The curvature tensor of almost cosymplectic and almost Kenmotsu ( κ, μ, ν ) -space

arXiv:1201.5565v2 [math.DG] 10 Feb 2012 THE CURVATURE TENSOR OF ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES ALFONSO CARRIAZO AND VER ´ ONICA MART´ IN-MOLINA Abstract. We study the Riemann curvature tensor of (κ, µ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, µ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases. 1. Introduction The study of the curvature tensor of a Riemannian manifold as a tool to classify it constitutes an important part of Differential Geometry. In particular, many advances have been made recently when the manifold is a generalized (κ, µ)-space or a (κ, µ, ν)-space with constant φ-sectional curvature, called generalized (κ, µ)-space form and (κ, µ, ν)-space form, respectively. These manifolds are considered of great importance by all researchers who are currently working on contact metric geometry and related topics. The contact metric (κ, µ)-spaces were originally introduced (under a different name) by D. E. Blair, T. Koufogiorgos and V. J. Papantoniou in [4] as those contact metric manifolds satisfying the equation (1.1) R(X, Y )ξ=κ{η(Y)X−η(X)Y}+µ{η(Y)hX −η(X)hY }, for every vector fields X, Y on M, where κand µare constants, h= 1/2Lξφand L is the usual Lie derivative. These spaces include the Sasakian manifolds (κ= 1 and h= 0), but the non-Sasakian examples have proven to be even more interesting. In [19], the contact metric generalized (κ, µ)-spaces were introduced as contact metric manifolds satisfying the equation (1.1) with κ, µ functions. The curvature tensor of a contact metric (κ, µ)-space form was shown by T. Koufogiorgos in [17] to have the form R=F+ 3 4R1+F−1 4R2+F+ 3 4−κR3+R4+1 2R5+ (1 −µ)R6, 2010 Mathematics Subject Classification. 53C15, 53C25. Key words and phrases. generalized (κ, µ)-space, (κ, µ, ν)-space, generalized Sasakian space form, almost cosymplectic, almost Kenmotsu. Both authors are partially supported by the MTM2011-22621 grant from the MEC (Spain) and by the PAI group FQM-327 (Junta de Andaluc´ıa, Spain). 1 2 A. CARRIAZO AND V. MART´ IN-MOLINA where R1,...,R6are the tensors (1.2) R1(X, Y )Z=g(Y, Z)X−g(X, Z)Y, R2(X, Y )Z=g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ, R3(X, Y )Z=η(X)η(Z)Y−η(Y)η(Z)X+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ. R4(X, Y )Z=g(Y, Z)hX −g(X, Z)hY +g(hY, Z)X−g(hX, Z)Y, R5(X, Y )Z=g(hY, Z)hX −g(hX, Z)hY +g(φhX, Z)φhY −g(φhY, Z)φhX, R6(X, Y )Z=η(X)η(Z)hY −η(Y)η(Z)hX +g(hX, Z)η(Y)ξ−g(hY, Z)η(X)ξ, for every vector fields X, Y, Z on M, where Fis the constant φ-sectional curvature. This result led the authors (jointly with M. M. Tripathi) to define in [6] a generalized (κ, µ)-space form as an almost contact metric manifold whose curvature tensor can be written as (1.3) R=f1R1+f2R2+f3R3+f4R4+f5R5+f6R6, where f1,...,f6are functions on Mand R1,...,R6the previously defined tensors. They denoted them by M(f1,...,f6) and shortened their name to g.(κ, µ)-s.f. In that same paper, they also studied these spaces with contact metric structure in every dimension, giving examples of them for all cases. In a later work, [5], they continued studying them when they have contact metric structure and showing what happens when they are Da-homothetically deformed, which preserves their structure in dimension 3, although with different functions f1,...,f6. In dimension greater than or equal to 5, a small change in the definition was needed, which meant the introduction of generalized (κ, µ)-spaces with divided R5, denoted by M(f1,...,f5,1, f5,2, f6), as almost contact metric manifolds with curvature tensor of the form R=f1R1+f2R2+f3R3+f4R4+f5,1R5,1+f5,2R5,2+f6R6, with f1,...,f6functions on Mand R5,1, R5,2the tensors R5,1(X, Y )Z=g(hY, Z)hX −g(hX, Z)hY,(1.4) R5,2(X, Y )Z=g(φhY, Z)φhX −g(φhX, Z)φhY.(1.5) It is obvious that R5=R5,1−R5,2, so the generalized (κ, µ)-spaces with divided R5include the generalized (κ, µ)-spaces. If we apply a Da-homothetic deformation to a contact metric generalized (κ, µ)-space with divided R5of any dimension, we obtain another one (with different functions), which provides us with infinitely many examples. Going beyond generalized (κ, µ)-spaces, T. Koufogiorgos, M. Markellos and V. J. Papantoniou introduced in [18] the notion of (κ, µ, ν)-contact metric manifold, where now the equation to be satisfied is R(X, Y )ξ=κ{η(Y)X−η(X)Y}+µ{η(Y)hX −η(X)hY } +ν{η(Y)φhX −η(X)φhY }, (1.6) for some smooth functions κ, µ, and νon M. They proved that this type of manifold is intrinsically related to the harmonicity of the Reeb vector on contact metric 3-manifolds. In dimension greater than or equal to 5, these manifolds must be (κ, µ)-spaces but in dimension 3 there are examples with non-zero νand non-constant κor µ. Some other authors have ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 3 also studied manifolds satisfying condition (1.6), but with a non-contact metric structure. Such is the case of P. Dacko and Z. Olszak, who in [8] defined an almost cosymplectic (κ, µ, ν)-space as an almost cosymplectic manifold that satisfies (1.6), but with κ, µ, ν functions varying exclusively in the direction of ξ. Later, they gave in [9] examples of this type of manifolds. G. Dileo and A. M. Pastore analysed in [15] the (κ, µ)-spaces and the (κ, 0,−µ)-spaces with almost Kenmotsu structure. Lastly, H. ¨ Ozt¨urk, N. Aktan and C. Murathan studied in [22] the almost α-cosymplectic (κ, µ, ν)-spaces under different conditions (like η-parallelism) and gave an interesting example in dimension 3. Some of their results are used in this paper but we later employ a different approach to the study of (κ, µ, ν)-spaces, concentrating on the writing of the curvature tensor and its relation to generalized (κ, µ, ν)-space forms. Therefore, from now on we will denote by (κ, µ)-space or (κ, µ, ν)-space an almost contact metric manifold satisfying equations (1.1) or (1.6), respectively, and we will specify its structure explicitly. Recently, the authors (jointly with K. Arslan and C. Murathan) proved in [2] that the curvature tensor of a (κ, µ, ν)-contact metric manifold of dimension 3 is not unique and can be written, among others, as R=FR1+ (F−κ)R3+µR4+νR7=F R1+ (F−κ)R3+µR4−νR8, where R7and R8are the tensors R7=g(Y, Z)φhX −g(X, Z)φhY +g(φhY, Z)X−g(φhX, Z), Y,(1.7) R8=η(X)η(Z)φhY −η(Y)η(Z)φhX +g(φhX, Zη(Y)ξ−g(φhY, Z)η(X)ξ.(1.8) It is important to note that R7=−R8in dimension 3 but not in general. This led to the introduction in the same paper of the generalized (κ, µ, ν)-space forms as those almost contact metric manifolds whose curvature tensor can be written as (1.9) R=f1R1+f2R2+f3R3+f4R4+f5R5+f6R6+f7R7+f8R8, where R1,...,R8are the tensors previously seen in (1.2), (1.7) and (1.8). They shortened their name to g.(κ, µ, ν)-s.f. and denoted them by M(f1,...,f8). They also studied these spaces with contact metric structure, giving examples or proving their non-existence in every dimension. Despite their technical appearance, there are good reasons for studying contact metric (κ, µ, ν)-spaces and, therefore, generalized (κ, µ, ν)-space forms. The first is that the condition (1.6) remains invariant under D-homothetic deformations, although the values κ,µand νmay change. Moreover, these manifolds provide non-trivial examples of some remarkable classes of contact Riemannian manifolds, like CR-integrable contact metric manifolds, H-contact manifolds and harmonic contact metric manifolds. It is worth noting that there are non-trivial examples of such Riemannian manifolds, the most important being the tangent sphere bundle of any Riemannian manifold of constant sectional curvature with its standard contact metric structure. Finally, in some cases, the formula (1.6) determines the curvature tensor field completely, which will be written in terms of some of the tensors R1,...,R8. This paper is organised in two additional sections. In the first one we present some background which is necessary in order to follow this work. In the second one we study the (κ, µ, ν)-spaces with almost cosymplectic and almost Kenmotsu structures, giving explicitly the writing of their curvature tensors. This will lead to 4 A. CARRIAZO AND V. MART´ IN-MOLINA the definition of g.(κ, µ, ν)-s.f. with divided R5, of which we will provide examples or obstruction results in all possible cases. 2. Preliminaries In this section, we recall some general definitions and basic formulas which will be used later. For more background on almost contact metric manifolds, we recommend the reference [3]. An odd-dimensional Riemann manifold (M, g) is said to be an almost contact metric manifold if there exist on Ma (1,1)-tensor field φ, a vector field ξ(called the structure vector field) and a 1-form ηsuch that η(ξ) = 1, φ2X=−X+η(X)ξand g(φX, φY ) = g(X, Y )−η(X)η(Y) for any vector fields X, Y on M. In particular, in an almost contact metric manifold we also have φξ = 0 and η◦φ= 0. Such a manifold is said to be a contact metric manifold if dη= Φ, where Φ(X, Y ) = g(X, φY ) is the fundamental 2-form of M. On the other hand, the almost contact metric structure of Mis said to be normal if the Nijenhuis torsion [φ, φ] of φequals −2dη⊗ξ. A normal contact metric manifold is called a Sasakian manifold. It can be proved that an almost contact metric manifold is Sasakian if and only if (2.1) (∇Xφ)Y=g(X, Y )ξ−η(Y)X for any vector fields X, Y on M. Moreover, for a Sasakian manifold the following equation holds: R(X, Y )ξ=η(Y)X−η(X)Y. Given an almost contact metric manifold (M, φ, ξ, η, g), a φ-section of Mat p∈Mis a section Π ⊆TpMspanned by a unit vector Xporthogonal to ξp, and φpXp. The φ-sectional curvature of Π is defined by K(X, φX) = R(X, φX, φX, X). A Sasakian manifold with constant φ-sectional curvature cis called a Sasakian space form. In such a case, its Riemann curvature tensor is given by equation R=f1R1+f2R2+f3R3with functions f1= (c+ 3)/4, f2=f3= (c−1)/4 and R1, R2and R3the tensors defined in (1.2). It is well known that on a contact metric manifold (M, φ, ξ, η, g), the tensor h, defined by 2h=Lξφ, satisfies the following relations [4]: (2.2) hξ = 0,∇Xξ=−φX −φhX, hφ =−φh, trh= 0, η ◦h= 0. Therefore, it follows that a contact metric manifold is K-contact if and only if h= 0. An almost contact metric manifold is said to be almost cosymplectic if dη = 0 and dΦ = 0. A normal almost cosymplectic manifold is cosymplectic. We will say that an almost contact metric manifold is almost Kenmotsu if dη = 0 and dΦ = 2η∧Φ. A normal almost Kenmotsu manifold is Kenmotsu. In [16], T. W. Kim and H. K. Pak defined the notion of almost α-cosymplectic manifold as such an almost contact metric manifold satisfying dη = 0 and dΦ = 2αη ∧Φ. These manifolds include trivially the almost cosymplectic (α= 0) and almost Kenmotsu ones (α= 1). A normal almost α-cosymplectic manifold is α-cosymplectic. Similar formulas to the ones we had in the contact metric case also hold on αcosymplectic manifolds, where we know that his a symmetric operator satisfying that [16]: (2.3) hξ = 0,∇Xξ=−αφX −φhX, hφ =−φh, trh = 0. ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 5 These results have also been proved when α= 0 (almost cosymplectic manifolds) in [10] and when α= 1 (almost Kenmotsu manifolds) in [14]. 3. Almost cosymplectic or almost Kenmotsu generalized (κ, µ, ν)-space forms In this section we will study how (κ, µ, ν)-spaces behave when they have almost cosymplectic or almost Kenmotsu structures. Firstly, we will present some results from [22] which are true for α-cosymplectic manifolds and therefore on almost cosymplectic (when α= 0) or almost Kenmotsu ones (when α= 1). Later, we will study both structures separately as we will use different approaches in order to obtain the writing of the curvature tensor of a (κ, µ, ν)-space. Proposition 3.1 ([22]).Given Man almost α-cosymplectic (κ, µ, ν)-space, then (3.1) h2= (κ+α2)φ2, hence κ≤ −α2and κ=−α2if and only if h= 0. Moreover, the next formulas are satisfied (3.2) ξ(κ) = 2(κ+α2)(ν−2α), (3.3) R(ξ, X)Y=κ(g(X, Y )ξ−η(Y)X) + µ(g(hX, Y )ξ−η(Y)hX) +ν(g(φhX, Y )ξ−η(Y)φhX), (3.4) (∇Yφh)X−(∇Xφh)Y= (κ+α2)(η(Y)X−η(X)Y) +µ(η(Y)hX −η(X)hY ) + (ν−α)(η(Y)φhX −η(X)φhY ), (∇Xφ)Y=g(αφX +hX, Y )ξ−η(Y)(αφX +hX),(3.5) (∇Yh)X−(∇Xh)Y= (κ+α2)(η(X)φY −η(Y)φX + 2g(X, φY )ξ), (3.6) +µ(η(X)φhY −η(Y)φhX) + (α−ν)(η(X)hY −η(Y)hX), for any X, Y vector fields on M. Theorem 3.2 ([22]).On an almost α-cosymplectic (κ, µ, ν)-space of dimension greater than or equal to 5, the functions κ,µand νonly vary in the direction of ξ, i.e. X(κ) = X(µ) = X(ν) = 0 for every vector field Xorthogonal to ξ. We will now focus on the almost cosymplectic (κ, µ, ν)-spaces, which have already been studied by other authors in some particular cases. For µ=ν= 0, P. Dacko published [7], where he proved that κmust be constant, and H. Endo presented multiple results in [10] and [11]. This last author also examined in [12] and [13] the (κ, µ, ν)-spaces with constant κ,µand ν= 0. Later, P. Dacko and Z. Olszak studied in [8] and [9] the almost cosymplectic (κ, µ, ν)-spaces with κ, µ and νfunctions that only vary on the direction of the vector field ξ. We now present a result that is valid for any functions κ, µ and ν. Proposition 3.3. Let M2n+1 be an almost cosymplectic (κ, µ, ν)-space. Then ∇ξφh =µh +νφh,(3.7) for every X, Y differentiable vector fields on M. If κ= 0, then h= 0 and Mis a cosymplectic manifold if its dimension is 3. 6 A. CARRIAZO AND V. MART´ IN-MOLINA If κ < 0, the eigenvalues of hare 0(with multiplicity 1) and ±λ=±√−κ (each one with multiplicity n). Moreover, µ=−2g(∇ξX, φX)holds for every X eigenvector of hassociated to the eigenvalue λor −λ. Proof. If κ= 0, then h= 0 and, by virtue of (2.3), we obtain that ∇ξ= 0. When Mis of dimension 3, it is enough to apply Corollary 5.6. of [20], which says that an almost contact metric manifold of dimension 3 is cosymplectic if and only if ∇ξ= 0. Choosing Y=ξin (3.4) (with α= 0), we deduce that (∇ξφh)X−(∇Xφh)ξ=−κφ2X+µhX +νφhX. On the other hand, using (2.3) and (3.1), then (∇Xφh)ξ=∇Xφhξ −φh∇Xξ=φhφhX =−φ2h2X=h2X=κφ2X, and substituting in the last equation we obtain (3.7). If κ < 0, by the properties of φand of hwe know that the eigenvalues of hare 0 (with multiplicity 1) and ±λ=±√−κ6= 0 (each one with multiplicity n). Let us take a unit vector field X, eigenvector of hassociated to the eigenvalue λ=√−κ (denoted by X∈D(λ)), which is orthogonal to ξand satisfies by (3.7) that (3.8) (∇ξφh)X=µhX +νφhX =λµX +λνφX. Taking the inner product of (3.8) with φX gives λν =ξ(λ) + λg(∇ξφX, φX)−g(φh∇ξX, φX) = ξ(λ) = −1 2λξ(κ), from where (3.2) follows with α= 0. Taking now the product of (3.8) with Xgives λµ =λg(∇ξφX, X)−g(∇ξX, φhX) = −2λg(∇ξX, φX). Moreover, we know by hypothesis that λ6= 0, so µ=−2g(∇ξX, φX) for every unit X∈D(λ). If we take Xa unit eigenvector of hassociated to the eigenvalue −λ, we get again the same two equations.  By Theorem 3.2, if an almost cosymplectic (κ, µ, ν)-space is of dimension greater than or equal to 5, the functions κ,µand νonly vary in the direction of ξ, hence we can use the results of [8] and [9], some of which will be summarised below. The case of dimension 3 will be studied apart later. Proposition 3.4 ([8]).Let M2n+1 be an almost cosymplectic (κ, µ, ν)-space, where κ, µ, ν only vary in the direction of ξ. If κ= 0 on some point of M, then κis the identically zero function on Mand h= 0, so R(X, Y )ξ= 0 for every X, Y on M. Moreover, Mis locally the product of an open interval and an almost K¨ahler manifold. If κ < 0, then the eigenvalues of hare 0(with multiplicity 1) and ±√−κ(each one with multiplicity n). In particular, κis constant if and only if ν= 0. We will consider a D-homothetic deformation of the almost contact metric structure (φ, ξ, η, g) defined as [8] (3.9) φ=φ, ξ =1 βξ, η=βη, g =αg + (β2−α)η⊗η, ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 7 where αis a positive constant and βa function that only varies in the direction of ξand is not zero on any point of the manifold. Proposition 3.5 ([8]).If (M, φ, ξ, η, g)is an almost cosymplectic manifold, the tensor hand the Levi-Civita connection ∇of the deformed manifold are related to the original ones the following way: h=1 βh,(3.10) ∇XY=∇XY−β2−α β2g(φhX, Y )ξ+ξ(β) βη(X)η(Y)ξ,(3.11) for all X, Y vector fields on M. Hence the almost cosymplectic (κ, µ, ν)-spaces with κ, µ, ν varying only in the direction of ξare deformed in almost cosymplectic (κ, µ, ν)-spaces with κ=κ β2,µ=µ β,ν=νβ −ξ(β) β2, where κ, µ, ν only vary in the direction of ξ. Therefore, every almost cosymplectic (κ, µ, ν)-spaces with κ < 0can be deformed in an almost cosymplectic (−1,µ, 0)-space with µ=µ/√−κ. We will now study the curvature tensor of an almost cosymplectic (κ, µ, ν)-space. If κ= 0, then we know its local structure by virtue of Proposition 3.4. If κ < 0, then it follows from Proposition 3.5 that we can obtain the writing of its curvature tensor by studying the form of an almost cosymplectic (κ, µ)-space. Using formula (3.2), we know that these latter spaces satisfy ξ(κ) = 0, so in dimensions greater than or equal to 5, κwould be constant and µwould only vary in the direction of ξ. This implies that we can also use [13] because, although in that article Endo focuses in almost cosymplectic (κ, µ)-spaces with κ, µ ∈R, a review of the proofs reveals that they are also true if µis not constant but only varies in the direction of ξ. Hence Theorem 3.1 of [13] would look like this in our case: Theorem 3.6. If Mis an almost cosymplectic (κ, µ)-space with κ < 0, where κ, µ only vary in the direction of ξ, then R(Xλ, Yλ)Z−λ=κ{g(φYλ, Z−λ)φXλ−g(φXλ, Z−λ)φYλ}, R(X−λ, Y−λ)Zλ=κ{g(φY−λ, Zλ)φX−λ−g(φX−λ, Zλ)φY−λ}, R(Xλ, Y−λ)Z−λ=−κg(Xλ, φZ−λ)φY−λ, R(Xλ, Y−λ)Zλ=−κg(Zλ, φY−λ)φXλ, R(Xλ, Yλ)Zλ= 0, R(X−λ, Y−λ)Z−λ= 0, where X±λ, Y±λ, Z±λare eigenvectors of hassociated to the eigenvalues ±λ= ±√−κ. Using the previous theorem and formula (3.3), we will give explicitly the form of the curvature tensor of an almost cosymplectic (κ, µ)-space with κ < 0. Theorem 3.7. Let Mbe an almost cosymplectic (κ, µ)-space of dimension greater than or equal to 5with κ < 0. Then its Riemann curvature tensor can be written as R=−κR3−R5,2−µR6, 8 A. CARRIAZO AND V. MART´ IN-MOLINA where R3, R6are the tensors defined in (1.2) and R5,2is the one in (1.5). Therefore, Mis a g.(κ, µ)-s.f. with divided R5M(f1,...,f5,1, f5,2, f6)with functions f1=f2= 0, f3=−κ, f4=f5,1= 0, f5,2=−1and f6=−µ, . Proof. As κ < 0, we know by Proposition 3.4 that T M =D(λ)⊕D(−λ)⊕< ξ >, where λ=√−κ > 0 . Given a vector field Xon M, we can write X=Xλ+X−λ+ η(X)ξ, where X±λis an eigenvector of hassociated to the eigenvalue ±λ. Then, by the properties of Rwe obtain that R(X, Y )Z=R(Xλ+X−λ, Yλ+Y−λ)(Zλ+Z−λ) + η(X)R(ξ, Y )Z +η(Y)R(X, ξ)Z+η(Z)R(Xλ+X−λ, Yλ+Y−λ)ξ, from which, using (1.1), we get R(X, Y )Z=R(Xλ+X−λ, Yλ+Y−λ)(Zλ+Z−λ) +η(X)R(ξ, Y )Z+η(Y)R(X, ξ)Z. (3.12) It follows from equation (3.3) (with ν= 0) and the definition of the tensors R1,...,R6that (3.13) η(X)R(ξ, Y )Z+η(Y)R(X, ξ)Z=−κR3(X, Y )Z−µR6(X, Y )Z. By Theorem 3.6, we obtain that R(Xλ+X−λ, Yλ+Y−λ)(Zλ+Z−λ) = κ{(g(Xλ, φZ−λ)−g(X−λ, φZλ)(φYλ−φY−λ) −(g(Yλ, φZ−λ)−g(Y−λ, φZλ))(φXλ−φX−λ)}. From the decomposition X=Xλ+X−λ+η(X)ξ, it can be deduced that Xλ= 1 2X−η(X)ξ+1 λhXand that X−λ=1 2X−η(X)ξ−1 λhX, hence (3.14) R(Xλ+X−λ, Yλ+Y−λ)(Zλ+Z−λ) = =κ λ2(−g(φhX, Z)φhY +g(φhY, Z)φhX) = −R5,2(X, Y )Z. Substituting (3.13) and (3.14) in (3.12), we conclude that R(X, Y )Z=−κR3(X, Y )Z−µR6(X, Y )Z−R5,2(X, Y )Z, for all X, Y, Z vector fields on M. Remark 3.8. By the previous theorem, every almost cosymplectic (κ, 0)-space with constant κ < 0has curvature tensor R=−κR3−R5,2,which coincides with Lemma 5 from [7]. Example 3.9. P. Dacko and Z. Olszak gave in [9] models of examples of almost cosymplectic (−1, µ, 0)-spaces, which they denoted by N(µ). By virtue of Theorem 3.7, these spaces are g.(κ, µ)-s.f.’s with divided R5M(f1,...,f5,1, f5,2, f6)with functions f1=f2= 0, f3= 1, f4=f5,1= 0, f5,2=−1and f6=−µ. We will use now the D-homothetic deformations given by (3.9) in order to obtain from the previous theorem the curvature tensor of an almost cosymplectic (κ, µ, ν)- space of dimension greater than or equal to 5 and κ < 0. ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 9 Corollary 3.10. If Mis an almost cosymplectic (κ, µ, ν)-space of dimension greater than or equal to 5and κ < 0, then its Riemannian curvature tensor can be written as R=−κR3−R5,2−µR6−νR8. Proof. If we decompose every vector field on Mas X=e X+η(X)ξ, where e Xis a vector field orthogonal to ξ, we obtain: R(X, Y )Z=R(e X, e Y)e Z+η(X)R(ξ, Y )Z+η(Y)R(X, ξ)Z. Using formula (3.3) and the definition of the tensors R1,...,R8, it follows from a direct computation that η(X)R(ξ, Y )Z+η(Y)R(X, ξ)Z=−κR3(X, Y )Z−µR6(X, Y )Z+νR8(X, Y )Z, which substituted in the previous equation gives (3.15) R(X, Y )Z=R(e X, e Y)e Z−κR3(X, Y )Z−µR6(X, Y )Z+νR8(X, Y )Z. We do not know, in general, R(e X, e Y)e Zon a (κ, µ, ν)-space, but if we use the D-homothetic deformations (3.9) with α= 1 and β=√−κ, we obtain that the deformed manifold is a (−1, µ)-space, with µ=µ/√−κ. This is thanks to Proposition 3.5, which can be applied because the functions κ, µ, ν only vary in the direction of ξ(Theorem 3.2). Given a vector field e X, orthogonal to ξwith respect to g, then it is also orthogonal to ξwith respect to g. Therefore, the next formula follows from Theorem 3.7 and the fact that hξ = 0: R(e X, e Y)e Z=R3(e X, e Y)e Z−R5,2(e X, e Y)e Z−µR6(e X, e Y)e Z=−R5,2(X, Y )Z, for every vector fields X, Y, Z on M. Moreover, if we use (3.9) and (3.10), we obtain that R5,2(X, Y )Z=−1/κR5,2(X, Y )Zfor every X, Y, Z, so (3.16) R(e X, e Y)e Z=1 κR5,2(X, Y )Z. It is now enough to see the relation between R(e X, e Y)e Zand R(e X, e Y)e Z. If we substitute α= 1 and β=√−κin the formula (3.11) and use (3.2), we obtain that ∇XY=∇XY−κ+ 1 κg(φhX, Y )ξ+νη(X)η(Y)ξ. By the definition of the Riemannian curvature tensor R(e X, e Y)e Z=∇e X∇e Ye Z−∇e Y∇e Xe Z−∇[ e X, e Y]e Z and the fact that e X(κ) = e Y(κ) = 0 (Theorem 3.2), after some computations we get (3.17) R(e X, e Y)e Z=R(e X, e Y)e Z+κ+ 1 κ(−g(φhe Y , e Z)∇e Xξ+g(φh e X, e Z)∇e Yξ + (g(φhe Y , ∇e Xe Z)−g(φh e X, ∇e Ye Z) + e Y(g(φh e X, e Z)) −e X(g(φhe Y , e Z)) +g(φh[e X, e Y],e Z))ξ). On the other hand, ∇e Xξ=−√−κ∇e Xξ=√−κφh e X=−φh e X, so −g(φhe Y , e Z)∇e Xξ+g(φh e X, e Z)∇e Yξ=g(φhe Y , e Z)φh e X−g(φh e X, e Z)φhe Y=R5,2(e X, e Y)e Z. 16 A. CARRIAZO AND V. MART´ IN-MOLINA so by a direct computation: g(Yλ, Zλ)Xλ−g(Xλ, Zλ)Yλ= =1 4(R1+R3) + 1 λ2R5,2+1 λ(R7+R8)(X, Y )Z, g(Y−λ, Z−λ)X−λ−g(X−λ, Z−λ)Y−λ= =1 4(R1+R3) + 1 λ2R5,2−1 λ(R7+R8)(X, Y )Z, g(Xλ, Zλ)Y−λ−g(Y−λ, Z−λ)Xλ−g(Yλ, Zλ)X−λ+g(X−λ, Z−λ)Yλ= =1 2−(R1+R3) + 1 λ2R5,2(X, Y )Z. Therefore, it follows from λ2=−(κ+ 1) that (3.26) R(Xλ+X−λ, Yλ+Y−λ)(Zλ+Z−λ) = (−R1−R3−R5,2+R7+R8)(X, Y )Z. Substituting (3.25) and (3.26) in (3.24) gives us R=−R1−(κ+ 1)R3−R5,2−µR6+R7−(ν−1)R8, the formula we were looking for.  Remark 3.24. By Theorem 3.2, we can omit in the previous theorem the hypothesis “κis a function that only varies in the direction of ξ” if the dimension is greater than or equal to 5. In dimension 3it is possible to simplify the writing of the curvature tensor, as we will see later in Corollary 3.32. Example 3.25. By the previous theorem, the examples that G. Dileo and A. M. Pastore gave in [15] of almost Kenmotsu (−1−λ2,0,2)-spaces, with λa positive real number, are in particular g.(κ, µ, ν)-s.f.’s with divided R5M(f1,...,f5,1, f5,2,...,f8) with functions f1=−1, f2= 0, f3=λ2>0, f4= 0, f5,1= 0, f5,2=−1, f6= 0, f7= 1, f8=−1. Moreover, as f7, f86= 0 and the writing of the curvature tensor is unique if these examples are of dimension greater than or equal to 5(Theorem 3.14), we know that they cannot be g.(κ, µ)-s.f.’s with divided R5M(f1,...,f5,1, f5,2, f6). The almost Kenmotsu (κ, µ, ν)-spaces of dimension greater than or equal to 5 with κ < −1 cannot be g.(κ, µ, ν)-s.f.’s, i.e. their curvature tensors cannot be written without dividing R5in R5,1and R5,2, as a consequence of Theorems 3.14 and 3.23. Moreover, we can prove a more general result when the dimension is greater than or equal to 5. Proposition 3.26. There are no almost Kenmotsu g.(κ, µ, ν)-s.f.’s M(f1,...,f8) of dimension greater than or equal to 5and κ=f1−f3<−1. Proof. We will prove the result by contradiction. Let us suppose that M(f1,...,f8) is an almost Kenmotsu g.(κ, µ, ν)-s.f. of dimension greater than or equal to 5 with κ < −1. Then Mis also a g.(κ, µ, ν)-s.f. with divided R5with f5,1=f5and f5,2=−f5, so R=f1R1+f2R2+f3R3+f4R4+f5R5,1−f5R5,2+f6R6+f7R7+f8R8. ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 17 On the other hand, Mis a (f1−f3, f4−f6, f7−f8)-space with κ=f1−f3<−1 a function that only varies in the direction of ξ(Proposition 3.12 and Theorem 3.2). Applying Theorem 3.23 we obtain that its curvature tensor can be written as R=−R1−(f1−f3+ 1)R3−R5,2−(f4−f6)R6+R7−(f7−f8−1)R8. By Theorem 3.14, the writing of the curvature tensor is unique, so we have in particular that f5= 0 and f5= 1, which is absurd.  We can also use Theorem 3.14 to determine some relations between the functions of an almost Kenmotsu g.(κ, µ, ν)-s.f. with divided R5M(f1,...,f5,1, f5,2,...,f8) of dimension greater than or equal to 5 with f1−f3<−1, as we did in Theorem 3.17 for the almost cosymplectic structure. Theorem 3.27. Let M(f1,...,f5,1, f5,2,...,f8)be an almost Kenmotsu g.(κ, µ)- s.f. with divided R5. If Mis of dimension greater than or equal to 5and satisfies f1−f3<−1, then Mverifies f1=−1, f2=f4=f5,1= 0, f5,2=−1, f7= 1, f3>0, and f3, f6, f8are functions that only vary in the direction of ξ, i.e. Mis a (−1− f3,−f6,1−f8)-space with f3>0. Proof. By Theorem 3.12, we know that Mis a (κ, µ, ν)-space with κ=f1−f3<−1, µ=f4−f6and ν=f7−f8. We can therefore apply Theorem 3.23, which says that the Riemann curvature tensor can be written as R=−R1−(f1−f3+ 1)R3−R5,2−(f4−f6)R6+R7−(f7−f8−1)R8. By the definition of g.(κ, µ, ν)-s.f. with divided R5and the uniqueness of the writing of the curvature tensor (Theorem 3.14), we obtain that f1=−1, f2=f4=f5,1= 0, f5,2=−1, f7= 1. Therefore, κ=−1−f3,µ=−f6and ν= 1−f8. By hypothesis, f1−f3=−1−f3< −1, so f3>0. The rest of the result is deduced from Theorem 3.2.  Finally, we will see what happens to an almost cosymplectic or almost Kenmotsu (κ, µ, ν)-space of dimension 3. Using formula (3.3), we can prove an analogous result to Theorem 3.1 of [2]. Theorem 3.28. Let M3be an almost cosymplectic (resp. almost Kenmotsu) (κ, µ, ν)-space with κ < 0(resp. κ < −1). Then its curvature tensor can be written as R=τ 2−2κR1+τ 2−3κR3+µR4+νR7, where τis the scalar curvature of Mand the tensors R1, R3, R4, R7are the ones that appear in (1.2) and (1.7). Proof. Firstly, we recall a formula that is valid for every Riemannian manifold of dimension 3: (3.27) R(X, Y )Z=g(Y, Z)QX −g(X, Z)QY +g(QY, Z)X−g(QX, Z)Y −τ 2(g(Y, Z)X−g(X, Z)Y), where Qis the Ricci operator and τ=trQ. 18 A. CARRIAZO AND V. MART´ IN-MOLINA We now take a φ-basis {E, φE, ξ}such that hE =λE, where λ=√−κ(resp. λ=√−1−κ), which is possible thanks to Proposition 3.3 (resp. Proposition 3.18). Using formula (3.3) and this basis, we can compute Qξ =R(ξ, E)E+R(ξ, φE)φE +R(ξ, ξ)ξ= (κ+λµ)ξ+ (κ−λµ)ξ= 2κξ. Making Y=Z=ξin (3.27) and using that Qξ = 2κξ and g(QX, Y ) = g(QY, X), we obtain: R(X, ξ)ξ=2κ−τ 2X+τ 2−4κη(X)ξ+QX. Using again formula (3.3), it follows that QX =R(X, ξ)ξ−2κ−τ 2X−τ 2−4κη(X)ξ =τ 2−κX+3κ−τ 2η(X)ξ+µhX +νφhX. Substituting this expression of QX in (3.27) and applying the definitions of R1, R3, R4and R7, we obtain the equation we were looking for.  We will now see a result that was proved for contact metric (κ, µ, ν)-spaces with κ < 1 in [2]: Proposition 3.29. Let M3be an almost cosymplectic (resp. almost Kenmotsu) (κ, µ, ν)-space with κ < 0(resp. κ < −1). Then its φ-sectional curvature is F= τ 2−2κ. Proof. Since the manifold is of dimension 3, the φ-sectional curvature does not depend on the choice of the φ-section. Hence we can take F=R(E, φE, φE, E), where Eis an eigenvector of hof eigenvalue λ > 0 thanks to equation (3.1) and the fact that κ < 0 (resp. equation (3.1) and κ < −1). It follows from Proposition 3.28 that F=R(E, φE, φE, E) = τ 2−2κR1(E, φE, φE, E) + τ 2−3κR3(E, φE, φE, E) +µR4(E, φE, φE, E) + νR7(E, φE, φE, E). A straightforward computation using the definition of the tensors R1,...,R7 gives us the formula we wanted.  Therefore, Theorem 3.28 can be rewritten as: Corollary 3.30. Let M3an almost cosymplectic (resp. almost Kenmotsu) (κ, µ, ν)- space with κ < 0(resp. κ < −1). Then its curvature tensor has the form R=FR1+ (F−κ)R3+µR4+νR7, where Fis the φ-sectional curvature and R1, R3, R4, R7are the tensors defined in (1.2) and (1.7). In particular, Mis a g.(κ, µ, ν)-s.f. M(F, 0, F −κ, µ, 0,0, ν, 0). Remark 3.31. It is worth noting that the expression of the curvature tensor given in the previous corollary coincides with the one obtained in Corollary 3.3 of [2] for contact metric (κ, µ, ν)-spaces of dimension 3. This is because the tensor hsatisfies the same properties in the three structures (almost cosymplectic, almost Kenmotsu and contact metric): it is symmetric and anticommutes with φ, by equations (2.2) and (2.3). ALMOST COSYMPLECTIC AND ALMOST KENMOTSU (κ, µ, ν)-SPACES 19 Under some extra hypotheses, we can write Fin terms of κ, so the curvature tensor would be completely determined by the functions κ, µ and ν. Corollary 3.32. Let M3be an almost cosymplectic (κ, µ, ν)-space. If κ < 0, µ, ν only vary in the direction of ξ, then its curvature tensor can be written as R=−κR1−2κR3+µR4+νR7, where R1, R2, R4, R7are the tensors defined in (1.2) and (1.7). Let M3be an almost Kenmotsu (κ, µ, ν)-space. If κ < −1is a function that only varies in the direction of ξ, then its curvature tensor can be written as R=−(κ+ 2)R1−2(κ+ 1)R3+µR4+νR7. Proof. Since the manifold is of dimension 3, it follows from Proposition 3.3 that when Mis almost cosymplectic we can take F=R(E, φE, φE, E), where Eis an eigenvector of hassociated to the eigenvalue λ=√−κ > 0. We do not know in general the φ-sectional curvature of a (κ, µ, ν)-space, but we can use a D-homothetic deformation (3.9) with α= 1 and β=√−κto obtain a (−1,µ)-space with µ=µ/√−κ. Reasoning analogously to the proof of Corollary 3.10, we get that, for every X, Y, Z vector fields on M R(e X, e Y)e Z=R(e X, e Y)e Z+κ+ 1 κR5,2(X, Y )Z, where X=e X+η(X)ξand e Xis orthogonal to ξ. Therefore, we have in particular that (3.28) F=R(E, φE, φE, E) = R(E, φE, φE, E)−κ+ 1 κR5,2(E, φE, φE, E). On the other hand, since Eis an unit vector field with respect to gand orthogonal to ξ, by (3.9) it is also unit with respect to gand orthogonal to ξ, so it follows from Theorem 3.6 that R(E, φE, φE, E) = g(E, φ2E)g(φ2E, E) = 1. By the definition of the tensor R5,2and the properties of h(formulas (1.5) and (2.3)): R5,2(E, φE, φE, E) = g(φhE, E)2+g(hE, E)2=λ2=−κ. Substituting the last two equations in (3.28) we obtain that F=−κ, which jointly with Corollary 3.30 gives the result we were looking for. If Mis almost Kenmotsu, then we can take Ean eigenvector of φh associated to the eigenvalue λ=√−1−λ6= 0 by virtue of Proposition 3.18. We know by Theorem 3.22 that F=−(κ+ 2)g(φE, φE)g(E, E) = −(κ+ 2) and we conclude that R=−(κ+ 2)R1−2(κ+ 1)R3+µR4+νR7. Remark 3.33. The expressions of the curvature tensor given in Corollary 3.10 and Theorem 3.23 coincide with that of the previous corollary in dimension 3. This is true because the following equations hold in dimension 3when the structure is almost cosymplectic or almost Kenmotsu: (3.29) R2= 3(R1+R3), R6=−R4, R8=−R7. 20 A. CARRIAZO AND V. MART´ IN-MOLINA If the manifold is an almost cosymplectic (κ, µ, ν)-space, it can also be proved that R5,2=κ(R1+R3), so the curvature tensor can be written as R=−κR3−R5,2−µR6−νR8=−κR1−2κR3+µR4+νR7. If the manifold is an almost Kenmotsu (κ, µ, ν)-space, it is also true that R5,2= (κ+ 1)(R1+R3), so the curvature tensor can be written as R=−R1−(κ+ 1)R3−R5,2−µR6+R7−(ν−1)R8 =−(κ+ 2)R1−2(κ+ 1)R3+µR4+νR7. Example 3.34. By the previous corollary, the examples of almost cosymplectic (−1, µ, 0)-spaces (with µvarying only in the direction of ξ) that appear in [9] are in dimension 3examples of g.(κ, µ)-s.f.’s M3(f1,...,f6)with functions f1= 1, f2= 0, f3= 2, f4=µ, f5=f6= 0. Analogously, the examples of almost Kenmotsu (−1−λ2,0,2)-spaces (with λa positive real number) that appear in [15] are examples of g.(κ, µ, ν)-s.f.’s M3(f1,...,f8) with functions: f1=−1 + λ2, f2= 0, f3= 2λ2, f4=f5=f6= 0, f7= 2, f8= 0. Finally, it remains to be seen what happens if a 3-dimensional manifold is almost cosymplectic and satisfies κ= 0 or almost Kenmotsu and satisfies κ=−1. In order to study the curvature tensor, we recall the next result: Proposition 3.35 ([21]).If M3is a trans-Sasakian manifold (0, β), then its curvature tensor can be written as R=τ 2+ 2β2+ 2β′R1+τ 2+ 3β2+ 3β′R3, where τis the scalar curvature of the manifold. If M3is an almost cosymplectic (κ, µ, ν)-space with κ= 0, then h= 0 and Mis cosymplectic by Proposition 3.3. By Proposition 3.35 (with β= 0), we have that the curvature tensor of the manifold can be written as R=τ 2R1+τ 2R3,which coincides with the result obtained in Theorem 3.28. 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Bull. 43 (2000), no. 4, 400–447. [20] Z. Olszak. On almost cosymplectic manifolds. Kodai Math. J. 4(1981), 239–250. [21] Z. Olszak and R. Rosca. Normal locally conformal almost cosymplectic manifolds. Publ. Math. Debrecen 39 (1991), no. 3-4, 315–323. [22] H. ¨ Ozt¨urk, N. Aktan and C. Murathan. Almost α-cosymplectic (κ, µ, ν)-spaces. arXiv:1007.0527v1. (Alfonso Carriazo and Ver´onica Mart´ın-Molina) Department of Geometry and Topology, Faculty of Mathematics, University of Sevilla, Aptdo. de Correos 1160, 41080 – Sevilla, SPAIN E-mail address, Alfonso Carriazo: carri[email protected] E-mail address, Ver´onica Mart´ın-Molina: veronicam[email protected]s