Contact metric (κ,μ) -spaces as bi-Legendrian manifolds
aa r X i v : . [ m a t h . DG ] F e b Contact metric ( κ, µ ) -spaces asbi-Legendrian manifolds Beniamino Cappelletti Montano, Luigia Di Terlizzi
Department of Mathematics, University of BariVia E. Orabona, 4I-70125 Bari (Italy) [email protected], [email protected]
Abstract
We regard a contact metric manifold whose Reeb vector field belongs to the( κ, µ )-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric ( κ, µ )-spaces in terms of acanonical connection which can be naturally defined on them.
Keywords and phrases.
Contact metric ( κ, µ )-manifolds, ( κ, µ )-nullity distribution,Legendrian foliations, bi-Legendrian structures.
Contact metric ( κ, µ )-spaces, introduced in [2] by D. E. Blair, T. Kouforgiorgos and B. J.Papantoniou, are those contact metric manifolds (
M, φ, ξ, η, g ) for which the Reeb vectorfield ξ belongs to the ( κ, µ )-nullity distribution, i.e. satisfies, for all vector fields V and W on M , R V W ξ = κ ( η ( W ) V − η ( V ) W ) + µ ( η ( W ) hV − η ( V ) hW ) , (1.1)for some real numbers κ and µ , where 2 h is the Lie derivative of φ in the direction of ξ . This definition can be regarded as a generalization both of the Sasakian condition R V W ξ = η ( W ) V − η ( V ) W and of those contact metric manifolds verifying R V W ξ = 0which were studied by D. E. Blair in [1].Recently contact metric ( κ, µ )-spaces have been studied by various authors ([4], [5], [6],[11], [14], etc.) and several important properties of these manifolds have been discovered.In fact there are many motivations for studying ( κ, µ )-spaces: the first is that, in the non-Sasakian case (that is for κ = 1), the condition (1.1) determines the curvature completely;1oreover, while the values of κ and µ change, the form of (1.1) is invariant under D -homothetic deformations; finally, there are non-trivial examples of these manifolds, themost important being the unit tangent sphere bundle of a Riemannian manifold of constantsectional curvature with the usual contact metric structure.A complete classification of contact metric ( κ, µ )-spaces has been given in [5] by E.Boeckx, who proved also that any non-Sasakian contact metric ( κ, µ )-space is locallyhomogeneous and strongly locally φ -symmetric ([4]).One of the peculiarities of these manifolds is that they give rise to three mutuallyorthogonal distributions D λ , D − λ and R ξ , corresponding to the eigenspaces of the operator h . In particular D λ and D − λ define two transverse Legendrian foliations of M so that thesemanifolds are endowed with a bi-Legendrian structure.In the same years the theory of Legendrian foliations has been developed by M. Y.Pang, P. Libermann and N. Jayne (cf. [16], [15], [13]), so it seems to be tempting to usethe techniques and the language of Legendrian foliations for the study of contact metric( κ, µ )-spaces and to begin the investigation of the interactions between these two areas ofthe contact geometry. This is what we set out to do in this article.The paper is organized as follows. After some preliminaries on contact metric manifoldsand Legendrian foliations, in § κ, µ )-space. We find, for both the foliations, an explicit formula of theinvariant Π introduced by Pang for classifying Legendrian foliations (cf. [16]) and we seethat the Legendrian foliations in question are, according to this classification, either non-degenerate or flat. Then we relate these invariants to the invariant I M used by Boeckx in[5] for classify contact metric ( κ, µ )-spaces. In § κ, µ )-space a linear connection in a canonical way. We study the properties of this connectionand, using it, we give an interpretation of the notion of contact metric ( κ, µ )-space interms of bi-Legendrian structures. In particular, we prove the following characterizationof contact metric ( κ, µ )-spaces. Theorem 1.1.
A contact metric manifold ( M, φ, ξ, η, g ) is a contact metric ( κ, µ ) -spaceif and only if M admits an orthogonal bi-Legendrian structure ( F , G ) such that the cor-responding bi-Legendrian connection ¯ ∇ satisfies ¯ ∇ φ = 0 and ¯ ∇ h = 0 . Furthermore, thebi-Legendrian structure ( F , G ) coincides with that one determined by the eigenspaces of h . This theorem should be compared with the well-known results, obtained by N. Tanaka(cf. [17]) and, independently, S. M. Webster ([22]). They proved that any strongly pseudo-convex CR-manifold admits a unique linear connection ˜ ∇ such that the tensors φ , η , g areall ˜ ∇ -parallel and whose torsion satisfies ˜ T ( Z, Z ′ ) = 2Φ ( Z, Z ′ ) ξ for all Z, Z ′ ∈ Γ ( D ) and˜ T ( ξ, φV ) = − φ ˜ T ( ξ, V ) for all V ∈ Γ (
T M ). In view of this remark and the mentionedtheorem of Boeckx that any contact metric ( κ, µ )-space is a strongly pseudo-convex CR-manifold, one can see that the connection mentioned in Theorem 1.1 plays the same role forcontact metric ( κ, µ )-space that the Tanaka-Webster connection has for CR-manifolds. As2e shall see, the connection ¯ ∇ uniquely determines a contact metric ( κ, µ )-space modulo D -homothetic deformations and it reveals very useful in the study of this kind of contactmetric manifolds. An almost contact metric manifold is a (2 n + 1)-dimensional Riemannian manifold ( M, g )which admits a tensor field φ of type (1 , η and a global vector field ξ ,called Reeb vector field , satisfying η ( ξ ) = 1 , φ V = − V + η ( V ) ξ, g ( φV, φW ) = g ( V, W ) − η ( V ) η ( W ) , (2.1)for all vector fields V and W on M . Given an almost contact metric manifold one can definea 2-form Φ, called the fundamental -form of the structure, by Φ ( V, W ) = g ( V, φW ). Thenwe say that (
M, φ, ξ, η, g ) is a contact metric manifold if the additional property dη = Φholds. From (2.1) it can be proven that (cf. [3]) (i) φξ = 0, η ◦ φ = 0, (ii) ∇ ξ φ = 0 and ∇ ξ ξ = 0, (iii) φ | D is an isomorphism,where ∇ denotes the Levi Civita connection and D = ker ( η ) is the 2 n -dimensional distri-bution orthogonal to ξ and called the contact distribution . It is also easy to prove that forany X ∈ Γ ( D ) the bracket [ X, ξ ] still belongs to D .In any contact metric manifold the 1-from η satisfies the relation η ∧ ( dη ) n = 0 (2.2)everywhere on M . Any (2 n + 1)-dimensional smooth manifold which carries a global1-form satisfying (2.2) is called a contact manifold . Thus any contact metric manifoldis a contact manifold. Conversely, it is well-known that any contact manifold admits acompatible contact metric structure ( φ, ξ, η, g ). It should be remarked that (2.2) impliesthat the contact distribution D is never integrable.Given a contact metric manifold, we can define a tensor field h by h = L ξ φ , L denoting the Lie differentiation. It can be shown (cf. [3]) that h is a trace-free, symmetricoperator verifying hξ = 0, φh = − hφ and ∇ V ξ = − φhV − φV (2.3)3or all V ∈ Γ (
T M ). Moreover ξ is Killing if and only if h vanishes identically; in this casewe say that ( M, φ, ξ, η, g ) is a K -contact manifold .On a contact metric manifold M one can define an almost complex structure J onthe product manifold M × R by setting J (cid:0) V, f ddt (cid:1) = (cid:0) φV − f ξ, η ( V ) ddt (cid:1) , where V is avector field tangent to M and f a function on M × R . If the almost complex structure J is integrable then ( M, φ, ξ, η, g ) is said to be
Sasakian . It is well-known that each of thefollowing conditions characterizes Sasakian manifolds( ∇ V φ ) W = g ( V, W ) ξ − η ( W ) V (2.4) R V W ξ = η ( W ) V − η ( V ) W (2.5)for all vector fields V and W on M . A generalization of the condition (2.5) leads tothe notion of ( κ, µ )-manifold. If the curvature tensor field of a contact metric manifoldsatisfies (1.1) for some real numbers κ and µ we say that ξ belongs to the ( κ, µ )-nullitydistribution or, simply, that ( M, φ, ξ, η, g ) is a contact metric ( κ, µ )-space. This manifoldswere introduced and deeply studied in [2]. Among other things, the authors proved thefollowing results.
Theorem 2.1 ([2]) . Let ( M, φ, ξ, η, g ) be a contact metric manifold with ξ belonging to the ( κ, µ ) -nullity distribution. Then κ ≤ . Moreover, if κ = 1 then h = 0 and ( M, φ, ξ, η, g ) isa Sasakian manifold; if κ < , the contact metric structure is not Sasakian and M admitsthree mutually orthogonal integrable distributions D = R ξ , D λ and D − λ corresponding tothe eigenspaces of h , where λ = √ − κ . Theorem 2.2 ([2]) . Let ( M, φ, ξ, η, g ) be a contact metric manifold with ξ belonging tothe ( κ, µ ) -nullity distribution. Then the following relation hold, for any X, Y ∈ Γ (
T M ) , ( ∇ X φ ) Y = g ( X, Y + hY ) ξ − η ( Y )( X + hX ) , ( ∇ X h ) Y = ((1 − κ ) g ( X, φY ) + g ( X, φhY )) ξ + η ( Y )( h ( φX + φhX )) − µφhY. Blair, Kouforgiorgos and Papantoniou proven also that the ( κ, µ )-nullity conditionremains unchanged under D -homothetic deformations. The concept of D -homothetic de-formation for a contact metric manifold ( M, φ, ξ, η, g ) has been introduced by S. Tanno in[18] and then intensively studied by many authors. We recall that, given a real positivenumber a , by a D -homothetic deformation of constant a we mean a change of the structuretensors in the following way:˜ φ = φ, ˜ η = aη, ˜ ξ = 1 a ξ, ˜ g = ag + a ( a − η ⊗ η. (2.6)In [2] the authors proven that if M is a contact metric manifold whose Reeb vector fieldbelongs to the ( κ, µ )-nullity distibution then for the contact metric manifold ( M, ˜ φ, ˜ ξ, ˜ η, ˜ g )4he same property holds. Precisely ˜ ξ belongs to the (˜ κ, ˜ µ )-nullity distribution where˜ κ = κ + a − a , ˜ µ = µ + 2 a − a . A Legendrian distribution on a contact manifold ( M n +1 , η ) is defined by an n -dimensionalsubbundle L of the contact distribution such that dη ( X, X ′ ) = 0 for all X, X ′ ∈ Γ ( L ).When L is integrable, it defines a Legendrian foliation of ( M n +1 , η ). Legendrian foliationshave been extensively investigated in recent years from various points of views (cf. [16],[15], [13], [8], etc.). In particular Pang provided a classification of Legendrian foliations bymeans of a bilinear symmetric form Π F on the tangent bundle of the foliation, defined byΠ F ( X, X ′ ) = − ( L X L X ′ η ) ( ξ ) = − η ([ X ′ , [ X, ξ ]]). He called a Legendrian foliation F non-degenerate , degenerate or flat according to the circumstance that the bilinear form Π F isnon-degenerate, degenerate or vanishes identically, respectively. In terms of an associatedmetric g , Π F is given by Π F (cid:0) X, X ′ (cid:1) = 2 g (cid:0) [ ξ, X ] , φX ′ (cid:1) . (2.7)The last formula provides a geometrical interpretation of this classification: Lemma 2.1 ([13]) . Let ( M, φ, ξ, η, g ) be a contact metric manifold and let F be a foliationon it. Then (i) F is flat if and only if [ ξ, X ] ∈ Γ ( T F ) for all X ∈ Γ ( T F ) , (ii) F is degenerate if and only if there exist X ∈ Γ ( T F ) such that [ ξ, X ] ∈ Γ ( T F ) , (iii) F is non-degenerate if and only if [ ξ, X ] / ∈ Γ ( T F ) for all X ∈ Γ ( T F ) . Given a compatible contact metric structure ( φ, ξ, η, g ) and a Legendrian distribution L on M , we may consider the distribution Q = φL . It can be proven (cf. [13]) that Q is a Legendrian distribution on M which in general is not integrable, even if L is; it iscalled the conjugate Legendrian distribution of L , and the tangent bundle of M splits asthe orthogonal sum T M = L ⊕ Q ⊕ R ξ . When both L and Q are integrable, they definestwo orthogonal Legendrian foliations F and G on M , and the pair ( F , G ) is an exampleof a bi-Legendrian structure on M . More in general a bi-Legendrian structure is a pair oftwo complementary, not necessarily orthogonal, Legendrian foliations on M .In [7] it has been attached to any contact manifold ( M n +1 , η ) endowed with a pairof two complementary Legendrian distributions ( L, Q ) a linear connection ¯ ∇ uniquely5etermined by the following properties:(i) ¯ ∇ L ⊂ L, ¯ ∇ Q ⊂ Q, ¯ ∇ ( R ξ ) ⊂ R ξ, (ii) ¯ ∇ dη = 0 , (2.8)(iii) ¯ T ( X, Y ) = 2 dη ( X, Y ) ξ, for all X ∈ Γ ( L ) , Y ∈ Γ ( Q ) , ¯ T ( V, ξ ) = [ ξ, V L ] Q + [ ξ, V Q ] L , for all V ∈ Γ (
T M ) , where ¯ T denotes the torsion tensor of ¯ ∇ and V L and V Q the projections of V onto thesubbundles L and Q of T M , respectively. Such a connection is called the bi-Legendrianconnection associated to the pair (
L, Q ) and it is defined as follows (cf. [7]). For all V ∈ Γ (
T M ), X ∈ Γ ( L ) and Y ∈ Γ ( Q ), ¯ ∇ V X := H ( V L , X ) L + [ V Q , X ] L + [ V R ξ , X ] L ,¯ ∇ V Y := H ( V Q , Y ) Q + [ V L , Y ] Q + [ V R ξ , Y ] Q and ¯ ∇ ξ = 0, where H denotes the oper-ator such that, for all V, W ∈ Γ (
T M ), H ( V, W ) is the unique section of D satisfying i H ( V,W ) dη | D = ( L V i W dη ) | D . Further properties of this connection are collected in thefollowing proposition. Proposition 2.1 ([7]) . Let ( M, η ) be a contact manifold endowed with two complemen-tary Legendrian distributions L and Q and let ¯ ∇ denote the corresponding bi-Legendrianconnection. Then the -form η and the vector field ξ are ¯ ∇ -parallel and the complete ex-pression of the torsion tensor field is given by ¯ T ( X, X ′ ) = − [ X, X ′ ] Q for all X, X ′ ∈ Γ ( L ) and ¯ T ( Y, Y ′ ) = − [ Y, Y ′ ] L for all Y, Y ′ ∈ Γ ( Q ) . Now consider a contact metric manifold (
M, φ, ξ, η, g ) endowed with two complemen-tary Legendrian distributions L and Q . The definition of the corresponding bi-Legendrianconnection does not involve the compatible metric g , however it makes sense to find condi-tions which ensure ¯ ∇ being a metric connection at least when Q is orthogonal to L . Thisproblem has been solved in [9] where the author proves the following result. Proposition 2.2.
Let ( M, φ, ξ, η, g ) be a contact metric manifold and L be a Legendriandistribution on M . Let Q = φL be the conjugate Legendrian distribution of L and ¯ ∇ theassociated bi-Legendrian connection. Then the following statements are equivalent: (i) ¯ ∇ g = 0 ; (ii) ¯ ∇ φ = 0 ; (iii) g is a bundle-like metric with respect both to the distribution L ⊕ R ξ and to Q ⊕ R ξ ; (iv) ¯ ∇ X X ′ = ( φ [ X, φX ′ ]) L for all X, X ′ ∈ Γ ( L ) , ¯ ∇ Y Y ′ = ( φ [ Y, φY ′ ]) Q for all Y, Y ′ ∈ Γ ( Q ) and the operator h maps the subbundle L onto L and the subbundle Q onto Q .Furthermore, assuming L and Q integrable, (i)–(iv) are equivalent to the total geodesicityof the Legendrian foliations defined by L and Q
6y a bi-Legendrian manifold we mean a contact manifold endowed with two transversalLegendrian foliations. In particular, in this paper we deal with contact metric manifoldsfoliated by two mutually orthogonal Legendrian foliations. With regard to this, it will beuseful in the sequel to prove the following lemma, which states essentially that in a bi-Legendrian manifold the operator h is deeply linked to the given bi-Legendrian structure.This is just the starting point of our work. Lemma 2.2.
Let F and G two mutually orthogonal Legendrian foliations on the contactmetric manifold ( M, φ, ξ, η, g ) . Then for all X, X ′ ∈ Γ ( T F )Π F (cid:0) X, X ′ (cid:1) − Π G (cid:0) φX, φX ′ (cid:1) = 4 g (cid:0) hX, X ′ (cid:1) . (2.9) Proof.
Since, by the orthogonality between F and G we have φ ( T F ) = T G , using (2.7)we have Π F (cid:0) X, X ′ (cid:1) − Π G (cid:0) φX, φX ′ (cid:1) = 2 g (cid:0) [ ξ, X ] , φX ′ (cid:1) − g (cid:0) [ ξ, φX ] , φ X ′ (cid:1) = 2 g (cid:0) [ ξ, φX ] , X ′ (cid:1) − g (cid:0) φ [ ξ, X ] , X ′ (cid:1) = 4 g (cid:0) hX, X ′ (cid:1) . Corollary 2.1. If M is K-contact then F and G belong to the same class according to theabove Pang’s classification. Corollary 2.2. If F and G are both flat then M is K-contact. ( κ, µ ) -space Let (
M, φ, ξ, η, g ) be a contact metric manifold such that ξ belongs to the ( κ, µ )-nullitydistribution. By Theorem 2.1 the orthogonal distributions D λ and D − λ defined by theeigenspaces of h are involutive and define on M two orthogonal Legendrian foliationswhich we denote by F λ and F − λ , respectively. In this section we begin the study of thebi-Legendrian manifold ( M, F λ , F − λ ). Proposition 3.1.
Let ( M, φ, ξ, η, g ) be a contact metric ( κ, µ ) -space which is not K-contact. Then the Legendrian foliations F λ and F − λ are either non-degenerate or flat.More precisely, F λ (respectively, F − λ ) is flat if and only if κ + µλ − ( λ + 1) = 0 (respec-tively, κ − µλ − ( λ − = 0 ), otherwise being non-degenerate. roof. Let X ∈ Γ ( D λ ). Then by (1.1) we have R Xξ ξ = κX + µhX = ( κ + µλ ) X. On the other hand, using (2.3), R Xξ ξ = −∇ ξ ∇ X ξ − ∇ [ X,ξ ] ξ = ∇ ξ φX + λ ∇ ξ φX + φ [ X, ξ ] + φh [ X, ξ ]= X − λX − [ φX, ξ ] + λX − λ X − λ [ φX, ξ ] + φ [ X, ξ ] + φh [ X, ξ ]= ( λ + 1) X − λφ [ X, ξ ] + φh [ X, ξ ] , so that φh [ X, ξ ] = λφ [ X, ξ ] + ( κ + µλ − ( λ + 1) ) X hence, applying φ and taking into account that [ X, ξ ] ∈ Γ ( D ), − h [ X, ξ ] = − λ [ X, ξ ] + ( κ + µλ − ( λ + 1) ) φX. Decomposing [
X, ξ ] in the directions of D λ and D − λ we obtain − h ([ X, ξ ] D λ + [ X, ξ ] D − λ ) = − λ [ X, ξ ] + ( κ + µλ − ( λ + 1) ) φX, from which it follows that2 λ [ X, ξ ] D − λ = ( κ + µλ − ( λ + 1) ) φX (3.1)and we conclude, according to Lemma 2.1, that F λ is either flat or non-degenerate. Thefirst case occurs if and only if κ + µλ − ( λ + 1) = 0 and the second when κ + µλ − ( λ + 1) =0. In a similar way one can prove the analogous results for F − λ . Remark 3.1.
From Corollary 2.1 it follows that the bi-Legendrian structure ( F λ , F − λ ) isflat if and only if κ = 1 and hence M is Sasakian. This can be also prove in a direct wayobserving that, according to Proposition 3.1, the functions f ( κ, µ ) = κ + µλ − λ ( λ + 1) =2 ( κ −
1) + ( µ − √ − κ and g ( κ, µ ) = κ − µλ − λ ( λ − = 2 ( κ −
1) + (2 − µ ) √ − κ both vanish if and only if κ = 1.Proposition 3.1 extends and improves the results obtained in [12] for contact metricmanifolds for which ξ belongs to the κ -nullity distribution (cf. [19]), i.e. the Levi Civitaconnection of g satisfies R V W ξ = κ ( η ( W ) V − η ( V ) W ). In [12] the author proven thatthe bi-Legendrian structure associated to such contact metric manifolds is non-degenerate;we recall that in his proof he used the fact that the non-degenerate plane sections contain-ing ξ have constant sectional curvature and this last property does not hold for contactmetric ( κ, µ )-spaces, as it is been proven in [2].We remark also that from the proof of Proposition 3.1 it follows an explicit expressionof the invariants Π F λ and Π F − λ of the Legendrian foliations F λ and F − λ . More precisely,from (3.1) and (2.7) one can prove the following proposition.8 roposition 3.2. Let ( M, φ, ξ, η, g ) be a contact metric ( κ, µ ) -space which is not K-contact. Then the canonical invariants associated to the Legendrian foliations F λ and F − λ are given by Π F λ = ( λ + 1) − κ − µλλ g | F λ ×F λ and Π F − λ = − ( λ − + κ − µλλ g | F − λ ×F − λ , (3.2) respectively. It should be remarked that the pair (cid:0) Π F λ , Π F − λ (cid:1) is an invariant of the contact metric( κ, µ )-space in question up to D -homothetic deformations. Indeed let ( ˜ φ, ˜ ξ, ˜ η, ˜ g ) be a D -homothetic deformation of ( φ, ξ, η, g ). Then first of all since ˜ h = L ˜ ξ ˜ φ = a h (cf. [2]), theeigenvalues of ˜ h are ± ˜ λ = ± a λ , apart from 0. It follows that the eigenspaces D ˜ λ and D − ˜ λ coincide with D λ and D − λ respectively. Next, for all X, X ′ ∈ Γ (cid:0) D ˜ λ (cid:1) = Γ ( D λ ) we haveΠ F ˜ λ ( X, X ′ ) = − ˜ η ([ X ′ , [ X, ˜ ξ ]]) = − aη (cid:0) a [ X ′ , [ X, ξ ]] (cid:1) = Π F λ ( X, X ′ ). Analogously one canprove that Π F − ˜ λ = Π F − λ . Moreover, it should be observed that the invariant Π F of anyLegendrian foliation F depend only on the Legendrian foliation and on the contact form η and not on the associated metric g . In particular the functionΠ F λ ( X, X ′ ) + Π F − λ ( φX, φX ′ )Π F λ ( X, X ′ ) − Π F − λ ( φX, φX ′ ) , (3.3)for all X, X ′ ∈ Γ ( D λ ) such that Π F λ ( X, X ′ ) = 0 (or, equivalently, g ( X, X ′ ) = 0), isan invariant of the bi-Legendrian manifold M up to D -homothetic deformations and itdoes not depend on the vector fields X, X ′ ∈ Γ ( D λ ). Indeed, after a straightforwardcomputation, taking into account Lemma 2.2, (3.2) and (2.1), one can find that (3.3) is aconstant and, more precisely, it is given byΠ F λ ( X, X ′ ) + Π F − λ ( φX, φX ′ )Π F λ ( X, X ′ ) − Π F − λ ( φX, φX ′ ) = 1 − µ √ − κ = 14 I M , where I M is the invariant introduced by Boeckx in [5] for classifying contact metric ( κ, µ )-spaces. In particular if F λ (respectively F − λ ) is flat then I M attains the value 4 (respec-tively − µ in terms ofLegedrian foliations µ = Π F λ ( X, X ′ ) g ( hX, X ′ ) = Π F λ ( X, X ′ ) λg ( X, X ′ ) (3.4)for all X, X ′ ∈ Γ ( D λ ) such that g ( X, X ′ ) = 0.9 An interpretation of contact metric ( κ, µ ) -spaces Let (
M, φ, ξ, η, g ) be a contact metric ( κ, µ )-space. We can attach to the bi-Legendrianstructure ( F λ , F − λ ) the corresponding bi-Legendrian connection ¯ ∇ , that is the uniquelinear connection on M such that (2.8) hold. Furthermore we have the following result. Proposition 4.1.
Let ( M, φ, ξ, η, g ) be a contact metric ( κ, µ ) -space and let ¯ ∇ be thebi-Legendrian connection associated to M . Then the tensors φ , h and g are ¯ ∇ -parallel.Moreover, for the torsion tensor of ¯ ∇ we have ¯ T ( Z, Z ′ ) = 2Φ ( Z, Z ′ ) ξ for all Z, Z ′ ∈ Γ ( D ) .Proof. A well-known property about F λ and F − λ is that they are totally geodesic foliations(cf. [2]). Thus applying Proposition 2.2 we get ¯ ∇ g = 0 and ¯ ∇ φ = 0. Next, for all V ∈ Γ (
T M ), X ∈ Γ ( D + ), Y ∈ Γ ( D − ), we have (cid:0) ¯ ∇ V h (cid:1) X = ¯ ∇ V hX − h ¯ ∇ V X = ¯ ∇ V ( λX ) − λ ¯ ∇ V X = 0 , (cid:0) ¯ ∇ V h (cid:1) Y = ¯ ∇ V hY − h ¯ ∇ V Y = ¯ ∇ V ( − λY ) + λ ¯ ∇ V Y = 0 , because ¯ ∇ preserves F λ and F − λ . Finally, for any f ∈ C ∞ ( M ), (cid:0) ¯ ∇ V h (cid:1) f ξ = ¯ ∇ V ( h ( f ξ )) − h (cid:0) ¯ ∇ V ( f ξ ) (cid:1) = − h (cid:0) f ¯ ∇ V ξ (cid:1) − V ( f ) hξ = 0because ¯ ∇ ξ = 0 and hξ = 0. It remains to prove the property about the torsion, but itfollows easily from Proposition 2.1 and from the integrability of D λ and D − λ . Corollary 4.1.
With the assumptions and the notation of Proposition 4.1, the connection ¯ ∇ is related to the Levi Civita connection of ( M, φ, ξ, η, g ) by the following formula, forall X, Y ∈ Γ ( D ) , ¯ ∇ X Y = ∇ X Y − η ( ∇ X Y ) ξ. (4.1) Proof.
Since ¯ ∇ is torsion free along the leaves of the foliations F λ and F − λ and, byProposition 4.1, it is metric, it coincides with the Levi Civita connection along the leavesof F λ and F − λ . Hence (4.1) holds for all X, Y ∈ Γ ( D λ ) or X, Y ∈ Γ ( D − λ ) because F λ and F − λ are totally geodesic foliations. Now let X ∈ Γ ( D λ ) and Y ∈ Γ ( D − λ ). It iswell-known (cf. [2]) that ∇ X Y ∈ Γ ( D − λ ⊕ R ξ ). For all Y ′ ∈ Γ ( D − λ ), using ¯ ∇ g = 0, wehave 2 g ( ∇ X Y, Y ′ ) = X ( g ( Y, Y ′ )) + Y ( g ( X, Y ′ )) − Y ′ ( g ( X, Y )) + g ([ X, Y ] , Y ′ )+ g ([ Y ′ , X ] , Y ) − g ([ Y, Y ′ ] , X )= X ( g ( Y, Y ′ )) + g ([ X, Y ] , Y ′ ) + g ([ Y ′ , X ] , Y )= X ( g ( Y, Y ′ )) − g ([ X, Y ′ ] D − λ , Y ) + g ([ X, Y ] D − λ , Y ′ )= 2 g ([ X, Y ] D − λ , Y ′ )= 2 g ( ¯ ∇ X Y, Y ′ ) , ∇ X Y = ( ∇ X Y ) D − λ and hence (4.1). Analogously one can prove(4.1) for X ∈ Γ ( D − λ ) and Y ∈ Γ ( D λ ).Now we examine in a certain sense an ”inverse” problem. We start with a bi-Legendrianstructure on an arbitrary contact metric manifold M and we ask whether M is a contactmetric ( κ, µ )-space for some κ, µ ∈ R . Theorem 4.1.
Let ( M, φ, ξ, η, g ) be a contact metric manifold, non K -contact, endowedwith two orthogonal Legendrian foliations F and G and suppose that the bi-Legendrianconnection corresponding to ( F , G ) satisfies ¯ ∇ φ = 0 and ¯ ∇ h = 0 . Then ( M, φ, ξ, η, g ) isa contact metric ( κ, µ ) -space. Furthermore, the bi-Legendrian structure ( F , G ) coincideswith that one determined by the eigenspaces of h .Proof. Firstly we prove that under our assumptions (4.1) holds. Since, by Proposition2.2, ¯ ∇ g = 0 and ¯ T ( X, X ′ ) = 0, ¯ T ( Y, Y ′ ) = 0 for all X, X ′ ∈ Γ ( T F ) and Y, Y ′ ∈ Γ ( T G ),it follows immediately that the bi-Legendrian connection and the Levi Civita connectioncoincide along the leaves of F and G . Moreover, for all X ∈ Γ ( T F ) and Y ∈ Γ ( T G ) ∇ X Y ∈ Γ ( T G ⊕ R ξ ) because for all X ′ ∈ Γ ( T F ) g (cid:0) ∇ X Y, X ′ (cid:1) = X (cid:0) g (cid:0) Y, X ′ (cid:1)(cid:1) − g (cid:0) Y, ∇ X X ′ (cid:1) = 0since F , as well as G , is totally geodesic by Proposition 2.2. Then one can argue as in theproof of Corollary 4.1 and prove that ∇ Z Z ′ = ¯ ∇ Z Z ′ + η (cid:0) ∇ Z Z ′ (cid:1) ξ (4.2)for all Z, Z ′ ∈ Γ ( D ). Now for all X, Y, Z ∈ Γ ( D ) we have, applying (4.2), g (( ∇ X h ) Y, Z ) = g ( ∇ X hY − h ∇ X Y, Z )= g (cid:0) ¯ ∇ X hY + η ( ∇ X hY ) ξ − h ¯ ∇ X Y − η ( ∇ X Y ) hξ, Z (cid:1) = g (cid:0)(cid:0) ¯ ∇ X h (cid:1) Y, Z (cid:1) + η ( ∇ X hY ) η ( Z )= g (cid:0)(cid:0) ¯ ∇ X h (cid:1) Y, Z (cid:1) = 0 , since, by assumption, ¯ ∇ h = 0. Thus the tensor field h is η -parallel and so, by [6, Theorem4], ( M, φ, ξ, η, g ) is a contact metric ( κ, µ )-space. For proving the last part of the theorem,suppose by absurd that F does not coincide with both F λ and F − λ . Let X be a vector fieldtangent to F and decompose it as X = X + + X − , with X + ∈ Γ ( D λ ) and X − ∈ Γ ( D − λ ).Then we have hX = h ( X + ) + h ( X − ) = λX + − λX − = λ ( X + − X − ), from which, sinceby Proposition 2.2 h preserves F , it follows that X + − X − ∈ Γ ( T F ). On the other handalso X + + X − = X ∈ Γ ( T F ), hence X + and X − are both tangent to F and this is acontradiction. 11rom Theorem 4.1 we get the following characterization of contact metric ( κ, µ )-spaces. Here, by an abuse of language, we call Legendrian distribution of an almostcontact manifold any n -dimensional subbundle L of the distribution D = ker ( η ) such that dη ( X, X ′ ) = 0 for all X, X ′ ∈ Γ ( L ) and, as in contact metric geometry, 2 h is defined asthe Lie differentiation of the tensor φ along the Reeb vector field ξ . Theorem 4.2.
Let ( M, φ, ξ, η, g ) be an almost contact metric manifold with ξ non-Killing.Then ( M, φ, ξ, η, g ) is a contact metric ( κ, µ ) -space if and only if it admits two orthogo-nal conjugate Legendrian distributions L and Q and a linear connection ˜ ∇ satisfying thefollowing properties: (i) ˜ ∇ L ⊂ L , ˜ ∇ Q ⊂ Q , (ii) ˜ ∇ η = 0 , ˜ ∇ dη = 0 , ˜ ∇ g = 0 , ˜ ∇ h = 0 , (iii) ˜ T ( Z, Z ′ ) = 2Φ ( Z, Z ′ ) ξ for all Z, Z ′ ∈ Γ ( D ) , ˜ T ( V, ξ ) = [ ξ, V L ] Q + [ ξ, V Q ] L for all V ∈ Γ (
T M ) ,where ˜ T denotes the torsion tensor field of ˜ ∇ . Furthermore ˜ ∇ is uniquely determined, L and Q are integrable and coincide with the eigenspaces of the operator h .Proof. The proof is rather obvious in one direction, it is sufficient to take as ˜ ∇ the bi-Legendrian connection associated to the bi-Legendrian structure defined by the eigenspacesof h . Now we prove the converse. Note that by (ii) it follows also that ξ is parallel withrespect to ˜ ∇ , because for any V ∈ Γ (
T M ) ( ˜ ∇ V η ) ξ = − η ( ˜ ∇ V ξ ) = 0, so ˜ ∇ V ξ ∈ Γ ( D ).On the other hand for any Z ∈ Γ ( D ), since ˜ ∇ is a metric connection and preserves thesubbundle D = L ⊕ Q , we have g ( ˜ ∇ V ξ, Z ) = V ( g ( ξ, Z )) − g ( ξ, ˜ ∇ V Z ) = 0 , from which ˜ ∇ V ξ is also orthogonal to D hence vanishes. Now we can prove the result.We show first that dη = Φ, so M is a contact metric manifold. For any X, X ′ ∈ Γ ( L )and Y, Y ′ ∈ Γ ( Q ) we have dη ( X, X ′ ) = 0 = g ( X, φX ′ ) and dη ( Y, Y ′ ) = 0 = g ( Y, φY ′ ).Moreover 2Φ ( X, Y ) ξ = ˜ T ( X, Y ) = ˜ ∇ X Y − ˜ ∇ Y X − [ X, Y ]from which 2Φ (
X, Y ) = g ( ˜ ∇ X Y, ξ ) − g ( ˜ ∇ Y X, ξ ) − g ([ X, Y ] , ξ ) . (4.3)Now, g ( ˜ ∇ X Y, ξ ) = X ( g ( Y, ξ )) − g ( Y, ˜ ∇ X ξ ) = 0 and, analogously, g ( ˜ ∇ Y X, ξ ) = 0, so that(4.3) becomes 2Φ (
X, Y ) = − η ([ X, Y ]) , dη ( X, Y ) = Φ (
X, Y ). For concluding that (
M, φ, ξ, η, g ) is acontact metric manifold it remains to check that dη ( Z, ξ ) = Φ (
Z, ξ ) for any Z ∈ Γ ( D ).Indeed dη ( Z, ξ ) = − η ([ Z, ξ ]) = 0 = Φ (
Z, ξ ) since[
Z, ξ ] = ˜ ∇ Z ξ − ˜ ∇ ξ Z − ˜ T ( Z, ξ ) = − ˜ ∇ ξ Z − [ ξ, Z L ] Q − [ ξ, Z Q ] L ∈ Γ ( D )because of (i). Therefore ( M, φ, ξ, η, g ) is a contact metric manifold endowed with twocomplementary (in particular orthogonal) Legendrian distributions L and Q , and since˜ ∇ ξ = 0 the connection ˜ ∇ coincides with the bi-Legendrian connection ¯ ∇ associated to( L, Q ). This fact and (iii) imply the integrability of L and Q . Indeed for any X, X ′ ∈ Γ ( L )we have [ X, X ′ ] Q = − ¯ T ( X, X ′ ) = − ˜ T ( X, X ′ ) = − dη ( X, X ′ ) ξ = 0and g ([ X, X ′ ] , ξ ) = η ([ X, X ′ ]) = − dη ( X, X ′ ) = 0 , hence [ X, X ′ ] ∈ Γ ( L ), and in a similar manner one can prove the integrability of Q . Thus L and Q define two orthogonal Legendrian foliations on M and now the result followsfrom Theorem 4.1.The connection ˜ ∇ is, under certain points of view, an ”invariant” of the contact metric( κ, µ )-space unless D -homothetic deformations. Indeed, by a direct computation, one hasthe following result. Proposition 4.2.
The bi-Legendrian connection associated to a contact metric ( κ, µ ) -space remains unchanged under a D -homothetic deformation. The connection stated in Theorem 4.2 should be compared to the Tanaka-Websterconnection of a non-degenerate integrable CR-manifold (cf. [17], [22]) and to the gener-alized Tanaka-Webster connection introduced by Tanno in [20]. This can be seen in thefollowing theorem, where we prove, using Theorem 4.2, the already quoted result that anycontact metric ( κ, µ )-space is a strongly pseudo-convex CR-manifold.
Corollary 4.2.
Any contact metric ( κ, µ ) -space is a strongly pseudo-convex CR-manifold.Proof. We define a connection on M as follows. We putˆ ∇ V W = (cid:26) ¯ ∇ V W, if V ∈ Γ ( D ); − φhW + [ ξ, W ] , if V = ξ .Then it easy to check that ˆ ∇ coincides with the Tanaka-Webster connection of M and sowe get the assertion. 13he above characterization may be also a tool for proving properties on ( κ, µ )-spaces.As an application we show in a very simple way that an invariant submanifold of a contactmetric metric ( κ, µ )-space, that is a submanifold N such that φT p N ⊂ T p N for all p ∈ N ,is in turn a contact metric ( κ, µ )-space (cf. [21]). Corollary 4.3.
Any invariant submanifold of a contact metric ( κ, µ ) -space is in turn a ( κ, µ ) -space.Proof. It is well-known (cf. [3]) that an invariant submanifold of a contact metric man-ifold inherits a contact metric structure by restriction. Now let N m +1 be an invariantsubmanifold of M n +1 and consider the distribution on N given by L x := T x N ∩ D λx and Q x := T x N ∩ D − λx for all x ∈ N . It is easy to check that L and Q define two mutuallyorthogonal Legendrian foliations of N m +1 and that the bi-Legendrian connection corre-sponding to ( L, Q ) is just the connection induced on N by the bi-Legendrian connectionassociated to ( D λ , D − λ ). The result now follows from Theorem 4.2.We conclude showing that the assumption in Theorem 4.2 that ξ must be not Killingis essential. This can be seen in the following example. Example 4.1.
Consider the sphere S = (cid:8) ( x , x , x , x ) ∈ R : x + x + x + x = 1 (cid:9) with the following Sasakian structure: η = x dx + x dx − x dx − x dx , ξ = x ∂∂x + x ∂∂x − x ∂∂x − x ∂∂x ,g = , φ = − −
11 0 0 00 1 0 0 . Set X := x ∂∂x − x ∂∂x − x ∂∂x + x ∂∂x and Y := φX = x ∂∂x − x ∂∂x + x ∂∂x − x ∂∂x ,and consider the 1-dimensional distributions L and Q on S generated by X and Y ,respectively. An easy computation shows that [ X, ξ ] = − Y , [ Y, ξ ] = 2 X , [ X, Y ] =2 ξ . Thus L and Q defines two non-degenerate, orthogonal Legendrian foliations on theSasakian manifold ( S , φ, ξ, η, g ). For the bi-Legendrian connection corresponding to thisbi-Legendrian structure, we have, after a straightforward computation, ¯ ∇ X X = ¯ ∇ X Y =¯ ∇ X ξ = 0 and ¯ ∇ Y X = ¯ ∇ Y Y = ¯ ∇ Y ξ = 0. Therefore ¯ ∇ φ = 0 and so, by Proposition 2.2,also ¯ ∇ g = 0. Moreover, as ξ is Killing obviously ¯ ∇ h = 0. References [1] D. E. Blair -
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