Almost hypercomplex manifolds with Hermitian-Norden metrics and 4-dimensional indecomposable real Lie algebras depending on two parameters

The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on two parameters. Some geometric characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics are obtained.


Introduction
Almost hypercomplex structure H on a 4n-dimensional manifold M is a triad of anticommuting almost complex structures such that each of them is a composition of the other two structures. In [7,8], this structure H is equipped with a metric structure of Hermitian-Norden type, generated by a pseudo-Riemannian metric g of neutral signature. In this case, one (resp., the other two) of the almost complex structures of H acts as an isometry (resp., act as anti-isometries) with respect to g in each tangent fibre. The metric g is Hermitian with respect to one of almost complex structures of H and g is a Norden metric regarding the other two almost complex structures of H. Then, The author was partially supported by the project MU21-FMI-008 of the Scientific Research Fund, University of Plovdiv, Bulgaria and National Scientific Program "Young Researchers and Post-Doctorants", Bulgaria.
According to [8], the fundamental tensors of such a manifold are the following three (0, 3)-tensors F α (x, y, z) = g ((∇ x J α ) y, z) = (∇ x g α ) (y, z) , (3) where ∇ is the Levi-Civita connection generated by g. These tensors have the following basic properties caused by the structures The following relations between the tensors F α are valid The corresponding Lee forms θ α are determined by where {e 1 , e 2 , . . . , e 4n } is an arbitrary basis of T p M, p ∈ M and g ij are the corresponding components of the inverse matrix of g.
Let us note that, according to (1), (M, J 1 , g) is an almost Hermitian manifold whereas the manifolds (M, J 2 , g) and (M, J 3 , g) are almost complex manifolds with Norden metric. The basic classes of these two types of manifolds are given in [6] and [4], respectively. In the case of the lowest dimension, dim M = 4, the four basic classes of almost Hermitian manifolds with respect to J 1 are restricted to two: the class of the almost Kähler manifolds W 2 (J 1 ) and the class of the Hermitian manifolds W 4 (J 1 ), determined by: where S is the cyclic sum by three arguments. The basic classes of the 4dimensional almost Norden manifolds (α = 2 or 3) are determined as follows: The Nijenhuis tensor in terms of the covariant derivatives of J α and the corresponding (0,3)-tensor for J α are defined by Moreover, the following properties of N α are valid [12]: [ , ] be the curvature (1,3)-tensor of ∇ and let the corresponding curvature (0, 4)-tensor with respect to g be denoted by the same letter: The following properties of R are well-known: The Ricci tensor ρ and the scalar curvature τ for R as well as their associated quantities ρ * α , τ * α and τ * * α are defined by: Every non-degenerate 2-plane μ with a basis {x, y} with respect to g in T p M, p ∈ M, has the following sectional curvature A 2-plane μ is said to be holomorphic (resp., totally real ) if the condition μ = J α μ (resp., μ ⊥ J α μ = μ with respect to g) holds. The sectional curvature k(μ; p) of a holomorphic (resp., totally real) 2-plane is called holomorphic (resp., totally real ) sectional curvature. Let {e 1 , e 2 , . . . , e 4n } be a basis of T p M. If μ has a basis {e i , e j } (i, j ∈ {1, 2, . . . , 4n}, i = j), then μ is called basic 2plane and k(μ; p) -a basic sectional curvature.
We introduce a pseudo-Euclidian metric g of neutral signature for x( Note that, using the latter equality, it is valid that According to (1) and (2), the metric g generates an almost hypercomplex structure with Hermitian-Norden metrics on l. Then, (L, H, G) is an almost hypercomplex manifold with Hermitian-Norden metrics.
A classification of real 4-dimensional indecomposable Lie algebras is given for instance in [16] and it can be found easily in [19] and [5]. In [1], it is given a classification of four dimensional solvable real Lie algebras and the authors establish the one-to-one correspondence between their classification and the classifications in [16] and [19]. The twelve basic classes of real 4dimensional indecomposable Lie algebras are described by the non-zero Lie brackets with respect to {e 1 , e 2 , e 3 , e 4 }. Five of the basic classes are determined by real parameters -two classes use two parameters and three classes use one parameter. In Table 1, it is shown the correspondence between the classes which depend on parameters in the different classifications.
In the present work, we use the notation of the classes from [5].
Our purpose is to investigate how the basic geometrical properties of the manifolds under study depend on these parameters. In [9], we study both the basic classes of the considered classification depending on two parameters. Now, we focus our investigations on the following basic classes g 4,2 , g 4,9 and g 4,11 which depend on one real parameter:  Let us note that further, the indices i, j, k, l run over the range {1, 2, 3, 4}.

The class g 4,2
Let us consider a manifold (L, G, H) with corresponding Lie algebra from g 4,2 .

The class g 4,9
In this subsection we focus our investigations on a manifold (L, G, H) with corresponding Lie algebra from g 4,9 .
By similar way as in the previous subsection, we obtain the following results for (L, G, H) in this case. The non-zero components of ∇ are: The non-zero of the components of F are determined by the following ones and properties (4)

The class g 4,11
Now, we focus our investigations on a manifold (L, G, H) with corresponding Lie algebra from g 4,11 .
By similar way as in the previous two subsections, we obtain the following results for (L, G, H) in the present case. The non-zero components of ∇ are: The non-zero of the components of F are determined by the following ones and properties (4) We generalize the obtained results in the three relevant classes by the following Moreover, we have: Proof. Using the results in (18), (19), (21), (23) and the classification conditions (6), (7) for dimension 4, we establish the truthfulness of the assertion in each case.

Some geometric characteristics of the considered manifolds
In this section we determine some geometric characteristics of the manifolds (L, G, H) considered in the previous section and we investigate the corresponding geometric properties in relation with the real parameters of the considered three classes of Lie algebras.