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Matrix Lie groups as 3-dimensional almost contact B-metric manifolds

The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some known Lie groups are equiped with almost contact B-metric structure of different types.


Introduction
The differential geometry of almost contact manifolds is well studied (e.g. [3]). In [5], the almost contact manifolds with B-metric are introduced and classified. These manifolds can be studied as the odd-dimensional counterpart of the almost complex manifolds with Norden metric [4,6].
As it is known from [10], every representation of a Lie algebra corresponds uniquely to a representation of the simply connected Lie group. This relation is one-to-one. Then, knowing the representation of a certain Lie algebra settles the question of the representation of its Lie group.
In [9], an arbitrary Lie group considered as a manifold is equiped with an almost contact B-metric structure. For the lowest dimension three, it is deduced an equivalence between the classification of considered manifolds given in [5] and the corresponding Lie algebra determined by commutators.
In the present work, it is found the explicit correspondence between all basicclass-manifolds of the classification in [5] and Lie groups given by their explicit matrix representation.
The paper is organized as follows. In Sect. 1 we recall some preliminary facts about the almost contact B-metric manifolds and Lie algebras related to them. In Sect. 2 we give the correspondence of all basic-class-manifolds and matrix Lie groups. Sect. 3 is devoted to some examples connected to the previous investigations.
1. Preliminaries 1.1. Almost contact B-metric manifolds. Let us denote an almost contact Bmetric manifold by (M, ϕ, ξ, η, g), i.e. M is an (2n + 1)-dimensional differentiable manifold, ϕ is an endomorphism of the tangent bundle, ξ is a Reeb vector field, η is its dual contact 1-form and g is a pseudo-Riemannian metric (called a B-metric) of signature (n + 1, n) such that [5] ϕξ = 0, 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 .
In [5], it is given a classification of almost contact B-metric manifolds, consisting of eleven basic classes F 1 , F 2 , . . . , F 11 . It is made with respect to the tensor F of type (0,3) defined by where ∇ stands for the Levi-Civita connection of g.
The special class F 0 , determined by the condition F (x, y, z) = 0, is the intersection of all basic classes and it is known as the class of the cosymplectic B-metric manifolds.
In this work, we consider the case of the lowest dimension of the considered manifolds, i.e. dim M = 3.

The main result
Theorem A yields an equivalence between the manifolds of the classification in [5] and the corresponding Lie algebra.
It is known (e.g. [10]) that for a real Lie algebra of finite dimension there is a corresponding connected simply connected Lie group, which is determined uniquely up to isomoprhism.
It is arised the problem for determination of the Lie group which is isomorphic to the given Lie group L equiped with a structure (ϕ, ξ, η, g) in the class F s . Theorem 2.1. Let (L, ϕ, ξ, η, g) be an almost contact B-metric manifold belonging to the class F s (s ∈ {1, 4, 5, 8, 9, 10, 11}). Then the compact simply connected Lie group G isomorphic to L, both with one and the same Lie algebra, has the form where E is the identity matrix and A is the matrix representation of the corresponding Lie algebra. The matrix form of A as well as the real parameters t and u for the different classes F s are given in Table 1, where a, b, c ∈ R and α, β are introduced in Theorem A.

2.1.
Proof of the theorem.
Lie groups as manifolds from the classes F 1 , F 5 , F 11 . Firstly, let us consider the case when (L, ϕ, ξ, η, g) is a F 1 -manifold. Then, according to Theorem A, the corresponding Lie algebra is determined by where α = 1 2 θ 1 , β = 1 2 θ 2 . From (2.1) we have the nonzero values of the commutation coefficients: According to [10], the commutation coefficients provide a matrix representation of the Lie algebra. This representation is obtained by the basic matrices M i , which entries are determined by
Let us consider the first case ∆ = 0. Then we have The Jordan matrix J is the diagonal matrix with elements J ii = λ i . It is known that e A = P e J P −1 . Then we obtain the matrix Lie group representation G 1 of the considered Lie algebra g 1 in this case as follows which can be written as In the second case ∆ = 0, the matrix P is non-invertible. Then A is nilpotent and e A can be computed directly from where q stands for the degree of A. Then, from the form of A in Table 1 we obtain that q = 2. Therefore, the matrix representation G 1 of the Lie group for the Lie algebra g 1 in this case is i.e.
Both the cases for F 1 can be generalized as it is shown in Table 1. By similiar considerations we obtain the results in Table 1 for the classes F 5 and F 11 .

2.1.2.
Lie groups as manifolds from the classes F 4 , F 9 , F 10 . Firstly, let us consider the case when (L, ϕ, ξ, η, g) is a F 4 -manifold. Then, according to Theorem A, the corresponding Lie algebra is determined by (2.4) [ where α = 1 2 θ 0 . By similar way as in §2.1.1, we obtain the form of A for the corresponding Lie algebra g 4 , given in Table 1. In this class tr A = 0 and we have two cases according to tr A 2 = −2α 2 c 2 . Then, the matrix representation is respectively Both the cases for F 4 can be generalized as it is shown in Table 1. By similiar considerations we obtain the results in Table 1 for the classes F 9 and F 10 .

2.1.3.
Lie groups as manifolds from the class F 8 . According to Theorem A, we have We get the matrix representation of A for the considered Lie algebra g 8 , given in Table 1.
In this class tr A = 0 and we have three cases according to the sign of tr A 2 = 2α 2 ∆, where ∆ = 2a 2 − 2b 2 + c 2 . They can be generalized as it is shown in Table 1.

Equipping of known Lie groups with almost contact B-metric structure
In this final section we equip some known Lie groups with almost contact Bmetric structures of different types. These considerations use results given in [8].
3.1. Examples 1, 2, 3. In [11], there are considered the matrix Lie groups G I , G II and G III of the following form where x, y, z ∈ R. These matrix groups and their automorphisms represent the only three types of generalized affine symmetric spaces denoted as I, II, III. The Lie group G I is the group of hyperbolic motions of the plane R 2 , G II is the group of isometries of the plane R 2 and G III is the Heisenberg group. The corresponding Lie algebras are determined by the commutators as follows: The types of the Lie algebras g I , g II , g III according to the well-known Bianchi classification in [1,2] of three-dimensional real Lie algebras are Bia(V), Bia(VII 0 ), Bia(II), respectively.
Bearing in mind Table 1 for F 9 and substituting X 1 = E 1 , X 2 = E 2 , X 3 = −E 0 , α = 1, we have that G I can be considered as an almost contact B-metric manifold of the class F 9 when the derived algebra is on ker(η), otherwise the manifold belongs to F 1 ⊕ F 11 . [8] Considering Table 1 for F 4 and substituting X 1 = E 1 , X 2 = E 2 , X 3 = E 0 , α = 1, we have that G II can be considered as an almost contact B-metric manifold of the class F 4 when the derived algebra is on ker(η), otherwise the manifold belongs to F 4 ⊕ F 8 or F 4 ⊕ F 8 ⊕ F 10 . [8] Depending on the way of equipping with an almost contact B-metric structure we can obtain G III as a manifold in F 4 ⊕ F 10 or F 8 ⊕ F 10 when the derived algebra is on ker(η) or span(ξ), respectively. [8] 3.2. Example 4. Let us consider the well-known Lie group SO(3) which is called the rotation group. The matrix representation A of the corresponding Lie algebra has the form where α is the angle of rotation and x, y, z ∈ R.
In terms of the matrix exponential and according to the Rodrigues rotational formula, the Lie group SO(3) consists the matrices of the following form e A = E + sin α A + (1 − cos α) A 2 .
Then, the type of the considered Lie algebra is Bia(IX), according to the Bianchi classification. Hence, bearing in mind [8], this Lie group can be considered as an almost contact B-metric manifold of the class F 4 ⊕ F 8 ⊕ F 10 .