Generalized S-space-forms

PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 151–161 DOI: 10.2298/PIM1308151P GENERALIZED S-SPACE-FORMS Alicia Prieto-Martín, Luis M. Fernández, and Ana M. Fuentes Abstract. We introduce and study generalized S-space-forms. Moreover, we investigate generalized S-space-forms endowed with an additional structure and we obtain some obstructions for them to be S-manifolds. 1. Introduction It is an interesting problem to analyze what kind of Riemannian manifolds may be determined by special pointwise expressions for their curvatures. For instance, it is well known that the sectional curvatures of a Riemannian manifold determine the curvature tensor field completely. So, if (M, g) is a connected Riemannian manifold with dimension greater than 2 and its curvature tensor field Rhas the pointwise expression R(X, Y )Z=λ{g(X, Z)Y−g(Y, Z)X}, where λis a differentiable function on M, then Mis a space of constant sectional curvature, that is, a real-space-form and λis a constant function. Further, when the manifold is equipped with some additional structure, it is sometimes possible to obtain conclusions from the special form of the curvature tensor field for this structure too. Thus, an almost-Hermitian manifold (M, J, g) is said to be a generalized complex-space-form [9] if its curvature tensor satisfies R(X, Y )Z=f1g(Y, Z)X−g(X, Z)Y (1.1) +f2g(X, JZ)JY −g(Y, JZ)JX + 2g(X, JY )JZ, where f1and f2are differentiable functions on M. This name derives from the fact that, when Mis a complex-space-form, that is, a Kaehlerian manifold of constant holomorphic curvature equal to c, the curvature tensor field of Msatisfies (1.1) with f1=f2=c/4. 2010 Mathematics Subject Classification: 53C25, 53C40. Key words and phrases: metric f-manifold, f-contact manifold, f-K-contact manifold, S- manifold, generalized S-space-form. The first and the second authors are partially supported by the PAI group FQM-327 (Junta de Andalucía, Spain, 2012) and by the MEC project MTM 2011-22621 (MEC, Spain, 2011). The first author is also supported by a grant of the Fundación Cámara, Spain, 2009-2013. 151 152 PRIETO-MARTÍN, FERNÁNDEZ, AND FUENTES Since Sasakian-spaces-forms play a similar role in contact metric geometry to that of complex-space-forms in complex geometry, Alegre, Blair and Carriazo have defined and studied generalized Sasakian-space forms [1] as those almost-contact metric manifolds (M, φ, ξ, η, g) whose curvature tensor field satisfies R(X, Y )Z=f1{g(Y, Z)X−g(X, Z)Y}+ +f2{g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ} +f3{η(X)η(Z)Y−η(Y)η(Z)X+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ}, f1, f2, f3being differentiable functions on M. If Mis actually a Sasakian-space- form, that is a Sasakian manifold with constant φ-sectional curvature equal to c, then f1=1 4(c+ 3), f2=f3=1 4(c−1). More in general, Yano [10] 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, Blair [2] defined K-manifolds (and particular cases of S-manifolds and C-manifolds) as the analogue of Kaehlerian manifolds in the almost complex geometry and of quasi-Sasakian manifolds in the almost contact geometry and he showed that the curvature of either S- manifolds or C-manifolds is completely determined by their f-sectional curvatures. Later, Kobayashi and Tsuchiya [8] got expressions of the curvature tensor field of S- manifolds and C-manifolds when their f-sectional curvature is constant depending on such a constant. For these reasons, we consider that it is interesting to introduce a notion of generalized S-space-form on metric f-manifolds (see Section 2 for a precise definition of these manifolds). We observe that this work was made in [5] for metric f-manifolds with two structure vector fields, giving some interesting examples. Now, we present the definition for any number of structure vector fields. To this end, we have followed the same procedure as in almost complex and almost contact cases, that is, we have substituted the constants in the expression of the curvature tensor field of an S-space-form (an S-manifold of constant f-sectional curvature) obtained in [8] by certain differentiable functions on the manifold. So, S-space-forms are natural examples of generalized S-space-forms. Furthermore, we check that C-space-forms are also generalized S-space-forms. We have organized the communication in the following way. In Section 2 we review definitions and formulas concerning metric f-manifolds which we shall use later. In Section 3 we define generalized S-space-forms and study the sectional curvatures of such manifolds. Moreover, we establish that the writing of the curvature tensor field is unique in terms of a family of differentiable functions on the manifold if and only if the dimension of the manifold is greater than 2 + s,sbeing the number of structure vector fields. In Section 4, we consider a different definition given by Falcitelli and Pastore in [6], comparing both definitions. Finally, in Section 5, we study generalized S-space-forms endowed with an additional structure and the relationships between the functions in such a case. Thus, we prove that any generalized S-space-form with a metric f-K-contact structure is actually GENERALIZED S-SPACE-FORMS 153 an S-manifold and we deduce an obstruction for a generalized S-space-form to be an S-manifold, depending on the functions. The same result holds for a metric fcontact structure with some additional conditions on the functions. We also study generalized S-space-forms with an underlying C-structure and, more in general, with a K-structure. 2. Metric f-manifolds A Riemannian manifold (M, g) of dimension 2n+sand endowed with an fstructure f(that is, a tensor field of type (1,1) and rank 2nsatisfying f3+f= 0 [10]) 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; g(X, Y ) = g(fX, fY ) + s X α=1 ηα(X)ηα(Y), for any X, Y ∈ X(M) and α= 1,...,s. The distribution on Mspanned by the structure vector fields is denoted by Mand its complementary orthogonal distribution is denoted by L. Consequently, T M =L ⊕ M. Moreover, if X∈ L, then ηα(X) = 0, for any α= 1,...,s and if X∈ M, then fX = 0. 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 and, particularly, Mis orientable. A metric f-manifold is said to be a metric f-contact manifold if F= dηα, for any α= 1,...,s. On the other hand, a metric f-contact manifold is said to be a metric f-K-contact manifold if the structure vector fields are Killing vector fields. When s= 1, metric f-contact manifolds correspond to contact manifolds and metric f-K-contact manifolds to K-contact manifolds. Furthermore, in a metric f-K-contact manifold it easy to show that: (2.1) ∇Xξα=−fX, X ∈ X(M), α = 1,...,s. The f-structure fis said to be normal if [f, f] + 2 Ps α=1 ξα⊗dηα= 0,where [f, f] denotes the Nijenhuis tensor of f. Then, a metric f-manifold is said to be aK-manifold [2] if it is normal and dF= 0. In a K-manifold M, the structure vector fields are Killing vector fields [2] and: (2.2) ∇ξαξβ= 0, α, β = 1,...,s. AK-manifold is called an S-manifold if F= dηα, for any α(that is, if it is also a metric f-K-contact manifold) and a C-manifold if dηα= 0, for any α. 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 [2, 3, 7]. Moreover, a K-manifold Mis an S-manifold if and only if ∇Xξα=−fX, X ∈ X(M), α = 1,...,s, 154 PRIETO-MARTÍN, FERNÁNDEZ, AND FUENTES and it is a C-manifold if and only if ∇f= 0 [2]. On the other hand, the curvature tensor field Rof a K-manifold Msatisfies (2.3) R(ξα, X, ξβ, Y ) = −g(∇Xξβ,∇Yξα), for any X, Y ∈ X(M) and α, β = 1,...,s [4]. A plane section πon a metric f-manifold Mis said to be an f-section if it is determined by a unit vector X∈ L and fX. The sectional curvature K(π) of πis called an f-sectional curvature. An S-manifold (resp., a C-manifold) is said to be an S-space-form (resp., a C-space-form) if it has a constant f-sectional curvature cand then, it is denoted by M(c). In such cases, the curvature tensor field Rof M(c) satisfies R(X, Y, Z, W) = X α,β g(fX, fW)ηα(Y)ηβ(Z)−g(fX, fZ)ηα(Y)ηβ(W) +g(fY, fZ)ηα(X)ηβ(W)−g(fY, fW)ηα(X)ηβ(Z) +c+ 3s 4g(fX, fW )g(fY, fZ)−g(fX, fZ)g(fY, fW) +c−s 4F(X, W)F(Y, Z)−F(X, Z)F(Y, W) −2F(X, Y )F(Z, W), (2.4) (resp., R(X, Y, Z, W) =c 4(g(fX, fW)g(fY, fZ)−g(fX, fZ)g(fY, fW)) +F(X, W)F(Y, Z)−F(X, Z)F(Y, W) −2F(X, Y )F(Z, W))), (2.5) for any X, Y, Z, W ∈ X (M) [8]. 3. Generalized S-space-forms A metric f-manifold (M, f, ξ1,...,ξs, η1,...,ηs, g) is said to be a generalized S-space-form if there exists a family of differentiable functions on M, {F1, F2, Fαβ, Gαβ, Hαβγ}, such that the curvature tensor field Rof Msatisfies (3.1) R=F1R1+F2R2+ s X α,β=1 FαβRαβ +X 1⩽α<β⩽s Gαβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α Hαβγ Rαβγ, GENERALIZED S-SPACE-FORMS 155 where R1(X, Y, Z, W) = g(X, W)g(Y, Z)−g(X, Z)g(Y, W); R2(X, Y, Z, W) = F(X, W)F(Y, Z)−F(X, Z)F(Y, W) −2F(X, Y )F(Z, W ); Rαβ(X, Y, Z, W ) = g(Y, W)ηα(X)ηβ(Z)−g(X, W )ηα(Y)ηβ(Z) +g(X, Z)ηα(Y)ηβ(W)−g(Y, Z)ηα(X)ηβ(W); e Rαβ(X, Y, Z, W ) = ηα(X)ηβ(Y)ηβ(Z)ηα(W)−ηβ(X)ηα(Y)ηβ(Z)ηα(W) +ηβ(X)ηα(Y)ηα(Z)ηβ(W)−ηα(X)ηβ(Y)ηα(Z)ηβ(W); Rαβγ (X, Y, Z, W ) = ηα(X)ηβ(Y)ηγ(Z)ηα(W)−ηβ(X)ηα(Y)ηγ(Z)ηα(W) +ηβ(X)ηα(Y)ηα(Z)ηγ(W)−ηα(X)ηβ(Y)ηα(Z)ηγ(W), (3.2) for any X, Y, Z, W ∈ X (M). This kind of manifold appears as a natural generalization of S-space-forms because a straightforward computation from (2.4) gives that any S-space-form M(c) is a generalized S-space-form with functions F1=1 4(c+ 3s); F2=1 4(c−s); Fαα =1 4(c+ 3s)−1; Fαβ =−1 (α6=β); Gαβ =1 4(c+ 3s)−2 (α < β); Hαβγ =−1 (α6=β6=γ6=α), where α, β, γ ∈ {1,...,s}. Moreover, any C-space-form M(c) is also a generalized S-space-form. In fact, from (2.5), we only have to take F1=F2=Fαα =Gαβ =c 4(α < β); Fαβ = 0 (α6=β); Hαβγ = 0 (α6=β6=γ6=α), where α, β, γ ∈ {1,...,s}. From (3.2) we easily deduce that e Rαα = 0; e Rαβ =e Rβα;Rαββ =e Rαβ;Rααα = Rααβ = 0, for any α, β = 1, . . . , s. Furthermore, from (3.1) we get that R(X, ξα, X, ξβ) = Fαβ,(3.3) R(ξα, ξβ, ξγ, ξα) = Hαβγ −Fβγ ,(3.4) for any unit vector field X∈ L and any α, β, γ = 1,...,s,α6=β6=γ6=α. Then, by using the symmetries of the curvature tensor field R, from (3.3) and (3.4) together, we obtain Fαβ =Fβα and Hαβγ =Hαγβ,α, β, γ = 1,...,s,α6=β6=γ6=α. Now, we observe that, if s= 2, (3.1) agrees with (3.1) of [5]. In that paper, more examples of generalized S-space-forms with two structure vector fields were given and they can be generalized to any s. Thus, pseudo-umbilical, totally contact-umbilical, totally contact-geodesic, totally umbilical and totally geodesic hypersurfaces of a generalized S-space-form are also generalized S-space-forms and, 156 PRIETO-MARTÍN, FERNÁNDEZ, AND FUENTES moreover, the bundle space of a principal toroidal bundle over a Kaehlerian manifold and the warped product of Rtimes a generalized S-space-form are generalized S-space-forms too. Next, for the sectional curvatures of a generalized S-space form and by using (3.1) and (3.2), we can prove the following proposition. Proposition 3.1.Let Mbe a generalized S-space-form with functions: {F1, F2, Fαβ, Gαβ, Hαβγ}. Then, for any orthonormal vector fields X, Y ∈ L and α, β ∈ {1,...,s}, we have (i) K(X, Y ) = R(X, Y, Y, X) = F1+ 3F2g(X, fY )2. (ii) H(X) = K(X, fX) = F1+ 3F2. (iii) K(X, ξα) = F1−Fαα. (iv) K(ξα, ξβ) = F1−Fαα −Fββ +Gαβ ,(α < β). We are going now to study if the writing of the curvature tensor field of a generalized S-space-form is unique. First, we can prove: Proposition 3.2.Let Mbe a (2n+s)-dimensional generalized S-space-form. If n⩾2, the writing of the curvature tensor field Rof Min terms of a family of functions is unique. Proof. Let us suppose that there exist two families of differentiable functions, {F1, F2, Fαβ, Gαβ, Hαβγ}and {F∗ 1, F∗ 2, F∗ αβ , G∗ αβ, H∗ αβγ }, such that (3.5) R=F1R1+F2R2+ s X α,β=1 FαβRαβ +X 1⩽α<β⩽s Gαβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α Hαβγ Rαβγ =F∗ 1R1+F∗ 2R2+ s X α,β=1 F∗ αβ Rαβ +X 1⩽α<β⩽s G∗ αβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α H∗ αβγ Rαβγ . Since n⩾2, we can consider a pair of orthonormal vector fields X, Y ∈ L such that g(X, fY ) = 0. From (3.5) we get that R(X, Y, fX, fY ) = F2=F∗ 2and so, R(X, Y, Y, X) = F1=F∗ 1. From (iii) and (iv) of Proposition 3.1 we deduce that Fαα =F∗ αα, for any α= 1, . . . , s and Gαβ =G∗ αβ, for any α, β = 1,...,s,α < β. Finally, if X∈ L is a unit vector field and α, β = 1,...,s,α6=β, from (3.5) again, we get that R(X, ξα, X, ξβ) = Fαβ =F∗ αβ and, by using (3.4), Hαβγ =H∗ αβγ , for any α, β, γ ∈ {1,...,s},α6=β6=γ6=α. Next, what about (2 + s)-dimensional generalized S-space-forms? In this case, the writing of the curvature tensor field is not unique. Actually, if Mis a generalized S-space-form of dimension 2 + ssuch that its curvature tensor field Rcan be simultaneously written as R=F1R1+F2R2+ s X α,β=1 Fαβ Rαβ +X 1⩽α<β⩽s Gαβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α Hαβγ Rαβγ GENERALIZED S-SPACE-FORMS 157 and R=F∗ 1R1+F∗ 2R2+ s X α,β=1 F∗ αβRαβ +X 1⩽α<β⩽s G∗ αβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α H∗ αβγ Rαβγ , then, given a unit vector field X∈ L and α, β, γ ∈ {1,...,s}, from (3.3), (3.4) and Proposition 3.1, we obtain the system F1−F∗ 1= 3(F∗ 2−F2); F1−F∗ 1=Fαα −F∗ αα; Fαβ −F∗ αβ = 0; (α6=β) Fαα −F∗ αα =Gαβ −G∗ αβ ; (α < β) Fββ −F∗ ββ =Gαβ −G∗ αβ ; (α < β) Hαβγ −H∗ αβγ = 0,(α6=β6=γ6=α) whose general solution is given by F∗ 1=F1+h, F∗ 2=F2−1 3h, F∗ αα =Fαα +h, G∗ αβ =Gαβ +h, F∗ αβ =Fαβ, H∗ αβγ =Hαβγ,(3.6) where his a differentiable function on M. Consequently, if h6= 0, the writing of R in not unique and the functions of two different writings are related by (3.6). On the other hand, if Mis a (2+s)-dimensional generalized S-space-form with functions {F1, F2, Fαβ , Gαβ, Hαβγ}and we define the functions {F∗ 1, F∗ 2, F∗ αβ, G∗ αβ, H∗ αβγ } as in (3.6), for any differentiable function hon M, then we deduce: R=F1R1+F2R2+ s X α,β=1 FαβRαβ +X 1⩽α<β⩽s Gαβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α Hαβγ Rαβγ, =F∗ 1R1+F∗ 2R2+ s X α,β=1 F∗ αβRαβ +X 1⩽α<β⩽s G∗ αβ e Rαβ + s X α,β,γ=1, α6=β6=γ6=α H∗ αβγ Rαβγ −hR1+h 3R2−h s X α=1 Rαα −hX 1⩽α<β⩽s e Rαβ. But it is straightforward to check that hR1−h 3R2+h s X α=1 Rαα +hX 1⩽α<β⩽se Rαβ = 0 and, consequently, Mis also a generalized S-space-form with functions {F∗ 1, F∗ 2, F∗ αβ, G∗ αβ, H∗ αβγ }. 158 PRIETO-MARTÍN, FERNÁNDEZ, AND FUENTES 4. A different definition In [6], Falcitelli and Pastore defined a generalized f.pk-space-form as a metric f.pk-manifold Mof dimension 2n+s(actually, a metric f-manifold) endowed with a family of differentiable functions {e F1,e F2,e Fαβ, α, β = 1,...,s}, such that e Fαβ =e Fβα, for any α, β ∈ {1,...,s}and such that the curvature tensor field Rof Mcan be written as R(X, Y )Z=e F1g(fX, fZ)f2Y−g(fY, fZ)f2X (4.1) +e F2g(X, fZ)fY +g(Y, fZ)fX + 2g(X, fY )fZ + s X α,β=1e Fαβηα(X)ηb(Z)f2Y−ηα(Y)ηb(Z)f2X +g(fY, fZ)ηα(X)ξβ−g(fX, fZ)ηα(Y)ξβ, for any X, Y, Z ∈ X(M). This definition is more restrictive than the one concerning generalized S-space-form. In fact, we observe that, from (4.1), R(ξα, ξβ)ξγ= 0, for any α, β, γ ∈ {1,...,s}(this means that the distribution Mis flat), but some examples of generalized S-space-forms not satisfying this condition were presented in [5]. Moreover, if Mis a generalized f.pk-space-form, a straightforward computation using (3.2) gives R=e F1R1+e F2R2+e F1s X α=1 Rαα −X 1⩽α<β⩽s e Rαβ − s X α,β=1e FαβRαβ − s X α,β=1e Fαα e Rαβ − s X α,β=1, α6=βe Fαβs X γ=1, α6=γ6=β Rγαβ. Consequently, Mis a generalized S-space form with functions F1=e F1;F2=e F2;Fαα =e F1−e Fαα;Fαβ =−e Fαβ (α6=β); Gαβ =e F1−e Fαα −e Fββ ;Hαβγ =−e Fβγ . Conversely, if Mis a generalized S-space-form with functions {F1, F2, Fαβ, Gαβ, Hαβγ } such that the distribution Mis flat, then, from (3.4) we get that Hαβγ =Fβγ , for any α, β, γ = 1,...,s,α6=β6=γ6=αand from (v) of Proposition 3.1, Gαβ = Fαα +Fββ −F1, 1 ⩽α < β ⩽s. Then, it is easy to check that Mis a generalized f.pk-space-form with functions: e F1=F1;e F2=F2;e Fαα =F1−Fαα;e Fαβ =−Fαβ (α6=β). 5. Generalized S-space-forms with additional structures Taking into account the results of the above section, if Mis a generalized S-space-form such that the distribution Mis flat (for instance, if Mis either a GENERALIZED S-SPACE-FORMS 159 metric f-K-contact manifold or a K-manifold), we can apply the results of [6] to it. Firstly, we can prove: Theorem 5.1.Let Mbe a (2n+s)-dimensional generalized S-space-form with functions {F1, F2, Fαβ, Gαβ, Hαβγ }, such that ∇ξα=−f, for any α= 1,...,s. Then, Mis an S-manifold and F1=1 4(c+ 3s); F2=1 4(c−s); Fαα =1 4(c+ 3s)−1; Fαβ =−1 (α6=β); Gαβ =1 4(c+ 3s)−2 (α < β); Hαβγ =−1 (α6=β6=γ6=α), where α, β, γ ∈ {1,...,s}and c=F1+3F2. In particular, any generalized S-space- form with a metric f-K-contact-structure is an S-manifold. Proof. Since, the condition of the statement implies that the distribution M is flat, we deduce that Mis a generalized f.pk-space-form and we apply Proposition 7 of [6]. For metric f-K-contact manifolds we only have to consider (2.1).  We point out here that, if n⩾2, cbecomes constant (see, for example, [7]) and Mis actually an S-space-form. Moreover, we deduce: Corollary 5.1.Let Mbe a (2n+s)-dimensional generalized S-space-form with functions {F1, F2, Fαβ , Gαβ, Hαβγ}. If Mis an S-manifold, then F1−F2=s. For C-manifolds, we have: Theorem 5.2.Let Mbe a (2n+s)-dimensional generalized S-space-form with functions {F1, F2, Fαβ , Gαβ , Hαβγ}and with an underlying C-structure. Then F1=F2=Fαα =Gαβ =c/4, α < β;(5.1) Fαβ =Hαβγ = 0, α 6=β6=γ6=α,(5.2) where α, β, γ ∈ {1,...,s}and c=F1+ 3F2. Moreover, if n > 1,Mis a C-space- form. Proof. Since Mis a C-manifold and so, a K-manifold, from (2.2), the distribution Mis flat and Mis also a generalized f.pk-space.form. Furthermore, the structure vector fields are parallel and, by using Proposition 8 and Remark 2 of [6] and applying the relationships obtained in the above section we get the desired results. Finally, from (3.1), the Ricci tensor field Sand the scalar curvature ρof Mare given by S(X, Y ) = (n+ 1)c 2g(X, Y )− s X α=1 ηα(X)ηα(Y) and ρ=n(n+ 1)c. Now, from the second Bianchi identity, ∇iρ= 2 X j ∇jSj i, where Sj idenotes the components of the Ricci tensor of type (1,1). Consequently, (n−1)dc = 0 and hence, dc = 0 if n > 1. 