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A Riemannian manifold with skew-circulant structures and an associated locally conformal K\"{a}hler manifold

A 4-dimensional Riemannian manifold M, equipped with an additional tensor structure S, whose fourth power is minus identity, is considered. The structure S has a skew-circulant matrix with respect to some basis of the tangent space at a point on M. Moreover, S acts as an isometry with respect to the metric g. A fundamental tensor is defined on such a manifold (M,g,S) by g and by the covariant derivative of S. This tensor satisfies a characteristic identity which is invariant to the usual conformal transformation. Some curvature properties of (M,g,S) are obtained. A Lie group as a manifold of the considered type is constructed. A Hermitian manifold associated with (M,g,S) is also considered. It turns out that it is a locally conformal K\"{a}hler manifold.


Introduction
The classification of almost Hermitian manifolds with respect to the covariant derivative of the almost complex structure J is made by Gray and Hervella in [6]. The Hermitian manifolds form a class of manifolds with an integrable almost complex structure J. Their subclass consists of the so-called locally conformal Kähler manifolds, determined by a special property of the covariant derivative of J. The class of Kähler manifolds is common to all classes in this classification and its manifolds have the richest geometry. The Kähler manifolds are extensively studied by many geometers. There is also a great interest in the study of locally conformal Kähler manifolds, since their structure J satisfies similar but weaker conditions than those of Kähler manifolds. Some of the recent investigations of locally conformal Kähler manifolds are made in [1,3,7,9,11,12].
Problems in differential geometry of a 4-dimensional Riemannian manifold M with a tensor structure S of type (1, 1), which satisfies S 4 = −id, are considered in [4]. The matrix of S in some basis of the tangent space T p M , at an arbitrary point p on M , is skew-circulant. Moreover, S is compatible with the metric g, so that an isometry is induced in T p M . A Hermitian manifold (M, g, J), where J = S 2 , is associated with such a manifold (M, g, S).
In the present work we continue the study of (M, g, S) and (M, g, J). In Section 2, we recall some necessary facts about these manifolds. In Section 3, we compute the components of the fundamental tensor F on (M, g, S) determined by the metric g and by the covariant derivative of S. We obtain an important characteristic identity for F . We establish that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. In Section 4, we find some curvature properties of (M, g, S). In Section 5, we establish that the associated manifold (M, g, J) belongs to the class of locally conformal Kähler manifolds. In Section 6, we construct a Lie group with a Lie algebra of a special class as a manifold with the structure (g, S).

Preliminaries
We consider a 4-dimensional Riemannian manifold M equipped with a tensor S of type (1, 1). The structure S has a skew-circulant matrix, with respect to some basis {e i } of T p M , given by Then S has the property S 4 = −id. The metric g and the structure S satisfy (2) g(Sx, Sy) = g(x, y), x, y ∈ X(M ).
The above condition and (1) imply that the matrix of g has the form: Here A and B are smooth functions of an arbitrary point p(x 1 , x 2 , x 3 , x 4 ) on M . It is supposed that A > √ 2B > 0 in order g to be positive definite. The manifold (M, g, S) is introduced in [4].
Anywhere in this work, x, y, z, u will stand for arbitrary elements of the algebra of the smooth vector fields X(M ) or vectors in the tangent space T p M . The Einstein summation convention is used, the range of the summation indices being always {1, 2, 3, 4}.
In [4], it is noted that the manifold (M, g, J), where J = S 2 , is a Hermitian manifold with an almost complex structure J. For such manifolds Grey-Hervella's classification is valid ( [6]). This classification is made with respect to the covariant derivative of the Kähler form J(x, y) = g(x, Jy). The almost Hermitian manifolds with an integrable structure J are called Hermitian manifolds. Their subclass of locally conformal Kähler manifolds is determined by the property Here ∇ is the Levi-Civita connection of g, and g ij are the components of the inverse matrix of (g ij ) with respect to the basis {e i } of T p M . It is known that the class of 4-dimensional locally conformal Kähler manifolds is non-trivial. Every Kähler manifold belongs to the class of locally conformal Kähler manifolds ( [6]). Now we consider an associated metricg with g on (M, g, S), determined by (5)g(x, y) = g(x, Sy) + g(Sx, y).
If F = 0, then the structure S is parallel with respect to ∇. The following necessary and sufficient conditions for S and also for J = S 2 to be parallel structures with respect to ∇ are established in [4].  In this section we obtain a characteristic property of the tensor F on (M, g, S), which is an analogue of the property (4) of ∇J on (M, g, J). For this purpose we calculate the components of F .
The inverse matrix of (g ij ) has the form: Using (1) and (3) we get that the matrixg, determined by (5), is of the type: Due to (6) the components of F are F ijk = ∇ igjk . We apply tog the following well-known formula for the covariant derivative of tensors: Here Γ s ij are the Christoffel symbols of ∇. They are determined by (12) 2Γ k ij = g ak (∂ i g aj + ∂ j g ai − ∂ a g ij ). Then, with the help of (3), (6), (7), (9), (10) and (11) we calculate the components of F , given in (8).
Immediately, we have the following Proof. The proof follows from (8) and (9) by direct computations.
Remark 3.7. According to Theorem 3.6, we can say that (M, g, S) and (M, g, S) belong to classes of the same type, defined by the equality (15) for the corresponding metric.
4. Some curvature properties of (M, g, S) It is well-known, that the curvature tensor R of ∇ is defined by The tensor of type (0, 4) associated with R is defined as follows: R(x, y, z, u) = g(R(x, y)z, u).
The Ricci tensor ρ and the scalar curvature τ with respect to g are as usually: (28) ρ(y, z) = g ij R(e i , y, z, e j ), τ = g ij ρ(e i , e j ).
In this section we investigate some curvature properties of (M, g, S), corresponding to the metric g and to the associated metricg.
LetΓ be the Christoffel symbols ofg and∇ the Levi-Civita connection ofg. Let R be the curvature tensor of∇. The Ricci tensorρ and the scalar curvatureτ with respect tog are given by (29)ρ(y, z) =g ijR (e i , y, z, e j ),τ =g ijρ (e i , e j ).
Further, we use the following statements for a special basis of T p M on (M, g, S), which are established in [4].
(i) A basis of type {S 3 x, S 2 x, Sx, x} of T p M exists and it is called an S-basis. In this case we say that the vector x induces an S-basis of T p M .
(ii) If a vector x induces an S-basis and ϕ is the angle between x and Sx, then and π 4 < ϕ < 3π 4 . (iii) An orthogonal S-basis of T p M exists. Now, we recall that the Ricci curvature, with respect to g, in the direction of a nonzero vector x is the value Due to Theorem 4.1 we establish the following (38)r(x) = 1 2 cos ϕ r(x) Proof. The proof follows directly from (31), (36) and (37).
In a similar way to Proposition 4.3 we prove the next g(x, y).

A locally conformal Kähler manifold (M, g, J)
The fundamental Kähler form of the structure (g, J) on an almost complex manifold (M, g, J) is determined by . Proof. Because of (1), (3) and (44), we get that the matrix of the fundamental Kähler form is of the type: Applying the Christoffel symbols Γ, obtained by (3), (9) and (12), and the components of the matrix (46) to equality ik J aj , we calculate the nonzero components of ∇J, given in (45).
Immediately, we have the following Corollary 5.2. The components ω k = g ij F (e i , e j , e k ) of the 1-form ω on the manifold (M, g, J) are expressed by the equalities Proof. The equalities (47) follow from (9) The latter identity is the local form of (4), which is a defining condition of a locally conformal Kähler manifold. 6. A Lie group with a structure (g, S) Let G be a 4-dimensional real connected Lie group. Let g be the corresponding Lie algebra with a basis {e 1 , e 2 , e 3 , e 4 } of left invariant vector fields. We introduce a skew-circulant structure S and a metric g as follows: Se 1 = −e 4 , Se 2 = e 1 , Se 3 = e 2 , Se 4 = e 3 ; (49) where δ ij is the Kronecker delta.
Consequently the used basis {e i } is an orthonormal S-basis. Obviously, (1) and (2) are valid and (g, S) is a structure of the considered type. We denote the corresponding manifold by (G, g, S). Then the associated manifold is (G, g, J), where J satisfies The real four-dimensional indecomposable Lie algebras are classified by Mubarakzyanov ( [10]). This scheme seems to be the most popular (see [2] and the references therein). We pay attention to the class {g 4,5 }, which represents an indecomposable Lie algebra, depending on two real parameters a and b. Actually, it induces a family of manifolds whose properties are functions of a and b.
According to the definition of the class {g 4,5 }, we have that the nonzero brackets are as follows ( [2]): Furthermore, with the help of the above equalities we compute the components of the tensor F on (G, g, S) and the components of the tensor ∇J on (G, g, J). We find conditions under which F satisfies (15) and ∇J satisfies (4). Proof. Bearing in mind (6), (49), (50) and (53) we get the components F ijk of F , θ i of θ and θ * i of θ * with respect to the basis {e i }. The nonzero of them are the following: . By equalities (5)   Then, from (50), we get ρ = τ 4 g. Consequently, due to (40), the manifold (G, g, S) is Einstein. Remark 6.6. Lie groups with a Lie algebra in {g 4,5 } are studied in [5] as an example of 4-dimensional Riemannian manifolds with circulant structures.
An example of a locally conformal Kähler manifold constructed on a Lie group with a Lie algebra in {g 4,5 } is considered in [8]. The metric of the manifold is indefinite but the condition a = b = 1 also exists.
The example of a 4-dimensional Riemannian manifold with skew-circulant structures, constructed in this section, has similar properties to the above examples.