FOUR-DIMENSIONAL ALMOST EINSTEIN MANIFOLDS WITH SKEW-CIRCULANT STRUCTURES

: We consider a four-dimensional Riemannian manifold M with an additional structure S , whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M . By a special identity for the curvature tensor, generated by the Riemannian connection of g , we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M . We consider a Hermitian manifold associated with the studied manifold and ﬁnd conditions for g , under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.


Introduction
The right circulant matrices and the right skew-circulant matrices are Toeplitz matrices, which are well-studied in [1] and [3]. The set of invertible circulant (skew-circulant) matrices form a group with respect to the matrix multiplication. Such matrices have application to geometry, linear codes, graph theory, vibration analysis (for example [2,7,9,11,13,14]).
A. Gray, L. Hervella and L. Vanhecke used curvature identities to classify and to study the almost Hermitian manifolds (for instance in [4][5][6]15]). The Hermitian manifolds form a class of manifolds with an integrable almost complex structure J. The class of the Kähler manifolds is their subclass and such manifolds have a parallel structure J. According to A. Gray, the Kähler manifolds have an especially rich geometric structure, due to the Kähler curvature identity R(·, ·, J·, J·) = R(·, ·, ·, ·). Some of the recent investigations on the curvature properties of the almost Hermitian manifolds are made in [8,10,12,16].
In the present work we study a four-dimensional differentiable manifold M with a Riemannian metric g. The manifold M is equipped locally with an additional structure S, which satisfies S 4 = −id. The component matrix of S is a special skew-circulant matrix, i.e., S is a skew-circulant structure. Moreover, S is compatible with g, such that an isometry is induced in every tangent space of M. Such a manifold (M, g, S) is associated with a Hermitian manifold (M, g, J), where J = S 2 is a complex structure.
The paper is organized as follows. In Sect. 2, we introduce a manifold (M, g, S) and give some necessary facts for our investigations. In Sect. 3, we obtain a class of almost Einstein manifolds (M, g, S) and a class of Einstein manifolds (M, g, S). In Sect. 4, we get conditions under which an orthogonal basis of type {S 3 x, S 2 x, Sx, x} exists in every tangent space of (M, g, S). In Sect. 5, we find some curvature properties of the considered Einstein and almost Einstein manifolds. In Sect. 6, we obtain a necessary and sufficient condition for S to be parallel with respect to the Riemannian connection of g. Also, we get conditions for (M, g, J) to be a Kähler manifold. In Sect. 7, we construct examples of the considered manifolds on Lie groups and find some of their geometric characteristics.

Preliminaries
Let M be a 4-dimensional Riemannian manifold equipped with an endomorphism S in every tangent space T p M at a point p on M. Let the coordinates of S, with respect to some basis {e i }, form a right skew-circulant matrix as follows (1) We use local coordinates to facilitate our calculations. According to (1) S has the property S 4 = −id.
We assume that the metric g and the structure S satisfy g(Sx, Sy) = g(x, y).
Here and anywhere in this work, x, y, z, u will stand for arbitrary elements of the algebra on smooth vector fields on M or vectors in T p M. The Einstein summation convention is used, the range of the summation indices being always {1, 2, 3, 4}. The conditions (1) and (3) imply that the matrix of g, with respect to the local basis {e i }, has the form i.e., it is right skew-circulant. Here A = A(p) and B = B(p) are smooth functions of an arbitrary point p(X 1 , X 2 , X 3 , X 4 ) on M. The determinant of the matrix (4) has the value det(g ij ) = ( in order g to be positive definite. A manifold M introduced in this way we denote by (M, g, S). Now, we consider an associated metricg with g, determined bỹ g(x, y) = g(x, Sy) + g(Sx, y).
Using (1), (4) and (6) we get that the matrix of its components is Two of the eigenvalues of (7) are 2B − √ 2A and the other two are 2B + √ 2A. Since inequalities (5) are valid,g has signature (2,2). Sog is an indefinite metric.
The inverse matrices of (g ij ) and (g ij ) are as follows: Let ∇ be the Riemannian connection of g. The curvature tensor R of ∇ is determined by The tensor of type (0, 4) associated with R is defined by The Ricci tensor ρ with respect to g is given by the well-known formula ρ(y, z) = g ij R(e i , y, z, e j ).
The scalar curvature τ with respect to g and its associated quantity τ * are determined by Now, we consider a manifold (M, g, S) with the condition i.e., S is a parallel structure with respect to ∇. Proposition 1. Every manifold (M, g, S) with a parallel structure S satisfies the curvature identity R(x, y, Sz, Su) = R(x, y, z, u).

Almost Einstein manifolds
In this section we consider manifolds (M, g, S) with the property (15). By R ijkh and ρ ij we will denote the components of the curvature tensor R and the components of the Ricci tensor ρ with respect to the local basis {e i }, respectively. Hence, we establish the following propositions.
Proposition 2. The property (15) of the curvature tensor R of (M, g, S) is equivalent to the conditions Proof. The local form of (15) is Then, using (1), we find the equalities By applying the Bianchi identity to the above components of R, we obtain (17). Vice versa, from (1) and (17) it follows (18), so (15) holds true.

Proposition 3.
If a manifold (M, g, S) has the property (15), then the components of the Ricci tensor ρ satisfy Proof. Due to Proposition 2, the components of the curvature tensor R satisfy (17). For brevity, we denote Thus, having in mind (8), (12), (17) and (20), we get the components of ρ as follows: So the equalities (19) are valid.
A Riemannian manifold is said to be Einstein if its Ricci tensor ρ is a constant multiple of the metric tensor g, i.e.
In [17], for locally decomposable Riemannian manifolds is defined a class of almost Einstein manifolds. For the considered in our paper manifolds, we give the following Definition 1. A Riemannian manifold (M, g, S) is called almost Einstein if the metrics g andg satisfy where α and β are smooth functions on M. Proof. According to Proposition 3, for (M, g, S) the equalities (19) are valid. Consequently, using (8), (9), (13) and (19), we get the values of the scalar curvature τ and τ * as follows: Immediately from the latter equalities we have and bearing in mind (4) and (7) we get Then, taking into account (4), (7), (19) and (24), we obtain i.e.
Let (M, g, S) satisfy the conditions of Theorem 1. If we suppose that (M, g, S) is an Einstein manifold, then its Ricci tensor ρ has the form (22). Hence (26) implies the following Corollary 1. If the manifold (M, g, S) with the property (15) is Einstein then In the next theorem, we express the curvature tensor R of an almost Einstein manifold (M, g, S) by both structures g and S.

Orthogonal S-basis of T p M
If x is a vector in a tangent space T p M of (M, g, S), then applying (1) we get the system of vectors {S 3 x, S 2 x, Sx, x}. We will use a basis and an orthogonal basis of the type {S 3 x, S 2 x, Sx, x} in T p M. Therefore, in this section we will consider the existence of such bases.
If x is a nonzero vector on (M, g, S), then according to (1) we have Sx = ±x. Thus the angle ϕ between x and Sx belongs to the interval (0, π). Evidently, the vectors x, Sx, S 2 x and S 3 x determine six angles, which belong to (0, π). For these angles we establish the next statement.
Proof. Let x = (x 1 , x 2 , x 3 , x 4 ) be a nonzero vector on (M, g, S). By using (1), we get Having in mind the components of x, also (4) and (32), we calculate From (2) and (3) it follows Now, due to (3) and (5), we can determine the angle ϕ between x and Sx, and the angle φ between x and S 2 x as follows: We apply (33) and (34) in (35) and find Then, bearing in mind (3) and (34), we get (31).

Definition 2.
A basis of type {S 3 x, S 2 x, Sx, x} of T p M is called an S-basis. In this case we say that the vector x induces an S-basis of T p M.
The following statements hold.
Theorem 4. Every nonzero vector x = (x 1 , x 2 , x 3 , x 4 ), which satisfies induces an S-basis of T p M.
Proof. If a nonzero vector x ∈ T p M has coordinates (x 1 , x 2 , x 3 , x 4 ), then using (32) we get the determinant formed by the coordinates of the vectors x, Sx, S 2 x and S 3 x. It is In case that (36) is valid, we have = 0, which implies that x, Sx, S 2 x and S 3 x form a basis.
Lemma 1. Let a vector x induce an S-basis and let ϕ be the angle between x and Sx. The following inequalities are valid: Proof. We suppose without loss of generality that g(x, x) = 1. Thus, because of (3), (34) and (35), we find We consider a nonzero vector y, such that Since g is a Riemannian metric we have g(y, y) > 0. Substituting (39) into the latter inequality, and using (38), we get 1 − 2 cos 2 ϕ > 0.
According to Theorem 4, there are many S-bases of T p M. Hence, bearing in mind Theorem 3 and Lemma 1, we arrive at the following Theorem 5. For every manifold (M, g, S) there exists an orthogonal S-basis of T p M.

Curvature properties of (M, g, S)
The sectional curvature of a non-degenerate 2-plane {x, y} spanned by the vectors x, y ∈ T p M is the value k(x, y) = R(x, y, x, y) g(x, x)g(y, y) − g 2 (x, y) . vectors equalities (31) are valid. Moreover, the angle ϕ = (x, Sx) satisfies (37). In the next theorem we establish the relations among the sectional curvatures of the 2-planes generated by an S-basis, the angle ϕ, the scalar curvature τ and τ * .
Now, we recall that the Ricci curvature in the direction of a nonzero vector x is the value where ϕ = (x, Sx).

Manifolds with parallel structures
In this section we study a manifold (M, g, S), whose structure S satisfies (14). Also, we consider an associated manifold (M, g, J) with a structure J = S 2 . Bearing in mind (1) and (3), we get that the manifold (M, g, J) is Hermitian and the structure J is complex. In case that J is parallel (M, g, J) is a Kähler manifold. The characteristic condition of a Kähler manifold is Evidently, for the structure J = S 2 , the equality (14) implies (47).
Theorem 8. Let (M, g, S) have the property (14). Then the scalar curvature τ and τ * satisfy where Proof. It is known that in a Riemannian manifold for the scalar curvature τ and the Ricci tensor ρ it is valid where ρ i k = ρ ak g ai . On the other hand, if (M, g, S) satisfies (14), then it satisfies (15). Therefore, the Ricci tensor has the expression (25). Hence, from (1), (4), (7), (8) and (25), we get where δ i k are the Kronecker symbols. Using the above equalities, (14) and (49) we obtain where because of (1) it follows (48).
Bearing in mind Theorem 9 and Theorem 10 we state the following Corollary 3. The structure S of (M, g, S) is parallel with respect to ∇ if and only if the structure J of (M, g, J) is parallel with respect to ∇.

Lie groups as 4-dimensional Riemannian manifolds with skew-circulant structures
Let G be a 4-dimensional real connected Lie group and g be its Lie algebra with a basis {x 1 , x 2 , x 3 , x 4 }. We introduce a tensor structure S and a left invariant metric g as follows: g(x i , x j ) = 0, i = j; 1, i = j.
Obviously (2) and (3) are valid. Therefore (G, g, S) is a Riemannian manifold of the considered type. If we suppose that S is an Abelian structure on a Lie group G, then the commutators [x i , The conditions (57), (59) and the Jacobi identity for [x i , where λ i ∈ R.
In this section we investigate a manifold (G, g, S) with a Lie algebra g determined by (60), i.e., a manifold (G, g, S) with an Abelian structure S. Theorem 11. Let (G, g, S) be a manifold with a Lie algebra g determined by (60). Then (G, g, S) has the property (14).