Sasaki-Einstein and paraSasaki-Einstein metrics from (κ,μ)-structures

arXiv:1109.6248v3 [math.DG] 16 Jun 2013 SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES BENIAMINO CAPPELLETTI-MONTANO, ALFONSO CARRIAZO, AND VER ´ ONICA MART´ IN-MOLINA Abstract. We prove that every contact metric (κ, µ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κand µfor which such metrics are Sasaki-Einstein and paraSasaki- Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (κ, µ)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of (κ, µ)-spaces related to the existence of Eistein- Weyl and Lorentzian Sasaki-Einstein structures. 1. Introduction It is well known that the tangent sphere bundle T1Nof a flat Riemannian manifold Ncarries a contact Riemannian structure such that R(X, Y )ξ= 0 for any vector fields X,Yon T1N, where the Reeb vector field ξis given by twice the geodesic flow. The class of contact metric manifolds satisfying the above condition, which were at first studied by Blair in [4], is not preserved by D-homothetic transformations. In fact, if one deforms D-homothetically the structure, one falls in the larger class of “contact metric (κ, µ)-spaces”, i.e. contact metric manifolds (M, ϕ, ξ, η, g) satisfying (1.1) R(X, Y )ξ=κ(η(Y)X−η(X)Y) + µ(η(Y)hX −η(X)hY ), for some constants κand µ, where 2hdenotes the Lie derivative of the structure tensor ϕin the direction of the Reeb vector field (see §2 for more details). This new class of Riemannian manifolds was introduced in [7] as a natural generalization of both the contact metric manifolds satisfying R(X, Y )ξ= 0 and the Sasakian condition R(X, Y )ξ=η(Y)X−η(X)Y. Despite the technical appearance of the definition, nowadays contact (κ, µ)-spaces are considered an important topic in contact Riemannian geometry because there are good reasons for studying them. The first is that, while the values κand µvary, one proves that the condition (1.1) remains unchanged under D-homothetic deformations. Next, in the non- Sasakian case (that is for κ6= 1), the condition (1.1) determines the curvature tensor field completely. Furthermore, (κ, µ)-spaces provide non-trivial examples of some remarkable classes of contact Riemannian manifolds, like CR-integrable contact metric manifolds ([27]), H-contact manifolds ([25]) and harmonic contact metric manifolds ([29]). Finally, 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. 2010 Mathematics Subject Classification. 53C15, 53C12, 53C25, 53B30, 57R30. Key words and phrases. contact metric manifold, Sasakian, paracontact, paraSasakian, nullity distribution, (κ, µ)-spaces, Einstein, η-Einstein, Legendre foliation, Weyl structure, Lorentzian Sasakian, tangent sphere bundle. This paper was started during a research visit of B.C.M. at the Department of Geometry and Topology of the University of Sevilla under a grant of the Institute of Mathematics of the University of Sevilla (IMUS). B.C.-M. acknowledges the support of IMUS and expresses his gratitude for the warm hospitality and good working conditions received at the Department of Geometry and Topology. The research of A.C. and V.M.-M. is partially supported by the PAI group FQM-327 (Junta de Andaluc´ıa, Spain) and by the MTM2011-22621 grant of the MEC, Spain. 1 2 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA In this paper we study the relations between the theory of (κ, µ)-spaces and two other important topics of contact geometry: Sasakian and paraSasakian manifolds. In fact, given a non-Sasakian (κ, µ)-space (M, ϕ, ξ, η, g), we describe a method for constructing a Sasakian or paraSasakian metric on Mcompatible with the same contact form η. The type of metric (Sasakian or paraSasakian) depends on the value of a well-known invariant introduced by Boeckx in [8] for classifying (κ, µ)-spaces, defined as (1.2) IM=1−µ 2 √1−κ. More precisely, we are able to define a Sasakian or paraSasakian metric if |IM|>1 or |IM|<1, respectively. Moreover, by using the aforementioned property that the (κ, µ)-nullity condition (1.1) determines the curvature completely, we find an explicit expression for the curvature tensor field of the above Sasakian and paraSasakian metrics. We obtain from it our main result, that such metrics are always η-Einstein and that for some values of κand µthey are Sasaki-Einstein and paraSasaki-Einstein, though the starting (κ, µ)-structure can never be Einstein in dimension greater than 3 ([6, p. 131]). Furthermore, we prove that in dimension greater than or equal to 5, every (κ, µ)-space such that IM>1 also carries an Einstein-Weyl structure. We then discuss some consequences of such results on the geometry of tangent sphere bundles T1N, which will accept η-Einstein Sasakian and paraSasakian metrics depending on the sign of c, the constant sectional curvature of the space form N. Moreover, these structures will be Sasaki-Einstein and paraSasaki-Einstein for certain values of c(which will depend only on n). Thus we extend the result of Tanno that T1S3≃S2×S3carries a Sasaki-Einstein metric and give (to the knowledge of the authors) the first non-trivial examples of η-Einstein (eventually Einstein) paraSasakian manifolds. In fact, while there has been an increasing interest in last years in paraSasakian geometry (see [2], [19], [30]), so far the only known examples of (η-)Einstein paracontact manifolds seem to be the hyperboloid H2n+1 n+1 (1) of constant curvature −1 ([19]) and R3 1with the flat metric ([31]), together with the Boothby-Wang fibrations with base a paraK¨ahler-Einstein manifold. Finally, in the last part of the paper we will give a geometric interpretation to the above canonical Sasakian and paraSasakian metrics. It is well known that any non-Sasakian (κ, µ)-space is foliated by two Legendre foliations, defined by the eigendistributions of the operator h, and that such a foliated structure plays an important role in the theory of (κ, µ)-spaces (cf. [13], [16]). We show that the geometry of these Legendre foliations, encoded by some invariants like the Pang invariant ([24]) and the Libermann map ([21]), is fully described by the above Sasakian and paraSasakian metrics. In this way we are able to find a sufficient condition for a K-contact (respectively, K-paracontact) manifold M, foliated by two mutually orthogonal, totally geodesic Legendre foliations, to admit a contact metric (κ, µ)-structure, compatible with the same underlying contact form, such that |IM|>1 (respectively, |IM|<1). 2. Preliminaries 2.1. Contact metric (κ, µ)-spaces. An almost contact structure on a (2n+ 1)-dimensional smooth manifold Mis a triplet (ϕ, ξ, η), where ϕis a tensor field of type (1,1), ηa 1-form and ξa vector field on Msatisfying the following conditions (2.1) ϕ2=−I+η⊗ξ, η(ξ) = 1, where Iis the identity mapping. From (2.1) it follows that ϕξ = 0, η◦ϕ= 0 and the (1,1)-tensor field ϕhas constant rank 2n([6]). Given an almost contact manifold (M, ϕ, ξ, η) one can define an almost complex structure Jon the product M×Rby setting JX, f d dt =ϕX −fξ, η (X)d dt for any X∈Γ (T M) and f∈C∞(M×R). Then the almost contact manifold is said to be normal if the almost complex structure Jis integrable. The condition for normality is given by the vanishing of the tensor field Nϕ:= [ϕ, ϕ] + 2dη ⊗ξ. Any almost contact manifold (M, ϕ, ξ, η) admits a compatible metric, i.e. a SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 3 Riemannian metric gsatisfying (2.2) g(ϕX, ϕY ) = g(X, Y )−η(X)η(Y) for all X, Y ∈Γ (T M). The manifold Mis said to be an almost contact metric manifold with structure (ϕ, ξ, η, g). From (2.2) it follows immediately that η=g(·, ξ) and g(·, ϕ·) = −g(ϕ·,·). Then one defines the 2-form Φ on Mby Φ (X, Y ) = g(X, ϕY ), called the fundamental 2-form of the almost contact metric manifold. If Φ = dη then ηbecomes a contact form, with ξand D:= ker(η) its corresponding Reeb vector field and contact distribution, respectively. Then (M, ϕ, ξ, η, g) is called a contact metric manifold. In a contact metric manifold one has (2.3) ∇ξ=−ϕ−ϕh where ∇is the Levi-Civita connection of (M, g) and hdenotes the (1,1)-tensor field defined by h:= 1 2Lξϕ. The tensor field his symmetric with respect to gand vanishes identically if and only if the Reeb vector field ξis Killing. In this last case the contact metric manifold is said to be K-contact. A normal contact metric manifold is called a Sasakian manifold. Any Sasakian manifold is K-contact and the converse holds in dimension 3. If the Ricci tensor of a contact metric manifold has the following form (2.4) Ric = ag +bη ⊗η for some functions aand b, we say that Mis η-Einstein. It is known that if Mis Sasakian and dim(M)≥5, then aand bare necessarily constants. This notion appears to be a good generalization of the concept of Einstein metrics in the context of contact Riemannian geometry. Many interesting geometric properties of η-Einstein metrics are presented in the recent paper [10]. Given a positive constant c > 0, a Dchomothetic deformation on a contact metric manifold (M, ϕ, ξ, η, g) is the change of the structure tensors of the following type (2.5) ϕ′:= ϕ, ξ′:= 1 cξ, η′:= cη, g′:= cg +c(c−1)η⊗η. Then (ϕ′, ξ′, η′, g′) is again a contact metric structure on M. A recent generalization of Sasakian manifolds is the notion of contact metric (κ, µ)-space ([7]). Let (M, ϕ, ξ, η, g) be a contact metric manifold. If the curvature tensor field of the Levi-Civita connection satisfies (1.1) for some κ, µ ∈R, we say that (M, ϕ, ξ, η, g) is a contact metric (κ, µ)-space (or that ξbelongs to the (κ, µ)-nullity distribution). This definition was introduced and deeply studied by Blair, Koufogiorgos and Papantoniou in [7]. Among other things, the authors proved the following result. Theorem 2.1 ([7]).Let (M, ϕ, ξ, η, g)be a contact metric (κ, µ)-space. Then (2.6) h2=−(1 −κ)ϕ2, so that necessarily κ≤1. Moreover, if κ= 1 then h= 0 and (M, ϕ, ξ, η, g)is Sasakian. If κ < 1, the contact metric structure is not Sasakian and Madmits three mutually orthogonal integrable distributions Dh(0) = Rξ,Dh(λ)and Dh(−λ)given by the eigenspaces of hcorresponding to the eigenvalues 0,λand −λ, where λ=√1−κ. The same authors also proved the following formulas for the covariant derivatives of the tensor fields ϕ,hand ϕh ([7]): (∇Xϕ)Y=g(X+hX, Y )ξ−η(Y) (X+hX),(2.7) (∇Xh)Y=((1 −κ)g(X, ϕY )−g(X, ϕhY ))ξ+η(Y)h(ϕX +ϕhX)−µη(X)ϕhY,(2.8) (∇Xϕh)Y=(g(X, hY )−(1 −κ)g(X, ϕ2Y))ξ+η(Y)(hX −(1 −κ)ϕ2X) + µη(X)hY.(2.9) Notice that while D-homothetic deformations preserve the state of being contact metric, K-contact, Sasakian or η-Einstein, they destroy conditions like R(X, Y )ξ= 0 or R(X, Y )ξ=k(η(Y)X−η(X)Y). 4 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA However, they preserve the class of contact metric (κ, µ)-structures. Indeed, if (ϕ, ξ, η, g) is a (κ, µ)- structure then the deformed structure (ϕ′, ξ′, η′, g′) is a (κ′, µ′)-structure with (2.10) κ′=κ+c2−1 c2, µ′=µ+ 2c−2 c. In [8] Boeckx provided a local classification of non-Sasakian (κ, µ)-spaces based on the number (1.2), which is an invariant of a contact metric (κ, µ)-structure up to D-homothetic deformations. He proved that two non-Sasakian contact metric (κ, µ)-spaces, denoted by (M1, ϕ1, ξ1, η1, g1) and (M2, ϕ2, ξ2, η2, g2), are locally isometric as contact metric manifolds, up to D-homothetic deformations, if and only if IM1=IM2. A geometric interpretation of the invariant IMand of the Boeckx’s classification was recently given in [13]. The standard example of (κ, µ)-spaces is given by the tangent sphere bundle T1Nof a manifold of constant curvature c6= 1 endowed with its standard contact metric structure. In this case κ=c(2 −c), µ=−2cand IT1N=1+c |1−c|. Other examples are given by certain Lie groups defined by Boeckx in [8]. We conclude the subsection by recalling the following formula for the Lie derivative of the operator h in any non-Sasakian (κ, µ)-space (cf. [17, Lemma 4.5]) (2.11) Lξh= (2 −µ)ϕh + 2(1 −κ)ϕ. 2.2. Paracontact geometry. An almost paracontact structure (cf. [20]) on a (2n+ 1)-dimensional smooth manifold Mis given by a (1,1)-tensor field eϕ, a vector field ξand a 1-form ηsatisfying the following conditions (i) η(ξ) = 1, eϕ2=I−η⊗ξ, (ii) the eigendistributions D+and D−of eϕcorresponding to the eigenvalues 1 and −1, respectively, have equal dimension n. As an immediate consequence of the definition, one has that eϕξ = 0, η◦eϕ= 0 and the field of endomorphisms eϕhas constant rank 2n. As for the almost contact case, one can consider the almost paracomplex structure on M×Rdefined by e JX, f d dt =eϕX +fξ, η(X)d dt , where Xis a vector field on Mand faC∞function on M×R. By definition, if e Jis integrable the almost paracontact structure (eϕ, ξ, η) is said to be normal. The computation of e Jin terms of the tensors of the almost paracontact structure leads us to define a tensor field Neϕof type (1,2) given by Neϕ:= [eϕ, eϕ]−2dη ⊗ξ. The almost paracontact structure is then normal if and only if Neϕvanishes identically (cf. [30]). Normality in paracontact geometry has the following geometric interpretation. Theorem 2.2 ([12]).An almost paracontact manifold (M, eϕ, ξ, η)is normal if and only if the eigendistributions D+and D−of eϕcorresponding to the eigenvalues ±1are integrable and the vector field ξis a foliated vector field with respect to both the foliations, i.e. [ξ, X]∈Γ(D±)for any X∈Γ(D±). If an almost paracontact manifold is endowed with a semi-Riemannian metric egsuch that (2.12) eg(eϕX, eϕY ) = −eg(X, Y ) + η(X)η(Y) for all X, Y ∈Γ(T M), then (M, eϕ, ξ, η, eg) is called an almost paracontact metric manifold. Notice that any such a semi-Riemannian metric is necessarily of signature (n, n + 1) and the above condition (ii) of the definition of almost paracontact structures is automatically satisfied. Moreover, as in the almost contact case, from (2.12) it follows easily that η= ˜g(·, ξ) and eg(·,eϕ·) = −eg(eϕ·,·). Hence one defines the fundamental 2-form of the almost paracontact metric manifold by e Φ(X, Y ) = eg(X, eϕY ). If dη =e Φ, η becomes a contact form and (M, eϕ, ξ, η, eg) is said to be a paracontact metric manifold. On a paracontact metric manifold one defines the tensor field eh:= 1 2Lξeϕ. It was proved in [30] that eh is a symmetric operator with respect to eg, it anti-commutes with eϕand it vanishes identically if and only if ξis a Killing vector field and in such case (M, eϕ, ξ, η, eg) is called a K-paracontact manifold. Moreover SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 5 the identity e ∇ξ=−eϕ+eϕehholds. A paracontact metric manifold is said to be integrable, or para-CR, if the following condition is satisfied, for any X, Y ∈Γ(T M), (e ∇Xeϕ)Y=η(Y)(X−ehX)−eg(X−ehX, Y )ξ. A normal paracontact metric manifold is said to be a paraSasakian manifold. Also in this context the paraSasakian condition implies the K-paracontact condition and the converse holds in dimension 3. In terms of the covariant derivative of eϕthe paraSasakian condition may be expressed by (e ∇Xeϕ)Y=−eg(X, Y )ξ+η(Y)X. Clearly, for K-paracontact manifolds the notion of integrability coincides with that of being paraSasakian. An equivalent definition of paraSasakian manifolds is presented in [2] in terms of pseudo-Riemannian cones. Standard examples of paraSasakian manifolds are the hyperboloid H2n+1 n+1 (1) = (x0, y0,...,xn, yn)∈R2n+2 |x2 0+...+x2 n−y2 0−...−y2 n= 1 and the hyperbolic Heisenberg group H2n+1 =R2n×Rwith the structures defined in [19]. Furthermore, let us recall that a notion of η-Einstein metric and Dc-homothetic deformation can be introduced also in paracontact metric geometry ([30]). The definition is the same as the one given in (2.4) and (2.5) for contact Riemannian manifolds, with the only change that the constant of homothety ccan be now any non-zero real number since the metric does not need to be positive definite. 3. The main results There is a strict relationship between the theory of contact metric (κ, µ)-spaces and of paracontact geometry, as shown in [15] and [17]. In fact, given a non-Sasakian contact metric (κ, µ)-space (M, ϕ, ξ, η, g), one can define canonically two integrable paracontact metric structures on M, (eϕ1, ξ, η, eg1) and (eϕ2, ξ, η, eg2), which are compatible with the same underlying contact form and Reeb vector field as the (κ, µ)-space M. They are defined by eϕ1:= 1 √1−κϕh, eg1:= 1 √1−κg(·, h·) + η⊗η, eϕ2:= 1 √1−κh, eg2:= 1 √1−κg(·, ϕh·) + η⊗η. The curvature tensor fields of such paracontact metric structures, in turn, satisfy a nullity-like condition e Rα(X, Y )ξ=eκα(η(Y)X−η(X)Y) + eµα(η(Y)ehαX−η(X)ehαY), where eκ1=1−µ 2−1, eµ1= 2 1−√1−κand eκ2=κ−2 + 1−µ 22,eµ2= 2. By (2.11) one can prove that eh1=−IMh. Hence, being integrable, the paracontact metric structure (eϕ1, ξ, η, eg1) is paraSasakian if and only if IM= 0. Whereas for no value of κand µ(eϕ2, ξ, η, eg2) is paraSasakian. Now we prove a much stronger result. Theorem 3.1. Let (M, ϕ, ξ, η, g)be a non-Sasakian contact metric (κ, µ)-space such that IM6=±1. (i) If |IM|>1, then Madmits a Sasakian structure, compatible with the contact form η, given by (3.1) ¯ φ:= ǫ1 (1 −κ)p(2 −µ)2−4(1 −κ)Lξh◦h, ¯g:= −dη(·,¯ φ·) + η⊗η, where (3.2) ǫ:= 1if IM>1 −1if IM<−1 6 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA (ii) If |IM|<1, then Madmits a paraSasakian structure, compatible with the contact form η, given by (3.3) e φ:= 1 (1 −κ)p4(1 −κ)−(2 −µ)2Lξh◦h, eg:= dη(·,e φ·) + η⊗η. Proof. By formula (2.11), using (2.6) and the anti-commutativity of ϕand h, one has that (3.4) Lξh◦h= (1 −κ)((2 −µ)ϕ+ 2ϕh). Hence (Lξh◦h)2= (2 −µ)2(1 −κ)2ϕ2−4(1 −κ)2ϕ2h2= (1 −κ)2((2 −µ)2−4(1 −κ))ϕ2. Therefore (Lξh◦h)2=λ4α(−I+η⊗ξ), where we recall that λ=√1−κand we have put α:= (2 −µ)2−4(1 −κ). Consequently, if α > 0 then ¯ φ:= ǫ1 λ2√αLξh◦hdefines an almost contact structure and if α < 0 then e φ:= 1 λ2√−αLξh◦hsatisfies the first of the two conditions defining an almost paracontact structure. Notice that, since κ≤1, α > 0 if and only if |IM|>1. (i) Let us assume that |IM|>1. We firstly show that ¯ φis a normal almost contact structure. Just by using the definition of the tensor field Nφ, one can prove that for any almost contact structure (φ, ξ, η) (same contact form and Reeb vector field as (ϕ, ξ, η, g)) the following identity holds (3.5) Nφ(X, Y ) = (∇φX φ)Y−(∇φY φ)X+ (∇Xφ)φY −(∇Yφ)φX −η(Y)∇Xξ+η(X)∇Yξ, where ∇is the Levi-Civita connection of the Riemannian metric g. By (3.4) we have that ¯ φ=ǫ1 √α((2 −µ)ϕ+ 2ϕh),(3.6) so that by (3.5) we find N¯ φ(X, Y ) =(2 −µ)2 α(∇ϕX ϕ)Y+2(2 −µ) α(∇ϕhX ϕ)Y+2(2 −µ) α(∇ϕX ϕh)Y+4 α(∇ϕhX ϕh)Y −(2 −µ)2 α(∇ϕY ϕ)X−2(2 −µ) α(∇ϕhY ϕ)X−2(2 −µ) α(∇ϕY ϕh)X−4 α(∇ϕhY ϕh)X +(2 −µ)2 α(∇Xϕ)ϕY +2(2 −µ) α(∇Xϕ)ϕhY +2(2 −µ) α(∇Xϕh)ϕY +4 α(∇Xϕh)ϕhY(3.7) −(2 −µ)2 α(∇Yϕ)ϕX −2(2 −µ) α(∇Yϕ)ϕhX −2(2 −µ) α(∇Yϕh)ϕX −4 α(∇Yϕh)ϕhX −η(Y)∇Xξ+η(X)∇Yξ. Then taking (2.3), (2.8) and (2.9) into account, and using (2.6) and the anti-commutativity of ϕand h, after very long computations one can prove that (3.7) reduces to N¯ φ(X, Y ) = 1−(2 −µ)2 α+4λ2(2 −µ) α−4λ2 α+4λ2µ α(η(Y)ϕX −η(X)ϕY ) +1 + (2 −µ)2 α−4(2 −µ) α+4λ2 α+2(2 −µ)µ α(η(Y)ϕhX −η(X)ϕhY ).(3.8) By substituting the values of λand αin (3.8) one can check that in fact N¯ φ(X, Y ) = 0. Next we prove that the tensor ¯gin (3.1) defines a Riemannian metric compatible with the almost contact structure SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 7 (¯ φ, ξ, η) and such that dη = ¯g(·,¯ϕ·). Firstly notice that from (3.1) and (3.4) it follows that ¯g(X, Y ) = ǫ1 p(2 −µ)2−4(1 −κ)((2 −µ)g(X, Y ) + 2g(X, hY )) + 1−ǫ2−µ p(2 −µ)2−4(1 −κ)!η(X)η(Y),(3.9) where ǫis given by (3.2). Since his a symmetric operator with respect to the Riemannian metric g, by (3.9) we see immediately that ¯gis a symmetric tensor. Next we prove that it is positive definite. First, by the very definition of ¯g, one has that ¯g(ξ, ξ) = 1. Moreover, for any non-zero tangent vector field X∈Γ(D), one has by (3.4) that ¯g(X, X) = ǫ1 p(2 −µ)2−4(1 −κ)((2 −µ)g(X, X) + 2g(X, hX)) . We can distinguish two cases: X∈Γ(Dh(λ)) or X∈Γ(Dh(−λ)). In the first case we obtain ¯g(X, X) = ǫ(2 −µ) + 2√1−κ p(2 −µ)2−4(1 −κ)g(X, X). The above formula implies that ¯g(X, X)>0, since (2 −µ) + 2√1−κis always positive when IM>1 and negative when IM<−1, the same as ǫ. The other case X∈Γ(Dh(−λ)) is similar. It remains to be seen that ¯gis an associated metric, that is dη = ¯g(·,¯ φ·). Indeed by the definition of ¯gand by the property of ¯ φof defining an almost contact structure we have ¯g(X, ¯ φY ) = −dη(X, ¯ φ2Y) + η(X)η(¯ φY ) = −dη(X, −Y+η(Y)ξ) = dη(X, Y ) + η(Y)dη(X, ξ) = dη(X, Y ). (ii) Now let us assume that |IM|<1. In this case e φis given by (3.10) e φ=1 √−α((2 −µ)ϕ+ 2ϕh) and the tensor egby eg(X, Y ) = −1 √−α((2 −µ)g(X, Y ) + 2g(X, hY )) + 1 + 2−µ √−αη(X)η(Y).(3.11) Equation (3.11) easily implies that egis a symmetric tensor. Moreover, one can prove as in the case (i) that dη(X, Y ) = eg(X, e φY ) for any X, Y ∈Γ(T M). This last condition, together with e φ2=I−η⊗ξ, implies that eg(e φX, e φY ) = −eg(X, Y ) + η(X)η(Y). Thus egis a semi-Riemannian metric of signature (n, n + 1) and (e φ, ξ, η, eg) is a paracontact metric manifold. So, once we have proved that the structure (e φ, ξ, η) is normal, we would have proved that (M, e φ, ξ, η, eg) is a paraSasakian manifold. In order to do that, we will check that the almost paracontact structure (e φ, ξ, η) satisfies the conditions of Theorem 2.2. First we prove that the eigendistributions D±are integrable. Let X, X′∈Γ(D+), i.e. e φX =X,e φX′=X′By the definition of e φand formulas (2.7) and (2.9), we get e φ[X, X′] =∇Xe φX′−∇X′e φX −2−µ √−αg(X+hX, X′)ξ+2−µ √−αg(X′+hX′, X)ξ −2 √−α(g(X, hX′)ξ+ (1 −κ)g(X, X′)ξ) + 2 √−α(g(X′, hX)ξ+ (1 −κ)g(X′, X)ξ) =∇XX′−∇X′X= [X, X′]. Hence [X, X′]∈Γ(D+) and we conclude that D+is an integrable distribution. It remains to be seen that ξis a foliated vector field with respect to the foliation defined by D+, i.e. [ξ, X]∈Γ(D+) for any 8 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA X∈Γ(D+). By using (2.3), ∇ξϕ= 0 and ∇ξϕh =µh we have, arguing as before, e φ[ξ, X] = ∇ξ2−µ √−αϕX +2 √−αϕhX−µ √−αhX +µ−2k √−αX.(3.12) On the other hand, 2−µ √−αϕX +2 √−αϕhX =e φX, and one can check that ∇e φX ξ=−ϕe φX −ϕhe φX = µ √−αhX −µ−2k √−αX. So that (3.12) yields e φ[ξ, X] = ∇ξe φX − ∇e φX ξ=∇ξX− ∇Xξ= [ξ, X], that is [ξ, X]∈Γ(D+). The same arguments also work for D−. Therefore, by Theorem 2.2 we conclude that the paracontact metric structure (e φ, ξ, η, g) is normal and hence paraSasakian.  Remark 3.2.One can see that the metrics stated in Theorem 3.1 extend the Riemannian metrics introduced, in a different context, in [15] and [17]. Remark 3.3.It is interesting to notice that the metrics ¯gand egare invariant under D-homothetic deformations. Indeed let us apply a Dc-homothetic deformation (2.5) to the non-Sasakian contact metric (κ, µ)- space (M, ϕ, ξ, η, g), for some c > 0. Then we obtain a new contact metric (κ′, µ′)-structure (ϕ′, ξ′, η′, g′) on M, with κ′and µ′given by (2.10) and the same Boeckx invariant as (ϕ, ξ, η, g). Thus (ϕ′, ξ′, η′, g′) in turn satisfies the assumptions of Theorem 3.1 and admits a Sasakian structure (¯ φ′, ξ′, η′,¯g′) or a paraSasakian structure (e φ′, ξ′, η′,eg′) according to the circumstance that |IM|>1 or |IM|<1, respectively. Let us assume for instance that |IM|>1. Since h′=1 chand a straightforward computation shows that α′= (2 −µ′)2−4(1 −κ′) = 1 c2α, from (3.6) it follows that ¯ φ′=ǫ1 √α′((2 −µ′)ϕ′+ 2ϕ′h′) = ǫ1 √α((2 −µ)ϕ+ 2ϕh) = ¯ φ. Moreover, due to ¯g′(·,¯ φ′·) = dη′, one easily proves that ¯g′=c¯g+c(c−1)η⊗η. Thus the Sasakian structure (¯ φ′, ξ′, η′,¯g′), associated via Theorem 3.1 to the (κ, µ)-space (M, ϕ′, ξ′, η′, g′), is nothing but the structure Dc-homothetic to the Sasakian structure (¯ φ, ξ, η, ¯g). The same arguments work in the case |IM|<1. We will now calculate the curvature tensor of the Sasakian and paraSasakian manifolds that appear in the previous theorem, which we will show to be always η-Einstein. Theorem 3.4. Let (M, ϕ, ξ, η, g)be a non-Sasakian contact metric (κ, µ)-space such that IM6=±1. (i) If |IM|>1then the Levi-Civita connection of (M, ¯ φ, ξ, η, ¯g), the Sasakian manifold defined in (3.1), relates to the original one in the following way: (3.13) ∇XY=∇XY+1−ǫ2−µ √α(η(Y)ϕX +η(X)ϕY ) +1−ǫ2 √α(η(Y)ϕhX +η(X)ϕhY )−g(X, ϕhY )ξ, where ǫis given by (3.2). Moreover, its curvature tensor has the form (3.14) R(X, Y )Z=ǫ√α 2(¯g(Y, Z)X−¯g(X, Z)Y) +ǫ√α 2−1(¯g(X, ¯ φZ)¯ φY −¯g(Y, ¯ φZ)¯ φX + 2¯g(X, ¯ φY )¯ φZ) +ǫ√α 2−1(η(X)η(Z)Y−η(Y)η(Z)X+ ¯g(X, Z)η(Y)ξ−¯g(Y, Z)η(X)ξ) +ǫ√α 2(1 −κ)¯g(hY, Z)hX −¯g(hX, Z)hY −¯g(hY, ¯ φZ)¯ φhX + ¯g(hX, ¯ φZ)¯ φhY . SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 9 The Ricci tensor of M2n+1 is (3.15) Ric = (ǫn√α−2)¯g+ (−ǫn√α+ 2n+ 2)η⊗η and therefore the Sasakian structure is always η-Einstein. (ii) If |IM|<1then the paraSasakian structure (e φ, ξ, η, ¯g)defined in (3.3) has the following Levi- Civita connection: (3.16) e ∇XY=∇XY+1−2−µ √−α(η(Y)ϕX +η(X)ϕY ) +1−2 √−α(η(Y)ϕhX +η(X)ϕhY )−g(X, ϕhY )ξ. Furthermore, the curvature tensor can be written as (3.17) e R(X, Y )Z=−√−α 2(eg(Y, Z)X−eg(X, Z)Y) +√−α 2−1(eg(X, e φZ)e φY −eg(Y, e φZ)e φX + 2eg(X, e φY )e φZ) −√−α 2−1(η(X)η(Z)Y−η(Y)η(Z)X+eg(X, Z)η(Y)ξ−eg(Y, Z)η(X)ξ) −√−α 2(1 −κ)eg(hY, Z)hX −eg(hX, Z)hY +eg(hY, e φZ)e φhX −eg(hX, e φZ)e φhY . The Ricci tensor of M2n+1 is (3.18) g Ric = (−n√−α+ 3)eg+ (n√−α−2n−3)η⊗η and therefore the paraSasakian structure is always η-Einstein. Proof. (i) Bearing in mind that ∇is a Levi-Civita connection, we can use Koszul’s formula, ∇g= 0 and formulas (2.8) and (3.9) to obtain 2¯g(∇XY, Z) =2¯g(∇XY, Z) + 2 1 + ǫµ−2κ √α(η(Y)g(X, ϕZ) + η(X)g(Y, φZ)) −2η(Z)g(X, φhY )−ǫ2µ √α(η(X)g(Y, φhZ) + η(Y)g(X, φhZ)). (3.19) We recall now that ηis the contact form of both structures, so g(X, ϕY ) = dη(X, Y ) = ¯g(X, ¯ φY ) and g(Y, φhZ) = −¯g(hY, ¯ φZ) is deduced. Substituting both formulas in (3.19), it follows that ∇XY=∇XY+1 + ǫµ−2κ √α(η(Y)¯ φX+η(X)¯ φY)−g(X, φhY )ξ−ǫµ √α(η(X)¯ φhY +η(Y)¯ φhX). This last equation together with (3.6) gives us (3.13). As Ris a Riemannian curvature tensor, using equations (2.7) and (2.9) we obtain after long computations that R(X, Y )Z=R(X, Y )Z+1−ǫ2−µ √α(g(X, ϕZ)ϕY −g(Y, ϕZ)ϕX(3.20) + 2g(X, ϕY )ϕZ) + g(Y, ϕhZ)ϕX −g(X, ϕhZ)ϕY +1−ǫ2 √α(g(ϕY, Z)ϕhX −g(ϕX, Z)ϕhZ + 2g(X, ϕY )ϕhZ) 16 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA for any Z∈Γ(T M), X∈Γ(TF). The operator ΛFis surjective and satisfies (ΛF)2= 0 and (4.2) ΛF[ξ, X] = 1 2X for all X∈Γ(TF). Then one can extend ΠFto a symmetric bilinear form defined on all T M by setting ΠF(Z, Z′) := ΠF(Z, Z′) if Z, Z′∈Γ(TF) ΠF(ΛFZ, ΛFZ′) otherwise. Now let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space. Being n-dimensional integrable subbundles of the contact distribution, the eigenspaces of h,Dh(λ) and Dh(−λ), define two mutually orthogonal Legendre foliations on M. Such a foliated structure of a (κ, µ)-space was studied in [13]. In particular, more explicit formulas for the Pang invariants of Dh(λ) and Dh(−λ) were found, namely (4.3) ΠDh(λ)= (2 −µ+ 2λ)g|Dh(λ)×Dh(λ),ΠDh(−λ)= (2 −µ−2λ)g|Dh(−λ)×Dh(−λ), from which it follows that Dh(λ) and Dh(−λ) are both non-degenerate if and only if IM6=±1. Let us assume that |IM|>1. By using (4.3) and (3.9) one has, for all X, X′∈Γ(Dh(λ)), ΠDh(λ)(X, X′) = (2 −µ)g(X, X′) + 2g(λX, X′) = (2 −µ)g(X, X′) + 2g(hX, X′) = ǫ√α¯g(X, X′), where α= (2 −µ)2−4(1 −κ) and ǫis the sign of IM, according to the notation of §3. Analogously, for all Y, Y ′∈Γ(Dh(−λ)): ΠDh(−λ)(Y, Y ′) = ǫ√α¯g(Y, Y ′) Whereas, if |IM|<1 one has ΠDh(λ)(X, X′) = −√−αeg(X, X′),ΠDh(−λ)(Y, Y ′) = −√−αeg(Y, Y ′). Therefore, geometrically the Sasakian metric ¯gin the case |IM|>1 and the paraSasakian metric egin the case |IM|<1 represent, up to a constant factor, the Pang invariant of the foliations Dh(λ) and Dh(−λ) when restricted to the leaves. Moreover, notice that since Dh(λ) and Dh(−λ) are totally geodesic foliations ([7, Proposition 3.7]) and because of (3.13) and (3.16), they are still totally geodesic with respect to ¯g and eg. Now we prove that the above properties determine uniquely the contact metric (κ, µ)-structure. Theorem 4.1. Let (M, ¯ φ, ξ, η, ¯g)be a K-contact manifold endowed with two totally geodesic, mutually orthogonal Legendre foliations F1and F2. Assume that (4.4) ΠF1=ǫ√ab¯g|F1×F1,ΠF2=ǫ√ab¯g|F2×F2 for some real numbers aand bsuch that a6=band a·b > 0, where ǫ=1,if a > 0,b > 0 −1,if a < 0,b < 0. Then Madmits a contact metric (κa,b, µa,b)-structure (ϕa,b, ξ, η, ga,b), compatible with the original contact form η, where (4.5) κa,b = 1 −(a−b)2 16 , µa,b = 2 −a+b 2. Furthermore, (M, ¯ φ, ξ, η, ¯g)is Sasakian and η-Einstein with Ricci tensor (4.6) Ric = (ǫn√ab −2)g+ (−ǫn√ab + 2n+ 2)η⊗η. Proof. First notice that, since F1and F2are mutually ¯g-orthogonal, ¯ φT F1⊂TF2and ¯ φT F2⊂TF1. Indeed let us fix X∈Γ(TF1). Then for any X′∈Γ(TF1) one has ¯g(¯ φX, X′) = −dη(X, X′) = 1 2η([X, X′]) = 0 since F1is a Legendre foliation. Moreover, obviously ¯g(¯ φX, ξ) = 0. Consequently, since by assumption (TF1⊕Rξ)⊥=TF2, we conclude that ¯ φX ∈Γ(TF2). In a similar way one can prove the other relation. SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 17 Now, taking into account the decomposition of the tangent bundle of Mas T M =TF1⊕TF2⊕Rξ, let us define a (1,1)-tensor field ϕa,b and a Riemannian metric ga,b on Mby (4.7) ϕa,b :=     qb a¯ φ, on TF1 pa b¯ φ, on TF2 0,on Rξ ga,b :=     qb a¯g, on TF1×TF1 pa b¯g, on TF2×TF2 η⊗η, otherwise and extend it by linearity. A straightforward computation shows that ϕ2 a,b =−I+η⊗ξand ga,b(ϕa,b·, ϕa,b·) = ga,b −η⊗η. Let us check that (ϕa,b, ξ, η, ga,b) is a contact metric structure, i.e. (4.8) dη(X, Y ) = ga,b(X, ϕa,bY) for all X, Y ∈Γ(T M). If either Xor Ybelongs to Rξ, (4.8) is trivially satisfied, both sides being zero. Let X, Y ∈Γ(TF1). Then dη(X, Y ) = −1 2η([X, Y ]) = 0 because F1is a Legendre foliation. On the other hand, since ¯ φY ∈Γ(TF2), then ga,b(X, ϕa,bY) = qb aga,b(X, ¯ φY ) = qb aη(X)η(¯ φY ) = 0 = dη(X, Y ). Analogously, if X, Y ∈Γ(TF2), then ga,b(X, ¯ φY ) = 0 = dη(X, Y ). It remains the case when X∈Γ(TF1) and Y∈Γ(TF2). In that case by using (4.7) one has ga,b(X, ϕa,bY) = pa bqb a¯g(X, ¯ φY ) = ¯g(X, ¯ φY ) = dη(X, Y ), since (¯ φ, ξ, η, ¯g) is a contact metric structure. Notice that, directly by the definition of ga,b, F1and F2are ga,b-orthogonal, so that the tensor field ϕa,b also maps TF1on TF2and TF2on TF1. Finally we prove that (M, ϕa,b, ξ, η, ga,b) is a (κ, µ)-space. First of all, we compute the operator ha,b of the contact metric structure (ϕa,b, ξ, η, ga,b). We have, for any X∈Γ(TF1), 2ha,bX= [ξ, ϕa,bX]−ϕa,b([ξ, X]F1)−ϕa,b([ξ, X]F2) =rb a[ξ, ¯ φX]−rb a¯ φ([ξ, X]F1)−ra b¯ φ([ξ, X]F2) =rb a[ξ, ¯ φX]F1+ 2rb a(¯ hX)F2−ra b¯ φ([ξ, X]F2) = rb a−ra b!(¯ φ[ξ, X])F1,(4.9) where we have used the K-contact condition ¯ h= 0 and we have decomposed the vector field [ξ, X] according to the decomposition T M =TF1⊕TF2⊕Rξ. In particular, from (4.9) it follows that ha,b maps TF1on TF1. Now we use the assumption (4.4) for finding a more explicit expression for the Liberman map ΛF1:T M −→ TF1. By using (4.1) we have, for any X∈Γ(TF1) and Z∈Γ(T M), that ǫ√ab¯g(ΛF1Z, X) = ΠF1(ΛF1Z, X) = dη(Z, X) = −¯g(X, ¯ φZ), from which it follows that (4.10) ΛF1Z=−ǫ1 √ab(¯ φZ)F1. Then, by using (4.2) and (4.10), we get (4.11) (¯ φ[ξ, X])F1=−ǫ√abΛF1[ξ, X] = −ǫ√ab 2X. Therefore, due to (4.11), formula (4.9) becomes (4.12) ha,bX=−ǫ√ab 4 rb a−ra b!X=ǫ|a|−|b| 4X=a−b 4X, since ǫ|a|=aand ǫ|b|=b. Due to ha,bϕa,b =−ϕa,bha,b, from (4.12) it follows that (4.13) ha,bY=−a−b 4Y 18 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA for any Y∈Γ(TF2). Thus we have proved that the tangent bundles of the foliations F1and F2coincide with the eigendistributions of ha,b corresponding to the constant eigenvalues ±λa,b, where λa,b := |a−b| 4. Let ∇bl be the bi-Legendrian connection associated to (F1,F2) (cf. [14]). By definition ∇bl is the unique linear connection on Msuch that the following conditions hold: (i) ∇blF1⊂ F1,∇blF2⊂ F2 (ii) ∇blη= 0, ∇bldη = 0, (iii) Tbl(X, Y ) = 2dη(X, Y )ξfor any X∈Γ(TF1), Y∈Γ(TF2), Tbl(Z, ξ) = [ξ, ZF1]F2+ [ξ, ZF2]F1for any Z∈Γ(T M), where Tbl denotes the torsion tensor field of ∇bl. Actually, because of the integrability of TF1and TF2, one has immediately that Tbl(X, X′) = 0 = 2dη(X, X′)ξfor any X, X′∈Γ(TF1) and Tbl(Y, Y ′) = 0 = 2dη(Y, Y ′)ξfor any Y, Y ′∈Γ(TF2). Thus the first identity in (iii) holds for any two vector fields on the contact distribution. Moreover, since F1and F2are assumed to be totally geodesic with respect to ¯g, due to [14, Proposition 2.9], we get that ∇bl ¯ φ= 0. Then, because of (i) and (4.7), one has that ∇bl preserves also ϕa,b. Next, as dη =ga,b(·, ϕa,b·), after a straightforward computation one obtains (∇bl Zga,b)(Z′, Z′′) = −(∇bl Zdη)(Z′, ϕa,bZ′′)−dη(Z′,(∇bl Zϕa,b)Z′′) + η(Z′′)(∇bl Zη)(Z′) + η(Z′)(∇bl Zη)(Z′′) for all Z, Z′, Z′′ ∈Γ(T M). Since ∇bl preserves η,dη and ϕa,b, the above equation implies that ∇blga,b = 0. Finally (i) and (4.12)–(4.13) easily ensure that ∇bl preserves also the tensor field ha,b. Therefore the bi- Legendrian connection ∇bl defined by the foliations F1,F2satisfies all the conditions required by [16, Theorem 4.4] and we can conclude that (ϕa,b, ξ, η, ga,b) is a (κ, µ)-structure. In order to find explicitly the constants κand µ(which of course will depend on aand b) we notice that, since ΠF1is an invariant of the Legendre foliation F1, the two expressions for ΠF1in (4.3) and (4.4) have to coincide. We thus have, for any 0 6=X∈Γ(TF1), (2 −µa,b + 2λa,b)ga,b(X, X) = ǫ√ab¯g(X, X) = ǫ|a|ga,b(X, X) = aga,b(X, X). Thus (4.14) 2 −µa,b + 2λa,b =a. Arguing in a similar way for F2one gets (4.15) 2 −µa,b −2λa,b =b. Then by using (4.14)–(4.15) we get the values (4.5) for κa,b and µa,b. We will now prove the last part of the statement. From (4.7) and (4.14)–(4.15) one has immediately that for any X, X′∈Γ(TF1) ¯g(X, X′) = ra bga,b(X, X′) =a √abga,b(X, X′) =1 p(2 −µa,b)2−4(1 −κa,b)((2 −µa,b)ga,b(X, X′) + 2ga,b(ha,bX, X′)) . Similarly, one has that ¯g(Y, Y ′) = 1 p(2 −µa,b)2−4(1 −κa,b)((2 −µa,b)ga,b(Y, Y ′) + 2ga,b(ha,bY, Y ′)) for any Y, Y ′∈Γ(TF2). Therefore, as TF1=Dha,b (λa,b) and TF2=Dha,b (−λa,b), we see that the metric ¯gcoincides with the Sasakian metric defined by (3.9). Then ¯gis also η-Einstein with Ricci tensor (4.6) because of Theorem 3.4.  SASAKI-EINSTEIN AND PARASASAKI-EINSTEIN METRICS FROM (κ, µ)-STRUCTURES 19 Notice that each contact metric (κa,b, µa,b)-structure (ϕa,b, ξ, η, ga,b) of the family stated in Theorem 4.1 has a Boeckx invariant whose absolute value is strictly greater than 1. Indeed from (4.5) it follows that IM=a+b |a−b|. Finally, we will examine the paracontact case. Theorem 4.2. Let (M, e φ, ξ, η, eg)be a K-paracontact manifold endowed with two totally geodesic, mutually orthogonal Legendre foliations F1and F2, such that eg|F1×F1is positive definite and eg|F2×F2is negative definite. If ΠF1=−√−abeg|F1×F1,ΠF2=−√−abeg|F2×F2 for some real numbers aand bsuch that a6=band a·b < 0, then Madmits a contact metric (κ, µ)- structure (ϕa,b, ξ, η, ga,b), compatible with the original contact form η, where κ= 1 −(a−b)2 16 , µ = 2 −a+b 2. Furthermore, (M, ¯ φ, ξ, η, ¯g)is paraSasakian and η-Einstein with Ricci tensor g Ric = (−n√−ab + 3)eg+ (n√−ab −2n−3)η⊗η. Proof. The proof is very similar to that of Theorem 4.1, the only difference is the definition of the structure tensors ϕa,b and ga,b, which now are given by (4.16) ϕa,b :=      √−ab ae φ, on TF1 √−ab be φ, on TF2 0,on Rξ ga,b :=      −√−ab aeg, on TF1×TF1 −√−ab beg, on TF2×TF2 η⊗η, otherwise The extra-assumption that eg|F1×F1is positive definite and eg|F2×F2is negative definite ensure that the symmetric tensor ga,b defined by (4.16) is a Riemannian metric. The rest of the proof goes as in Theorem 4.1, once one notices that [14, Proposition 2.9], which is used for proving that (ϕ, ξ, η, g) is a (κ, µ)- structure, straightforwardly holds also in the context of paracontact metric geometry.  References [1] P. Alegre and A. Carriazo, Structures on generalized Sasakian-space-forms, Differential Geom. Appl. 26(2008), no. 6, 656–666. [2] D. V. Alekseevsky, V. Cort´es, A. Galaev and T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy, J. Reine Angew. Math. 635(2009) 23–69. [3] H. Baum, Twistor and Killing spinors in Lorentzian geometry, Global analysis and harmonic analysis (Marseille- Luminy, 1999), 35-52, S´emin. Congr., 4, Soc. Math. France, Paris, 2000. 35-52. [4] D. E. Blair, Two remarks on contact metric structures, Tˆohoku Math. J. 28(1976), 373–379. [5] D. E. Blair, On the non-regularity of tangent sphere bundles, Proc. Royal Soc. Edinburgh 82(1978), 13–17. [6] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Second Edition. Progress in Mathematics 203, Birkh¨auser, Boston, 2010. [7] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisfyng a nullity condition, Israel J. Math. 91(1995), 189–214. [8] E. Boeckx, A full classification of contact metric (κ, µ)-spaces, Illinois J. Math. 44(2000), 212–219. [9] C. Bohle, Killing spinors on Lorentzian manifolds, J. Geom. Phys. 45(2003), 285-308. [10] C. P. Boyer, K. Galicki and P. Matzeu, On η-Einstein Sasakian geometry, Commun. Math. Phys. 262(2006), 177–208. [11] D. M. J. Calderbank and H. Pedersen, Einstein-Weyl geometry. Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., VI, Boston, MA: Int. Press (1999), 387-423. [12] B. Cappelletti-Montano, Bi-Legendrian structures and paracontact geometry, Int. J. Geom. Met. Mod. Phys. 6(2009), 487–504. [13] B. Cappelletti-Montano, The foliated structure of contact metric (κ, µ)-spaces, Illinois J. Math. 53(2009), 1157–1172. [14] B. Cappelletti-Montano, Some remarks on the generalized Tanaka-Webster connection of a contact metric manifold, Rocky Mountain J. Math. 40(2010), 1009–1037. [15] B. Cappelletti-Montano, Bi-paracontact structures and Legendre foliations, Kodai Math. J. 33(2010), 473–512. 20 B. CAPPELLETTI-MONTANO, A. CARRIAZO, AND V. MART´ IN-MOLINA [16] B. Cappelletti-Montano and L. Di Terlizzi, Contact metric (κ, µ)-spaces as bi-Legendrian manifolds, Bull. Austral. Math. Soc. 77(2008), 373–386. [17] B. Cappelletti-Montano and L. Di Terlizzi, Geometric structures associated with a contact metric (κ, µ)-space, Pacific J. Math. 246(2010), 257–292. [18] A. Gosh, EinsteinWeyl structures on contact metric manifolds, Ann. Glob. Anal. Geom. 35(2009), 431-441. [19] S. Ivanov, D. Vassilev and S. Zamkovoy, Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata 144(2010), 79-100. [20] S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99(1985), 173–187. [21] P. Libermann, Legendre foliations on contact manifolds, Differential Geom. Appl. 1(1991), 57–76. [22] P. Matzeu, Closed Einstein-Weyl structures on compact Sasakian and cosymplectic manifolds, Proc. Edinb. Math. Soc. 54(2011), 149–160. [23] F. Narita, Riemannian submersions and Riemannian manifolds with Einstein-Weyl structures, Geom. Dedicata 65(1997), no. 1, 103-116. [24] M. Y. Pang, The structure of Legendre foliations, Trans. Amer. Math. Soc. 320(1990), no. 2, 417–453. [25] D. Perrone, Contact metric manifolds whose characteristic vector field is a harmonic vector field, Different. Geom. Appl. 20(2004), no. 3, 367–378. [26] S. Tanno, Harmonic forms and Betti numbers of certain contact Riemannian manifolds, J. Math. Soc. Japan 19(1967), 308–316. [27] S. Tanno, Variational problems on contact manifolds, Trans. Amer. Math. Soc. 314(1989), 349–379. [28] P. Tondeur, Geometry of foliations, Monographs in Mathematics 90, Birkh¨auser, Boston, 1997. [29] E. Vergara-Diaz and C. M. Wood, Harmonic contact metric structures, Geom. Dedicata 123(2006), 131–151. [30] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36(2009), 37–60. [31] S. Zamkovoy and V. Tzanov, Non-existence of flat paracontact metric structures in dimension greater than or equal to five, arXiv:0910.5838v1. Dipartimento di Matematica e Informatica, Universt` a degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari E-mail address:b.cappellettimontan[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Sevilla, Aptdo. de Correos 1160, 41080 Sevilla, SPAIN E-mail address:[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Sevilla, Aptdo. de Correos 1160, 41080 Sevilla, SPAIN E-mail address:veronicam[email protected]s