Four-dimensional almost Einstein manifolds with skew-circulant stuctures

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 [1,3]. The set of invertible circulant (skewcirculant) 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 We use local coordinates to facilitate our calculations.
According to (2.1) S has the property 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}.
Vol. 111 (2020) Four-dimensional almost Einstein manifolds Page 3 of 18 9 The conditions (2.1) and (2.3) 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 (2.4) has the value det(g ij ) = (A 2 − 2B 2 ) 2 . It is supposed that 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). (2.6) Using (2.1), (2.4) and (2.6) we get that the matrix of its components is Two of the eigenvalues of (2.7) are 2B − √ 2A and the other two are 2B + √ 2A. Since inequalities (2.5) 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 R(x, y, z, u) = g(R(x, y)z, u). (2.11) 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 ). (2.12) The scalar curvature τ with respect to g and its associated quantity τ * are determined by τ = g ij ρ(e i , e j ), τ * =g ij ρ(e i , e j ). (2.13) Now, we consider a manifold (M, g, S) with the condition ∇S = 0. (2.14) i.e., S is a parallel structure with respect to ∇. Due to the last proposition, we note that the identity (2.15) defines a more general class of manifolds (M, g, S) than the class with the condition (2.14). Farther in this paper, we will investigate the properties of manifolds in these two classes.

Almost Einstein manifolds
In this section we consider manifolds (M, g, S) with the property (2.15).
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.
Proof. The local form of (2.15) is Vol. 111 (2020) Four-dimensional almost Einstein manifolds Page 5 of 18 9 Then, using (2.1), we find the equalities By applying the Bianchi identity to the above components of R, we obtain (3.1).
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 where α and β are smooth functions on M .
Let (M, g, S) satisfy the conditions of Theorem 3.4. If we suppose that (M, g, S) is an Einstein manifold, then its Ricci tensor ρ has the form (3.6). Hence (3.10) implies the following 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 (2.1) 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 (2.1) 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.

From (2.2) and (2.3) it follows
Now, due to (2.3) and (2.5), we can determine the angle ϕ between x and Sx, and the angle φ between x and S 2 x as follows: We apply (4.3) and (4.4) in (4.5) and find Then, bearing in mind (2.3) and (4.4), we get (4.1).

Definition 4.2. 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 4.3. Every nonzero vector
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 (4.2) 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 (4.6) is valid, we have = 0, which implies that x, Sx, S 2 x and S 3 x form a basis.

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 .  Sx).
Due to Theorem 5.1 and Corollary 3.5 we establish the following Now, we recall that the Ricci curvature in the direction of a nonzero vector x is the value  Proof. In the course of the proof of Theorem 3.4, we find that ρ is given by (3.10). Then, using (2.3), we obtain Proof. The above equalities follow directly by substituting τ * = 0 into (5.6).

Manifolds with parallel structures
In this section we study a manifold (M, g, S), whose structure S satisfies (2.14). Also, we consider an associated manifold (M, g, J) with a structure J = S 2 . Bearing in mind (2.1) and (2.3), 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 (2.14) implies (6.1).
Theorem 6.1. Let (M, g, S) have the property (2.14). Then the scalar curvature τ and τ * satisfy 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 (2.14), then it satisfies (2.15). Therefore, the Ricci tensor has the expression (3.9). Hence, from (2.1), (2.4), (2.7), (2.8) and (3.9), we get where δ i k are the Kronecker symbols. Using the above equalities, (2.14) and (6.3) we obtain where because of (2.1) it follows (6.2).

Conditions for parallel structures
Theorem 6.2. The manifold (M, g, S) satisfies (2.14) if and only if Proof. If Γ s ij are the Christoffel symbols of ∇, then Together with (2.14), (6.5) yields From (2.1) and (6.6) we get we obtain conditions (6.4).
Bearing in mind Theorems 6.2 and 6.3 we state the following Corollary 6.4. 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: where λ i ∈ R.
In this section we investigate a manifold (G, g, S) with a Lie algebra g determined by (7.4), i.e., a manifold (G, g, S) with an Abelian structure S.