitelli and A. M. Pastore, f-structures of Kenmotsu type, Mediterr. J. Math. 3(2006), 549-564. [6] L.M. Fern´andez, “Variedades con K-estructuras. Subvariedades”, Ph.D. Thesis, University of Sevilla, Sevilla, 1987. [7] S.I. Goldberg and K. 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Yano, On a structure defined by a tensor field fof type (1,1) satisfying f3+f= 0, Tensor 14 (1963), 99-109. Departamento de Econom´ ıa, M´ etodos Cuantitativos e Historia de la Econom´ ıa, Universidad Pablo de Olavide, Ctra. de Utrera, km. 1, 41013-Sevilla, Spain E-mail address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, C./ Tarfia, s.n., 41012-Sevilla, Spain. E-mail address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Matem´ aticas, Universidad de Sevilla, C./ Tarfia, s.n., 41012-Sevilla, Spain. E-mail address:[email protected]