Invariant pseudo-Sasakian and K-contact structures on seven-dimensional nilpotent Lie groups
aa r X i v : . [ m a t h . DG ] J a n N. K. SmolentsevInvariant pseudo-Sasakian and K -contact structures onseven-dimensional nilpotent Lie groups Abstract
We study the question of the existence of left-invariant Sasaki contact structures on theseven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasakistructure with a positive definite metric tensor is the Heisenberg group. We find a completelist of the 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Sasakistructure. We also present a list of 25 classes of seven-dimensional nilpotent Lie groupsadmitting a K -contact structure, but not the pseudo-Sasaki structure. All the contactstructures considered are central extensions of six-dimensional nilpotent symplectic Liegroups and are established formulas that connect the geometrical characteristics of the six-dimensional nilpotent almost pseudo-K¨ahler Lie groups and seven-dimensional nilpotentcontact Lie groups. It is known that for the six-dimensional nilpotent pseudo-K¨ahler Liegroups the Ricci tensor is always zero. Unlike the pseudo-K¨ahlerian case, it is shown thaton contact seven-dimensional algebras the Ricci tensor is nonzero even in directions of thecontact distribution. A left-invariant
K¨ahler structure on a Lie group H is a triple ( h, ω, J ) , consisting of a left-invariant Riemannian metric h , a left-invariant symplectic form ω and an orthogonal left-invariant complex structure J , where h ( X, Y ) = ω ( X, J Y ) for any left-invariant vector fields X and Y on H . Therefore, such a structure on the group H can be defined as a pair ( ω, J ) ,where ω is the symplectic form, and J is a complex structure compatible with ω , i.e., suchthat ω ( J X, J Y ) = ω ( X, Y ) . If ω ( X, J X ) > , ∀ X = 0 , it becomes a K¨ahler metric, and ifthe positivity condition is not met, then h ( X, Y ) = ω ( X, J Y ) is a pseudo-Riemannian metric.Then, ( h, ω, J ) is called the pseudo-K¨ahler structure on the Lie group H . The left-invariance ofthese objects implies that the (pseudo) K¨ahler structure ( h, ω, J ) can be defined by the valuesof the ω , J and h on the Lie algebra h of the Lie group H . Then, the ( h , ω, J, h ) is calleda pseudo-K¨ahler Lie algebra . Conversely, if ( h , J, h ) is a Lie algebra endowed with a complexstructure J , orthogonal with respect to the pseudo-Riemannian metric h , then the equality ω ( X, Y ) = h ( J X, Y ) determines the (fundamental) two-form ω , which is closed if and only ifthe J is parallel [9].Classification of the six-dimensional real nilpotent Lie algebras admitting invariant complexstructures and estimation of the dimensions of moduli spaces of such structures is obtained in[11]. In article [7], classification is obtained of symplectic structures on six-dimensional nilpotentLie algebras. The condition of existence of left-invariant positive definite K¨ahler metric on theLie group H imposes strong restrictions on the structure of its Lie algebra h . For example,Benson and Gordon have shown [1] that this Lie algebra cannot be nilpotent except in theabelian case. The nilpotent Lie groups and nilmanifolds (except tora) do not allow the left-invariant K¨ahler metrics, however, such manifolds may exist as a pseudo-left-invariant K¨ahlermetric. The article in [5] provides a complete list of the six-dimensional nilpotent Lie algebrasadmitting pseudo-K¨ahler structures. A more complete study of the properties of the curvatureof pseudo-K¨ahler structures is carried out in [12].1he analogues of symplectic structures in an odd case are contact structures [2]. It isknown [7] that the contact Lie algebra g is a central extension of symplectic Lie algebras ( h , ω ) with the help of a non-degenerate cocycle ω . In this case, a contact Lie algebra g admits Sasakian structure only if the Lie algebra ( h , ω ) admits a K¨ahler metric. Therefore, thequestion of the existence of Sasakian structures on the seven-dimensional nilpotent contact Liealgebra g is reduced to the question of the existence of K¨ahler structures on six-dimensionalnilpotent symplectic Lie algebras h = g /Z ( g ) , where Z ( g ) is the center of contact Lie algebra g . The classification of seven-dimensional nilpotent contact Lie algebras is obtained in [10].The authors of [4] found examples of K -contact but not Sasakian structures on the seven-dimensional nilpotent contact Lie algebras. Just as in the even-dimensional nilpotent case, thereare topological obstructions to the existence of Sasakian structures on nilpotent Lie algebra,based on the strong Lefschetz theorem for contact manifolds [3].In this paper, we show that on seven-dimensional nilpotent Lie algebras the invariant contactSasakian structures exist only in the case of the Heisenberg algebra, and pseudo-Sasakian (i.e.,having a pseudo-Riemannian metric tensor) there are only 22 classes of central extensions ofsix-dimensional pseudo K¨ahler nilpotent Lie algebras. This article provides a comprehensivelist of such pseudo-Sasakian structures and their curvature properties are investigated. Wealso obtain the list of 25 classes of seven-dimensional nilpotent contact Lie algebras that allow K -contact structure, but do not allow pseudo-Sasakian structures. The basic concepts of contact manifolds and contact Lie algebra are summarized here.
In this section, we recall some basic definitions and properties in contact Riemannian geometry.For further details we refer the reader to the monographs in [2]. Let M be a smooth manifoldof dimension n + 1 . A 1-form η on M is called a contact form if η ∧ ( dη ) n = 0 is a volume form.Then the pair ( M, η ) is called a (strict) contact manifold . In any contact manifold one provesthe existence of a unique vector field ξ , called the Reeb vector field , satisfying the properties η ( ξ ) = 1 and dη ( ξ, X ) = 0 for all vector fields X on M .The contact form η on M determines the distribution of D = { X ∈ T M | η ( X ) = 0 } ofdimension n , which is called the contact distribution . It is easy to see that L ξ η = 0 . If M is a contact manifold with a contact form η , then the contact metric structure is called thequadruple ( η, ξ, φ, g ) , where ξ is the Reeb field, g is the Riemannian metric on M and φ is theaffinor on M , and for which there are the following properties [2]:1. φ = − I + η ⊗ ξ ;2. dη ( X, Y ) = g ( X, φY ) ;3. g ( φX, φY ) = g ( X, Y ) − η ( X ) η ( Y ) ,where I is the identity endomorphism of the tangent bundle. The Riemannian metric g is calledthe contact metric structure associated with the contact structure η . From the third propertyit follows immediately that the associated metric g for the contact structure η is completelydetermined by the affinor φ : g ( X, Y ) = dη ( φX, Y ) + η ( X ) η ( Y ) .2he contact metric manifold whose Reeb vector field ξ is Killing, L ξ g = 0 , is called K -contact [2]. This last property is equivalent to the condition L ξ φ = 0 . Recall from [2] that an almost contact structure on a manifold M is a triple ( η, ξ, φ ) , where η is a 1-form, ξ is a vectorfield and φ is an affinor on M with properties: η ( ξ ) = 1 and φ = − I + η ⊗ ξ .Suppose that M is an almost contact manifold. Consider the direct product M × R . Avector field on M × R is represented in the form ( X, f d/dt ) , where X is the tangent vector to M , t is the coordinate on R and f is a function of class C ∞ on M × R . We define an almost complexstructure J on the direct product M × R as follows [2]: J ( X, f d/dt ) = ( φX − f ξ, η ( X ) d/dt ) .An almost contact structure ( η, ξ, φ ) is called normal if the almost complex structure J isintegrable. A four tensor N (1) , N (2) , N (3) and N (4) is defined [2] on an almost contact manifold M by the following expressions: N (1) ( X, Y ) = [ φ, φ ]( X, Y ) + dη ( X, Y ) ξ, N (2) ( X, Y ) = ( L φX η )( Y ) − ( L φY η )( X ) ,N (3) ( X, Y ) = ( L ξ φ ) X, N (4) ( X, Y ) = ( L ξ η )( X ) . An almost contact structure ( η, ξ, φ ) is a normal, if these tensors vanish. However, it canbe shown from the vanishing tensor N (1) that the remaining tensors N (2) , N (3) and N (4) alsovanish. Therefore, the condition of normality is only the following: N (1) ( X, Y ) = [ φ, φ ]( X, Y ) + dη ( X, Y ) ξ = 0 . Thus, a Sasaki manifold is a normal contact metric manifold.
Definition 2.1.
A pseudo-Riemannian contact metric structure ( η, ξ, φ, g ) is called a K -contact,if the vector field ξ is a Killing vector field. A pseudo-Riemannian contact metric structure ( η, ξ, φ, g ) is called pseudo-Sasakian if N (1) ( X, Y ) = 0 . Remark.
In this paper, we assume that the exterior product and exterior differential isdetermined without a normalizing factor. Then, in particular, dx ∧ dy = dx ⊗ dy − dy ⊗ dx and dη ( X, Y ) = Xη ( Y ) − Y η ( X ) − η ([ X, Y ]) . We also assume that the curvature tensor R is definedby the formula: R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z . The Ricci tensor Ric is defined asa contraction of the curvature tensor in the first and the fourth (top) indices:
Ric ij = R kkij . In[2] Blair shows a different formula for the tensor N (1) : N (1) ( X, Y ) = [ φ, φ ]( X, Y ) + 2 dη ( X, Y ) ξ .In [2], a different formula is used for the exterior derivative d η ( X, Y ) = ( Xη ( Y ) − Y η ( X ) − η ([ X, Y ])) , which corresponds to the selection of wedge product 1-forms as dx ∧ dy = ( dx ⊗ dy − dy ⊗ dx ) . This is evident then in determining the associated metrics g ( X, Y ) = dη ( φX, Y )+ η ( X ) η ( Y ) . In our case, the associated Riemannian metric g will be different along the contactdistribution D from of the associated Riemannian metric g adopted in [2]: if X, Y ∈ D , then g ( X, Y ) = 2 g ( X, Y ) . It further appears that the formula ∇ X ξ = − φX in our case looks like: ∇ X ξ = − φX . In addition in [2], the sectional curvature for the K -contact metric manifoldin the direction of the 2-plane, containing ξ , is equal to 1 (p. 113, Theor. 7.2), and in ourcase, the sectional curvature is equal to . Consequently this changes the formula for the Riccicurvature, Ric ( ξ, ξ ) = n/ .When a manifold is taken Lie group G , it is natural to consider the left-invariant contactstructure. In this case, the contact form η , Reeb vector field ξ , affinor φ and associated metric g are defined by its values in the unit e , that is, on the Lie algebra g . We will call the g a contact Lie algebra if it is defined in a contact form η ∈ g ∗ and the vector ξ ∈ g , such that η ∧ ( dη ) n = 0 , η ( ξ ) = 1 and dη ( ξ, X ) = 0 . Note that dη ( x, y ) = − η ([ x, y ]) . In a similar sense, itis considered a contact metric Lie algebra, symplectic
Lie algebra, pseudo-K¨ahler
Lie algebra, (pseudo) Sasakian
Lie algebra, etc. 3et ∇ be the Levi-Civita connection corresponding to the (pseudo) Riemannian metric g .It is determined from the six-membered formula [9], which for the left-invariant vector fields X, Y, Z on the Lie group takes the form: g ( ∇ X Y, Z ) = g ([ X, Y ] , Z )+ g ([ Z, X ] , Y )+ g ( X, [ Z, Y ]) .Recall also that if R ( X, Y ) Z is the curvature tensor, then the Ricci tensor Ric ( X, Y ) for thepseudo-Riemannian metric g is defined by the formula: Ric ( X, Y ) = n +1 X i =1 ε i g ( R ( e i , Y ) Z, e i ) , where { e i } is a orthonormal frame on g and ε i = g ( e i , e i ) . Contact Lie algebra can be obtained as a result of the central expansion of the symplecticLie algebra h . Recall this procedure. If there is a symplectic Lie algebra ( h , ω ) , the centralextension g = h × ω R is a Lie algebra in which the Lie brackets are defined as follows: [ X, ξ ] g = 0 , [ X, Y ] g = [ X, Y ] h + ω ( X, Y ) ξ for any X, Y ∈ h , where ξ = d/dt is the unit vector in R .On the Lie algebra g = h × ω R contact form given by the form η = ξ ∗ , and ξ = d/dt isthe Reeb field. If x = X + λξ and y = Y + µξ , where X, Y ∈ h , λ, µ ∈ R , then: dη ( x, y ) = − η ([ x, y ]) = − ξ ∗ ([ X, Y ] h + ω ( X, Y ) ξ ) = − ω ( X, Y ) .As is known, the isomorphism classes of the central extensions of the Lie algebra h are inone-to-one correspondence with the elements of H ( h , R ) . Non-degenerate elements of H ( h , R ) (the symplectic Lie algebra) define the contact structures on h × ω R .To define the affinor φ on g = h × ω R we can use an almost complex structure J on h as follows: if x = X + λξ , where X ∈ h , then φ ( x ) = J X . If this almost complex structure J on h is also compatible with the ω , that is, it has the property of ω ( J X, J Y ) = ω ( X, Y ) ,we will get the contact (pseudo) metric structure ( η, ξ, φ, g ) on g = h × ω R , where g ( X, Y ) = dη ( φX, Y ) + η ( X ) η ( Y ) . Let h ( X, Y ) = ω ( X, J Y ) be a associated (pseudo) Riemannian metricon the symplectic Lie algebra ( h , ω ) . Then for x = X + λξ and y = Y + µξ , we have: g ( x, y ) = − ω ( J X, Y ) + λµ = h ( X, Y ) + λµ.
As is known [9], an almost complex structure J is integrable (complex), if the Nijenhuis tensorvanishes, N J ( X, Y ) = [
J X, J Y ] − [ X, Y ] − J [ X, J Y ] − J [ J X, Y ] = 0 . An analogue of the Nijenhuis tensor of the almost complex structure in the case of any tensorfield T of type (1,1) is the Nijenhuis torsion [2]: [ T, T ]( X, Y ) = T [ X, Y ] + [
T X, T Y ] − T [ X, T Y ] − T [ T X, Y ] . Proposition 2.2.
A central extension g = h × ω R of almost pseudo-K¨ahler Lie algebra h is K -contact Lie algebra.Proof. As is well known [2], a contact manifold is called K -contact if L ξ φ = 0 . For left-invariantfields of the form x = X + λξ and y = Y + µξ where X, Y ∈ h , we have: g (( L ξ φ ) x, y ) = g ( L ξ ( φ x ) − φ ( L ξ x ) , y ) = 0 , because the L ξ x = [ ξ, X + λξ ] = 0 and L ξ ( φ x ) = 0 . Therefore L ξ φ = 0 .4 roposition 2.3. Let ( h , ω, J ) be a almost (pseudo) K¨ahler Lie algebra h and ( η, ξ, φ, g ) bethe corresponding contact metric structure on the central extension of g = h × ω R . Then theNijenhuis torsion [ φ, φ ] on g is expressed in terms of the Nijenhuis tensor N J almost complexstructure J on h as: [ φ, φ ]( x, y ) = N J ( X, Y ) − dη ( x, y ) ξ, where x = X + λξ , y = Y + µξ and X, Y ∈ h .Proof. Direct calculations: [ φ, φ ]( x, y ) = [ φ, φ ]( X + λξ, Y + µξ ) == φ [ X + λξ, Y + µξ ] + [ φ ( X + λξ ) , φ ( Y + µξ )] − φ [ X + λξ, φ ( Y + µξ )] − φ [ φ ( X + λξ ) , Y + µξ ] == φ ([ X, Y ] h ) + [ J ( X ) , J ( Y )] − φ [ X + λξ, J ( Y )] − φ [ J ( X ) , Y + µξ ] == − [ X, Y ] h + [ J ( X ) , J ( Y )] h + ω ( J X, J Y ) ξ − φ [ X, J ( Y )] − φ [ J ( X ) , Y ] == − [ X, Y ] h +[ J ( X ) , J ( Y )] h + ω ( X, Y ) ξ − φ ([ X, J ( Y )] h + ω ( X, J Y ) ξ ) − φ ([ J ( X ) , Y ] h + ω ( J X, Y ) ξ ) == − [ X, Y ] h + [ J ( X ) , J ( Y )] h + ω ( X, Y ) ξ − J ([ X, J ( Y )] h ) − J ([ J ( X ) , Y ] h ) == N J ( X, Y ) + ω ( X, Y ) ξ = N J ( X, Y ) − dη ( x, y ) ξ. Corollary 2.4.
The tensor N (1) ( x, y ) of the contact metric structure ( η, ξ, φ, g ) on the centralexpansion g = h × ω R expressed in terms of the Nijenhuis tensor N J of almost complex structure J on h using the formula: N (1) ( x, y ) = N J ( X, Y ) , where x = X + λξ , y = Y + µξ and X, Y ∈ h .Proof. N (1) ( x, y ) = [ φ, φ ]( x, y ) + dη ( x, y ) ξ = N J ( X, Y ) − dη ( x, y ) ξ + dη ( x, y ) ξ = N J ( X, Y ) . Corollary 2.5.
Contact metric structure ( η, ξ, φ, g ) on the central expansion g = h × ω R is(pseudo) Sasakian if and only if the symplectic algebra ( h , ω, J ) is a (pseudo) K¨ahler. As mentioned in the introduction, left-invariant K¨ahler (i.e. positive-definite) structuresdo not exist on nilpotent Lie groups other than the torus [1]. However, the pseudo-K¨ahlerstructure of these Lie groups may exist and in [5] a complete list of these is given. For eachLie algebra on the list in [5] an example is selected of a nilpotent complex structure, and theagreed symplectic forms are found for this. It is more correct to rely on the classification listproduced by Goze, Khakimdjanov and Medina [7], which lists all symplectic six-dimensionalLie algebras and shows that every nilpotent Lie algebra symplecto-isomorphic is on this list.From this point of view, the author in [12] studied the above Lie algebra with the symplecticstructure from the list [7] and found all compatible complex structures. Explicit expressions ofcomplex structures were obtained and the curvature properties of the corresponding pseudo-Riemannian metrics investigated. It was found that there are multi-parameter familyies ofcomplex structures. However, they all share a number of common properties: the associatedpseudo K¨ahler metric is Ricci-flat, the Riemann tensor has zero pseudo-norm, and the Riemanntensor has a few nonzero components, which depend only on two or, at most, three parameters.Recall also that [11] presents a classification of the six-dimensional real nilpotent Lie algebrasadmitting an invariant complex structure and an estimation of the dimension of the modulispaces of such structures is given. 5 xample
Consider the Lie algebra h with commutation relations [ e , e ] = e , [ e , e ] = e , [ e , e ] = e and symplectic form ω = − e ∧ e + e ∧ e + e ∧ e . In [12] it is shown that there is asix-parametric family of pseudo-K¨ahler metrics on h , each of which is defined by the operatorof the complex structure of the form: J = ψ ψ − ψ +1 ψ − ψ ψ ( ψ +1) − ψ ψ ψ ψ − ψ − ψ − ψ +1 ψ ψ ψ ψ ψ ψ J ψ ψ ψ ψ ψ − ψ − ψ ψ ( ψ +1) − ψ ψ ψ ψ − ψ +1 ψ − ψ , where J = − ψ ψ ( ψ ψ − ψ ψ )+ ψ ( ψ +1)+ ψ ( ψ + ψ ψ )( ψ +1) ψ and ψ = 0 . The correspondingpseudo-Riemannian metric is easily obtained in the form h ( X, Y ) = ω ( X, J Y ) . The curvaturetensor is zero for all values of the parameters ψ ij . In this section, we will determine when the seven-dimensional nilpotent Lie algebra there areSasakian and pseudo-Sasakian structure and expresses explicitly the properties of contact struc-tures on a central expansion g = h × ω R through the corresponding properties of the symplecticLie algebra ( h , ω ) . Theorem 3.1.
The only seven-dimensional nilpotent Lie algebra admitting a positive definiteSasakian structure is the Heisenberg algebra. A seven-dimensional nilpotent Lie algebra admitspseudo-Sasakian structure if and only if it is one of the algebras on the following list: g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e η = e , dη = e ∧ e + e ∧ e − e ∧ e g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e η = e , dη = − ( − e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = ( λ − e η = e , dη = − ( e ∧ e + λe ∧ e + ( λ − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( − e ∧ e + e ∧ e + e ∧ e + e ∧ e ) , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = − e η = e , dη = − e ∧ e + e ∧ e + e ∧ e + e ∧ e g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = λe , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e + λe ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − λe , η = e , dη = e ∧ e + e ∧ e − e ∧ e + λe ∧ e g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , η = e , dη = e ∧ e + e ∧ e − e ∧ e − e ∧ e g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = e , [ e , e ] = ( λ + 1) e , η = e , dη = − ( λe ∧ e + e ∧ e + ( λ + 1) e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = − e , η = e , dη = e ∧ e + e ∧ e + e ∧ e g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , η = e , dη = − ( e ∧ e − e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = ( λ − e , η = e , dη = − ( e ∧ e + λe ∧ e + ( λ − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = − λe , [ e , e ] = e , [ e , e ] = − λe , η = e , dη = − ( e ∧ e + λe ∧ e − λe ∧ e + e ∧ e − λe ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = 2 e , [ e , e ] = e , η = e , dη = − ( − e ∧ e + e ∧ e + 2 e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) Proof.
It is well known that the Heisenberg algebra h n +1 is a central extension of the symplecticabelian Lie algebra, which admits the K¨ahler metric. Therefore, h n +1 admits Sasaki structure.According to Corollary 2 of the previous section, pseudo-Sasakian structures are central ex-tensions of pseudo-K¨ahler Lie algebra ( h , ω, J ) . The authors of [5] found all six-dimensionalnilpotent Lie algebras which admit pseudo-K¨ahler structure. In [7], the classification of sym-plectic six-dimensional nilpotent Lie algebras up to symplecto-isomorphisms is obtained. Theauthor of [12] found all the symplectic Lie algebras on the listing by Goze, Khakimdjanov andMedina which admit compatible complex structures, i.e., which are pseudo-K¨ahler. The Liealgebras of the list of the theorem are a central extension of the pseudo-K¨ahler Lie algebrasfound in [12], based on the classification by Goze, Khakimdjanov and Medina. The indicesof the Lie algebras correspond to their numbers in the classification list [7], and their ordercorresponds to their properties specified in [12].Since the contact metric structure ( η, ξ, φ, g ) is a central extension g = h × ω R of almostpseudo-K¨ahler structures on h , then it is clear that a seven-dimensional nilpotent Lie algebracannot be (pseudo) Sasakian when the symplectic algebra ( h , ω, J ) is not (pseudo) K¨ahler, i.e.,does not admit a compatible integrable almost complex structure J . Since the complete list ofthe six-dimensional nilpotent symplectic Lie algebras admitting compatible complex structure J is given in [5], then the remaining 12 classes of symplectic six-dimensional nilpotent Lie alge-bras listed in [7] do not admit Sasaki structures even with the pseudo-metric. In addition, manyof the Lie algebras in [5] can have several non-isomorphic symplectic structures [7]. Moreover,some symplectic forms of the same Lie algebra allow compatible complex structures, while oth-ers do not. The central extensions of the last Lie algebras also do not admit pseudo-Sasakianstructures. Therefore, we have the following theorem in which the central extensions of sym-plectic Lie algebras are listed first which do not allow complex structures in general, and thenthe central extension of symplectic Lie algebras is specified, which may have complex structuresbut which do not allow the compatible complex structures of the considered symplectic form. Theorem 3.2.
The following nilpotent Lie algebras admit K -contact structure, but do notadmit Sasaki structures even with pseudo-Riemannian metric: g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = (1 − λ ) e , [ e , e ] = λe , η = e , dη = − ( e ∧ e + (1 − λ ) e ∧ e + λe ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = − λe , η = e , dη = − λ ( e ∧ e + e ∧ e + e ∧ e − e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , η = e , dη = − ( e ∧ e − e ∧ e + e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , η = e , dη = − λ e ∧ e − λ ( e ∧ e + e ∧ e + e ∧ e + e ∧ e ) , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e , η = e , dη = − λ e ∧ e − λ ( e ∧ e + e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = − λe , [ e , e ] = − λe , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = λe , η = e , dη = − λ ( e ∧ e − e ∧ e − e ∧ e + e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = − λe , [ e , e ] = λe , [ e , e ] = − λe , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = λe , η = e , dη = − λ ( e ∧ e − e ∧ e + e ∧ e − e ∧ e + e ∧ e + e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = 2 λe , [ e , e ] = λe , [ e , e ] = 2 λe , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = λe , η = e , dη = − λ (2 e ∧ e + e ∧ e + 2 e ∧ e + e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , η = e , dη = e ∧ e + e ∧ e + e ∧ e − e ∧ e g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = − λe , η = e , dη = − λ ( e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , η = e , dη = e ∧ e + e ∧ e − e ∧ e g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = λe , [ e , e ] = − λe , η = e , dη = − λ ( e ∧ e + e ∧ e − e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e ) g [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( − e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e , η = e , dη = − ( e ∧ e + e ∧ e − e ∧ e − e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = λe , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + λe ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) g , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , η = e , dη = − ( e ∧ e + e ∧ e + e ∧ e ) In [12] it is shown that the pseudo-K¨ahler structure on a six-dimensional nilpotent Lie algebrahas a zero Ricci tensor. However, for Sasaki structures the Ricci tensor is non-zero, even ifsuch a structure is obtained from a pseudo-K¨ahler Lie algebra by central extension. Therefore,in this section we establish the formulas, the binding properties of the curvature of almostpseudo-K¨ahler Lie algebras and the contact structures obtained by central extensions.
Lemma 3.3.
Let ( ω, J, h ) be a almost (pseudo) K¨ahler structure on the Lie algebra h and ( η, ξ, φ, g ) be the corresponding contact metric structure on the central extension g = h × ω R .Then the covariant derivative ∇ on g expressed in terms of the covariant derivative D on h ,symplectic form ω and almost complex structure J on h are as follows: ∇ X Y = D X Y + 12 ω ( X, Y ) ξ, ∇ X ξ = ∇ ξ X = − J X и ∇ ξ ξ = 0 . where X, Y ∈ h .Proof. Direct calculations using a six-membered formula [9], which is for the left-invariantvector fields
X, Y, Z on the Lie group takes the form: g ( ∇ X Y, Z ) = g ([ X, Y ] , Z )+ g ([ Z, X ] , Y )+ g ( X, [ Z, Y ]) . Take two left-invariant vector fields of the form x = X + λξ and y = Y + µξ , where X, Y ∈ h . Then, using the formula [ X, Y ] g = [ X, Y ] h + ω ( X, Y ) ξ , [ X, ξ ] g = 0 and orthogonality h to ξ , we obtain for X, Y ∈ h : g ( ∇ X Y, Z ) = h ([ X, Y ] h , Z ) + h ([ Z, X ] h , Y ) + h ( X, [ Z, Y ] h ) = 2 h (D X Y, Z ) , g ( ∇ X Y, ξ ) = g ([ X, Y ] g , ξ ) + g [ ξ, X ] g , Y ) + g ( X, [ ξ, Y ] g ) = ω ( X, Y ) . Consequently, ∇ X Y = D X Y + ω ( X, Y ) ξ . Further, considering h ( X, Y ) = ω ( X, J Y ) : g ( ∇ X ξ, Z ) = g ([ X, ξ ] g , Z ) + g [ Z, X ] g , ξ ) + g ( X, [ Z, ξ ] g ) = ω ( Z, X ) == − ω ( Z, J J X ) = − h ( Z, J X ) = − g ( J X, Z ) , g ( ∇ X ξ, ξ ) = g ([ X, ξ ] g , ξ ) + g [ ξ, X ] g , ξ ) + g ( X, [ ξ, ξ ] g ) = 0 . Therefore, ∇ X ξ = − J X ∈ h . Similarly, we obtain: ∇ ξ X = − J X . Theorem 3.4.
Let ( ω, J, h ) be a almost (pseudo) K¨ahler structure on the Lie algebra h and ( η, ξ, φ, g ) be the corresponding contact metric structure on the central extension g = h × ω R .Then the curvature tensor R of g expressed in terms of the curvature tensor R h of h , symplecticform ω and almost complex structure J on h are given by the formulas: R ( X, Y ) Z = R h ( X, Y ) Z −
12 (D Z ω )( X, Y ) ξ −
14 ( ω ( Y, Z ) J X − ω ( X, Z ) J Y ) + 12 ω ( X, Y ) J Z,R ( X, Y ) ξ = −
12 ((D X J ) Y − (D Y J ) X ) ,R ( X, ξ ) Z = −
12 (D X J ) Z − g ( X, Z ) ξ, R ( X, ξ ) ξ = 14 X, where X, Y ∈ h .Proof. This consists of direct calculation using the formula R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z −∇ [ X,Y ] Z using formulas for covariant derivatives obtained in Lemma 3.3. For example, letus take first the left-invariant vector field of the form X, Y, Z ∈ h . Then, using the formula [ X, Y ] g = [ X, Y ] h + ω ( X, Y ) ξ , [ X, ξ ] g = 0 , orthogonality h to ξ , and ∇ X Y = D X Y + ω ( X, Y ) ξ ,we obtain: R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z == ∇ X { D Y Z + 12 ω ( Y, Z ) ξ } − ∇ Y { D X Z + 12 ω ( X, Z ) ξ } − ∇ [ X,Y ] h + ω ( X,Y ) ξ Z == D X D Y Z + 12 ω ( X, D Y Z ) ξ + 12 ω ( Y, Z )( − J X ) −{ D Y D X Z + 12 ω ( Y, D X Z ) ξ + 12 ω ( X, Z )( − J Y ) }−− { D [ X,Y ] Z + 12 ω ([ X, Y ] , Z ) ξ } − ω ( X, Y ) ∇ ξ Z = R h ( X, Y ) Z ++ 12 { ω ( X, D Y Z ) − ω ( Y, D X Z ) − ω ([ X, Y ] , Z ) } ξ − { ω ( Y, Z ) J X − ω ( X, Z ) J Y } + 12 ω ( X, Y ) J Z = To simplify ω ( X, D Y Z ) − ω ( Y, D X Z ) − ω ([ X, Y ] , Z ) in the last expression, we use the properties D X Z = D Z X + [ X, Z ] and D Y Z = D Z Y + [ Y, Z ] . Then, considering the closedness of theform ω , ω ( X, [ Y, Z ]) + ω ( Y, [ Z, X ]) + ω ( Z, [ X, Y ]) = dω ( X, Y, Z ) = 0 and equality Zω ( X, Y ) =(D Z ω )( X, Y ) + ω (D Z X, Y ) + ω ( X, D Z Y ) = 0 , we obtain: ω ( X, D Y Z ) − ω ( Y, D X Z ) − ω ([ X, Y ] , Z ) == ω ( X, D Z Y + [ Y, Z ]) − ω ( Y, D Z X + [ X, Z ]) − ω ([ X, Y ] , Z ) == ω ( X, D Z Y ) − ω ( Y, D Z X ) + ω ( X, [ Y, Z ]) − ω ( Y, [ X, Z ]) − ω ([ X, Y ] , Z ) == ω (D Z X, Y ) + ω ( X, D Z Y ) + ω ( X, [ Y, Z ]) + ω ( Y, [ Z, X ]) + ω ( Z, [ X, Y ]) = − (D Z ω )( X, Y ) . = R h ( X, Y ) Z −
12 (D Z ω )( X, Y ) ξ −
14 ( ω ( Y, Z ) J X − ω ( X, Z ) J Y ) + 12 ω ( X, Y ) J Z.
Consider the case of R ( X, Y ) ξ , where X, Y ∈ h . Then, considering equalities ∇ X ξ = − J X and ∇ ξ ξ = 0 we get: R ( X, Y ) ξ = ∇ X ( ∇ Y ξ ) − ∇ Y ( ∇ X ξ ) − ∇ [ X,Y ] ξ = −∇ X ( 12 J Y ) + ∇ Y ( 12 J X ) − ∇ [ X,Y ] h + ω ( X,Y ) ξ ξ == 12 ( −∇ X ( J Y ) + ∇ Y ( J X ) + J [ X, Y ]) == 12 ( − D X J Y − ω ( X, J Y ) ξ + D Y J X + 12 ω ( Y, J X ) ξ + J [ X, Y ]) == 12 ( − (D X J ) Y − J (D X Y ) − ω ( X, J Y ) ξ + (D Y J ) X + J (D Y X ) + 12 ω ( Y, J X ) ξ + J [ X, Y ]) == 12 ( − (D X J ) Y + (D Y J ) X − J (D X Y − D Y X ) − ω ( X, J Y ) ξ + 12 ω ( Y, J X ) ξ + J [ X, Y ]) == 12 ( − (D X J ) Y + (D Y J ) X − J ([ X, Y ]) − g ( X, Y ) ξ + 12 g ( Y, X ) ξ + J [ X, Y ]) == −
12 ((D X J ) Y − (D Y J ) X ) . Now consider the case R ( X, ξ ) Y , where X, Y ∈ h . Similarly, we obtain: R ( X, ξ ) Z = ∇ X ( ∇ ξ Z ) − ∇ ξ ( ∇ X Z ) − ∇ [ X,ξ ] Z == −∇ X ( 12 J Z ) − ∇ ξ (D X Z + 12 ω ( X, Z ) ξ ) = −∇ X ( 12 J Z ) + 12 J (D X Z ) == 12 ( − D X J Z − ω ( X, J Z ) ξ + J (D X Z )) = 12 ( − (D X J ) Z − J (D X Z ) − ω ( X, J Z ) ξ + J (D X Z )) == −
12 (D X J ) Z − ω ( X, J Z ) ξ = −
12 (D X J ) Z − g ( X, Z ) ξ. Similarly, established the last formula, R ( X, ξ ) ξ = ∇ X ( ∇ ξ ξ ) − ∇ ξ ( ∇ X ξ ) − ∇ [ X,ξ ] ξ = −∇ ξ ( − J X ) = 12 ∇ ξ ( J X ) = −
12 12 J ( J X ) = 14 X. In the case of a (pseudo) K¨ahler structure on the Lie algebra h we have Dω = 0 and DJ = 0 .Therefore, the formulas for the curvature will have a simpler form. If additionally submit anexpression of ω ( Y, Z ) J X − ω ( X, Z ) J Y as h ( Z, J Y ) J X − h ( Z, J X ) J Y , we get:
Corollary 3.5.
Let ( ω, J, h ) be a (pseudo) K¨ahler complex structure on the Lie algebra h and ( η, ξ, φ, g ) be the corresponding (pseudo) Sasakian structure at the central expansion g = h × ω R .Then the curvature tensor R on g is expressed in terms of the curvature tensor R h on h , thesymplectic form ω and an almost complex structure J on h as follows: R ( X, Y ) Z = R h ( X, Y ) Z −
14 ( h ( Z, J Y ) J X − h ( Z, J X ) J Y ) + 12 ω ( X, Y ) J Z,R ( X, Y ) ξ = 0 , R ( X, ξ ) Z = − g ( X, Z ) ξ, R ( X, ξ ) ξ = 14 X, where X, Y ∈ h . .2 Ricci tensor Recall that the Ricci tensor
Ric in a pseudo-Riemannian case is defined by the formula:
Ric ( X, Y ) = n +1 X i =1 ε i g ( R ( e i , Y ) Z, e i ) , where { e i } is the orthonormal basis on g and ε i = g ( e i , e i ) . We choose a basis for g in the form { e , . . . , e n , e n +1 } = { E , . . . , E n , ξ } , where E i ∈ h and ξ is the Reeb field. The followingcalculations assume that the index i changes from 1 to n + 1 , and the index j changes from 1to n . In addition, we consider that an almost complex structure on h is integrable, so that h is the (pseudo) K¨ahler Lie algebra. Theorem 3.6.
Let ( ω, J, h ) be (pseudo) K¨ahler structure on the Lie algebra h , and ( η, ξ, φ, g ) corresponds to a contact (pseudo) metric Sasaki structure on a central expansion g = h × ω R .Then the Ricci tensor on g expressed by the Ricci tensor Ric h on h forms ω and an almostcomplex structure J on h as follows: Ric ( Y, Z ) =
Ric h ( Y, Z ) − h ( Y, Z ) ,Ric ( Y, ξ ) = 0 , Ric ( ξ, ξ ) = n/ . where X, Y ∈ h .Proof. In the basis { e , . . . , e n , e n +1 } = { E , . . . , E n , ξ } of the algebra g , where E j ∈ h , forthe Y, Z ∈ h we obtain: Ric ( Y, Z ) = X i ε i g ( R ( e i , Y ) Z, e i ) == X j (cid:18) ε j h ( R h ( E j , Y ) Z, E j ) − ε i h ( ω ( Y, Z ) J E j − ω ( E j , Z ) J Y, E j ) + 12 ε j ω ( E j , Y ) h ( E j , J Z ) (cid:19) ++ g ( R ( ξ, Y ) Z, ξ ) ==
Ric h ( Y, Z ) − X j ε j ω ( Y, Z ) h ( J E j , E j ) + 14 X j ε j ω ( E j , Z ) h ( J Y, E j )++ 12 X j ε j ω ( E j , − J J Y ) g ( E j , J Z ) + g ( 14 g ( Y, Z ) ξ, ξ ) == Ric h ( Y, Z ) + 14 X j ε j ω ( E j , − J J Z ) h ( J Y, E j ) + 14 h ( Y, Z ) − X j ε j g ( E j , J Y ) g ( E j , J Z ) == Ric h ( Y, Z ) − X j ε j h ( E j , J Z ) h ( E j , J Y ) + 14 h ( Y, Z ) − X j ε j g ( E j , J Y ) g ( E j , J Z ) == Ric h ( Y, Z ) − h ( J Y, J Z ) + 14 h ( Y, Z ) − h ( J Y, J Z ) =
Ric h ( Y, Z ) − h ( Y, Z ) . Further,
Ric ( Y, ξ ) = X i g ( R ( e i , Y ) ξ, e i ) = X j ε j g ( R ( E j , Y ) ξ, E j ) + g ( R ( ξ, Y ) ξ, ξ ) = − g ( Y, ξ ) = 0 . ic ( ξ, ξ ) = X i ε i g ( R ( e i , ξ ) ξ, e i ) = X j ε j g ( R ( E j , ξ ) ξ, E j ) + g ( R ( ξ, ξ ) ξ, ξ ) == 14 X j ε j g ( E j , E j ) = 14 X j ε j ε j = n/ . In [12], the invariant pseudo-K¨ahler structures on six-dimensional nilpotent Lie algebrawith the symplectic structure of the classification list [7] are studied in detail. For each of themthere are many compatible complex structures and corresponding pseudo-Riemannian metrics.It was found that they all have common properties: the associated pseudo-Kahler metric isRicci-flat, the Riemann tensor has zero pseudo-norm, and the Riemann tensor has severalnonzero components which depend only on two or, at most, three parameters. The author of[12] found a pseudo-K¨ahler structure depending only on the parameters that have an impact oncurvature. Such metrics are called canonical. The curvature properties of the almost pseudo-K¨ahler structures on six nilpotent Lie algebra are obtained in [13]. Each (almost) pseudo-K¨ahlerstructure on a six-dimensional nilpotent Lie algebra determines the pseudo-Riemannian ( K -contact) Sasaki structure on the seven-dimensional nilpotent Lie algebra. The formulas ofTheorems 3.4 and 3.6 allow the use of the properties of the pseudo-K¨ahler structures for theproperties of the corresponding contact ( K -contact) Sasaki structures. This is demonstratedby the following example. Example
Consider the Lie algebra g with the commutators [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e and contact form η = e . It is central extensionof the algebra h , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , using the symplectic form ω = − e ∧ e + e ∧ e + e ∧ e . As noted in the example in the previous section, there is [12] afamily of pseudo-K¨ahler metrics on h which depend on six parameters ψ ij . The curvaturetensor on h is zero for all values of the parameters ψ ij . Therefore, in this case, we obtain afamily of contact pseudo-Sasaki structures ( η, ξ, φ, g ) on g , where ξ = e and φ ( x ) = J X , if x = X + λξ , X ∈ h . The curvature tensor and Ricci tensor have the form: R ( X, Y ) Z = −
14 ( g ( Z, J Y ) J X − g ( Z, J X ) J Y ) + 12 ω ( X, Y ) J Z,R ( X, Y ) ξ = 0 , R ( X, ξ ) Z = − g ( X, Z ) ξ, R ( X, ξ ) ξ = 14 X,Ric ( Y, Z ) = − g ( Y, Z ) , Ric ( Y, ξ ) = 0 , Ric ( ξ, ξ ) = n/ . Remark.