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Almost contact B-metric structures and the Bianchi classification of the three-dimensional Lie algebras

The object of investigation are the almost contact manifolds with B-metric in the lowest dimension three, constructed on Lie algebras. It is considered a relation between the classes in the Bianchi classification of three-dimensional real Lie algebras and the classes of a classification of the considered manifolds. There are studied some geometrical characteristics in some special classes.


INTRODUCTION
The differential geometry of the manifolds equipped with an almost contact structure is well studied (e.g. [3]). The almost contact manifolds with B-metric are introduced and classified in [6]. These manifolds are the odd-dimensional counterpart of the almost complex manifolds with Norden metric [5,7].
An object of special interest is the case of the lowest dimension of the considered manifolds. We investigate the almost contact B-metric manifolds in dimension three and get explicit results. Some curvature identities of the three-dimensional manifolds of this type are studied in [11,12].
Almost contact manifolds with B-metric can be constructed on Lie algebras. It is known that all three-dimensional real Lie algebras are classified in [1,2]. The main goal of this paper is to find a relation between the classes in the Bianchi classification and the classification of almost contact B-metric manifolds given in [6]. Moreover, the present work gives some geometrical characteristics of considered manifolds in certain special classes.
The present paper is organized as follows. In Sect. 2 we recall some preliminary facts about the almost contact B-metric manifolds. In Sect. 3 we equip each Bianchi-type Lie algebra with an almost contact B-metric structure. In Sect. 4 we give the relation between the Bianchi classification and the classification given in [6]. Sect. 5 is devoted to the curvature properties of some of the considered manifolds.

PRELIMINARIES
Let (M, ϕ, ξ, η, g) be an almost contact manifold with B-metric or an almost contact B-metric manifold, where M is a (2n + 1)-dimensional differentiable manifold, (ϕ, ξ, η) is an almost contact structure consisting of an endomorphism ϕ of the tangent bundle, a Reeb vector field ξ and its dual contact 1-form η. Moreover, M is equipped with a pseudo-Riemannian metric g, called a B-metric, such that the following algebraic relations are satisfied [6]: where Id is the identity. In the latter equalities and further, x, y, z, w will stand for arbitrary elements of the algebra of the smooth vector fields on M or vectors in the tangent space T p M of M at an arbitrary point p in M .
The associated B-metricg of g is determined byg(x, y) = g(x, ϕy) + η(x)η(y). The manifold (M, ϕ, ξ, η,g) is also an almost contact B-metric manifold. The signature of both metrics g andg is necessarily (n + 1, n). We denote the Levi-Civita connection of g andg by ∇ and∇, respectively.
A classification of almost contact B-metric manifolds, consisting of eleven basic classes F 1 , F 2 , . . . , F 11 , is given in [6]. This classification is made with respect to the tensor F of type (0,3) defined by and having the following properties: The special class determined by the condition F (x, y, z) = 0 is denoted by F 0 . This class is the intersection of all the basic classes. Hence F 0 is the class of almost contact B-metric manifolds with ∇-parallel structures, i.e. ∇ϕ = ∇ξ = ∇η = ∇g = ∇g = 0. Therefore F 0 is the class of the cosymplectic manifolds with B-metric.
According to [10], the square norm of ∇ϕ is defined by: It is clear, ∇ϕ 2 = 0 is valid if (M, ϕ, ξ, η, g) is a cosymplectic manifold with Bmetric, but the inverse implication is not always true. An almost contact B-metric manifold having a zero square norm of ∇ϕ is called an isotropic-cosymplectic Bmetric manifold.
If {e i ; ξ} (i = 1, 2, . . . , 2n) is a basis of T p M and g ij is the inverse matrix of (g ij ), then the 1-forms θ, θ * , ω, called Lee forms, are associated with F and defined by: Let now consider the case of the lowest dimension of the almost contact B-metric manifold M , i.e. dim M = 3.
We introduce an almost contact structure (ϕ, ξ, η) on M defined by Let us denote the components F ijk = F (e i , e j , e k ) of F with respect to a ϕ-basis {e 1 , e 2 , e 3 } of T p M .
According to [8], the components of the Lee forms are Then, if F s (s = 1, 2, . . . , 11) are the components of F in the corresponding basic classes F s and x = x i e i , y = y j e j , z = z k e k for arbitrary vectors in T p M , we have [8]: Obviously, the class of three-dimensional almost contact B-metric manifolds is The corresponding curvature (0, 4)-tensor is denoted by the same letter: R(x, y, z, w) = g(R(x, y)z, w). The following properties are valid: It is known from [11] that every 3-dimensional cosymplectic B-metric manifold is flat, i.e. R = 0.
The Ricci tensor ρ and the scalar curvature τ for R as well as their associated quantities are defined respectively by where {e 1 , e 2 , . . . , e 2n+1 } is an arbitrary basis of T p M . Let α be a non-degenerate 2-plane (section) in T p M . It is known that the special 2-planes with respect to (ϕ, ξ, η, g) are: a totally real section if α is orthogonal to its ϕ-image ϕα, a ϕ-holomorphic section if α coincides with ϕα and a ξ-section if ξ lies on α.
The sectional curvature k(α; p)(R) of α with an arbitrary basis {x, y} at p is defined by According to [9], it is reasonable to call a manifold M whose Ricci tensor satisfies the condition ρ = λg + µg + νη ⊗ η an η-complex-Einstein manifold.

EQUIPPING OF EACH BIANCHI-TYPE LIE ALGEBRA WITH ALMOST CONTACT B-METRIC STRUCTURE
It is known that L. Bianchi has categorized all three-dimensional real (and complex) Lie algebras. He proved that every three-dimensional Lie algebra is isomorphic to one, and only one, Lie algebra of his list (cf. [1,2]). These isomorphism classes form the so-called Bianchi classification and are noted by Bia(I), Bia(II), Bia(IV), where o is the zero vector of l.
The geometrization conjecture, associated with W. Thurston, states that every closed manifold of dimension three could be decomposed in a canonical way into pieces, connected to one of eight types of Thurston's geometric structures ( [13]): Euclidean geometry E 3 , Spherical geometry S 3 , Hyperbolic geometry H 3 , the geometry of S 2 × R, the geometry of H 2 × R, the geometry of the universal cover SL(2, R) of the special linear group SL(2, R), the N il geometry, the Solv geometry.
Seven of the eight Thurston geometries can be associated to a class of the Bianchi classification as it is shown in the following table. The Thurston geometry on S 2 × R has no such a realization (e.g. [4]).
Let us consider each Lie algebra from the Bianchi classification, equipped with an almost contact structure (ϕ, ξ, η) and a B-metric g as in (2.3) and (2.4).
The presence of the structure (ϕ, ξ, η, g) gives us a reason to consider the relation between the Bianchi types and the classification of almost contact B-metric manifolds in [6].
We obtain immediately the following Proposition 3.1. Some Bianchi types can be equipped with a structure (ϕ, ξ, η, g) in several ways. In the cases Bia(I) and Bia(IX) there are only one variant. In the rest cases, there are three possible subtypes of each type differing each other by a cyclic change of the basic vectors e 1 , e 2 , e 3 . All subtypes are determined as follows:

ALMOST CONTACT B-METRIC MANIFOLDS OF EACH BIANCHI TYPE
Let us consider the Lie group L corresponding to the given Lie algebra l. Each definition of a Lie algebra for the different subtypes in Proposition 3.1 generates a corresponding almost contact B-metric manifold denoted by (L, ϕ, ξ, η, g). In this section we characterize the obtained manifolds with respect to the classification in [6].
Using (2.5), we obtain the corresponding components of F in each subtypes (1), (2), (3) in Proposition 3.1 and determine the corresponding class of almost contact B-metric manifolds. The results are given in the following Theorem 4.1. The manifold (L, ϕ, ξ, η, g), determined by each type of Lie algebra given in Proposition 3.1, belongs to a class of the classification in [6] as it is given in the following table:  TABLE 3. Relations between the Bianchi types and the classes in [6] Bia(I) Proof. We give our arguments for the case of Bia(II). In a similar way we prove the other cases.

CURVATURE PROPERTIES OF THE CONSIDERED MANIFOLDS IN SOME BIANCHI CLASSES
Now, we focuss our considerations on the Bianchi classes depending on real parameter h. They are Bia(VI h ) and Bia(VII h ). Actually, these two classes are families of manifolds whose properties are functions of h. The classes regarding F corresponding to Bia(VI h ), h < 0 and Bia(VII h ), h > 0, according to Theorem 4.1, can not be restricted for special values of h.
In this section an object of special interest are the curvature properties of these manifolds in relation with h.
Having in mind Proposition 3.1, it is reasonable to investigate all three subtypes of the Bianchi classes Bia(VI h ), h ≤ 0 and Bia(VII h ), h ≥ 0. The nonzero components of ∇ for Bia(VI h ) are calculated: Further, there are computed the basic components R ijkl = R(e i , e j , e k , e l ) of the curvature tensor R, ρ jk = ρ(e j , e k ) of the Ricci tensor ρ, ρ * jk = ρ * (e j , e k ) of the associated Ricci tensor ρ * , the values of the scalar curvatures τ and τ * and of the sectional curvatures k ij = k(e i , e j ) as follows: Using the latter equalities we can conclude the following By similar way we obtain the corresponding results to (5.2) and (5.3) for the rest cases. For the subtype (2) we get: These results imply the following In the case of subtype (3) we obtain: ∇ϕ 2 = 10(h 2 + 1); The latter equalities imply the following In this subsection we focus our investigations on the three subtypes of Bia(VII h ). Firstly, let us consider the subtype (1). By similar way as the previous subsection, we obtain: 6. (L, ϕ, ξ, η, g) is an η-complex-Einstein manifold.