On some curvature conditions of pseudosymmetry type

It is known that the difference tensor $$R \cdot C - C \cdot R$$R·C-C·R and the Tachibana tensor $$Q(S,C)$$Q(S,C) of any semi-Riemannian Einstein manifold $$(M,g)$$(M,g) of dimension $$n \ge 4$$n≥4 are linearly dependent at every point of $$M$$M. More precisely $$R \cdot C - C \cdot R = (1/(n-1))\, Q(S,C)$$R·C-C·R=(1/(n-1))Q(S,C) holds on $$M$$M. In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies $$R \cdot C = C \cdot R = Q(S,C) = 0$$R·C=C·R=Q(S,C)=0. Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors $$R \cdot C - C \cdot R$$R·C-C·R and $$Q(S,C)$$Q(S,C) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.


Introduction
Let ∇, R, S, κ and C be the Levi-Civita connection, the Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature tensor and the Weyl conformal curvature tensor of a semi-Riemannian manifold (M, g), n = dim M ≥ 2, respectively.
It is well-known that the manifold (M, g), n ≥ 3, is said to be an Einstein manifold ( [1]) if at every point of M its Ricci tensor S is proportional to the metric tensor g, i.e., S = κ n g on M. In particular, if S vanishes on M then it is called Ricci flat. We denote by U S the set of all points of (M, g) at which S is not proportional to g, i.e., U S := {x ∈ M | S − κ n g = 0 at x}. The manifold (M, g), n ≥ 3, is said to be a quasi-Einstein manifold if at every point x ∈ U S we have rank (S − α g) = 1, for some α ∈ R, i.e., S = α g + ε w ⊗ w, for some α ∈ R, where w is a non-zero covector at x and ε = ±1. We mention that quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations and investigation on quasi-umbilical hypersurfaces of conformally flat spaces, see, e.g., [2] and references therein.
An extension of the class of semi-Riemannian Einstein manifolds is formed by the manifolds for which we have ∇ S = 0. Manifolds satisfying the last condition are called Riccisymmetric. Locally symmetric manifolds, for which we have ∇ R = 0, constitute an important subclass of the class of Ricci-symmetric manifolds. The last relation implies the integrability condition where R(X, Y )· denotes the derivation obtained from the curvature endomorphism R(X, Y ) and X, Y are vector fields on M. We refer to Sect. 2 for the precise definitions of the symbols used here. Manifolds satisfying (1.1) are called semisymmetric manifolds ( [3]). Semisymmetric manifolds form a subclass of the class of pseudosymmetric manifolds. A semi-Riemannian manifold (M, g), n ≥ 3, is said to be pseudosymmetric ( [2,4,5]) if the tensors R(X, Y ) · R and (X ∧ g Y ) · R are linearly dependent at every point of M. This is equivalent to on U R := {x ∈ M | R − κ (n−1)n G = 0 at x}, where L R is some function on this set and the tensor G is defined by G(X, Y, W, Z ) = g(X ∧ g Y (Z ), W ). A geometric interpretation of the notion of pseudosymmetry is given in [6]. Further, a semi-Riemannian manifold (M, g), n ≥ 4, is said to be a manifold with pseudosymmetric Weyl tensor ( [2,4,7]) if the tensors C(X, Y ) · C and (X ∧ g Y ) · C are linearly dependent at every point of M. This is equivalent to on U C := {x ∈ M | C = 0 at x}, where L C is some function on this set, C(X, Y )· denotes the derivation obtained from the Weyl conformal curvature endomorphism C(X, Y ), and the Weyl conformal curvature tensor C is defined by C(X, Y, W, Z ) = g(C(X, Y )(Z ), W ). It is known that (1.3) is invariant under the conformal deformations of the metric tensor g. We also note that In what follows, for a (0, k)-tensor T and a symmetric (0, 2)-tensor A on a manifold (M, g) we will denote the tensors R(X, Y ) · T , C(X, Y ) · T and (X ∧ A Y ) · T by R · T , C · T and Q(A, T ), respectively. The tensor Q(A, T ) is called the Tachibana tensor (see, e.g., [8]). In particular, we have the following (0, 6)-tensors: R · R, R · C, C · R, C · C and the Tachibana tensors: Q(g, R), Q(S, R), Q(g, C), Q(S, C). Then we can write (1.2) and (1.3) in the form and C · C = L C Q(g, C), (1.5) respectively. We note that if (1.4) and (1.5) hold on the subset U = U S ∩ U C of a semi-Riemannian manifold (M, g), n ≥ 4, then on this set, where λ is some function ([7] Theorem 3.1). In addition, if (M, g) is a nonquasi-Einstein manifold then from (1.6) it follows that on some open subset U 1 ⊂ U its curvature tensor R is a linear combination of the Kulkarni-Nomizu products S ∧ S, g ∧ S and G = 1 2 g ∧ g, i.e., on U 1 , where φ, μ and η are some functions on this set ([7] Theorem 3.2 (ii)). A semi-Riemannian manifold (M, g), n ≥ 4, satisfying (1.7) on U S ∩ U C ⊂ M is called a Roter type manifold ( [9]). We refer to [10] for a survey on that class of manifolds. We can check that on any Einstein manifold (M, g), n ≥ 4, the tensors Q(g, R), Q(S, R), Q(g, C) and Q(S, C) satisfy Further, in [11](Theorem 3.1) it was stated that on every Einstein manifold (M, g), n ≥ 4, the following identity is satisfied: The remarks above lead to the problem of investigation of curvature properties of non-Einstein and non-conformally flat semi-Riemannian manifolds (M, g), n ≥ 4, satisfying at every point of M the curvature condition, of the following form: the difference tensor R · C − C · R is proportional to Q(g, R), Q(S, R), Q(g, C) and Q(S, C). Such conditions are strongly related to some pseudosymmetry type curvature conditions, see, e.g., [2] and references therein. In Sect. 2 we present the definitions of the most important conditions of pseudosymmetry type. For instance, (1.2) and (1.3) are conditions of this kind. We also note that there are manifolds for which the difference tensor R · C − C · R is a linear combination of the Tachibana tensors above, see, e.g., Sect. 5 of the present paper, [12](Theorem 5.1) and [13](Propositions 2.1 and 3.2). In this paper we will investigate semi-Riemannian manifolds (M, g), n ≥ 4, satisfying at every point of M the following condition: the tensors R · C − C · R and Q(S, C) are linearly dependent.
( * ) It is obvious that ( * ) is satisfied at every point of M at which C vanishes. It is also clear that (1.8) and (1.9) imply that holds on any Einstein manifold (M, g), n ≥ 4. Therefore we will restrict our considerations to manifolds (M, g), n ≥ 4, satisfying ( * ) on the set U = U S ∩ U C ⊂ M. We will investigate on U the condition where L is some function on this set. We mention that if the tensor R · C − C · R vanishes on U, then on this set we have ([11] Theorem 4.1) The main result of Sect. 3 (Theorem 3.4) states that pseudosymmetric manifolds satisfying some additional curvature conditions are quasi-Einstein manifolds satisfying the conditions: C · C = 0, C · R = 0 and (1.10) with the function L = 1 n−1 . In that section an example of warped product manifolds satisfying assumptions of Theorem 3.4 is also given.
In Sect. 5 Roter type manifolds satisfying (1.10) are investigated. We prove (Theorem 5.2) that if (M, g), n ≥ 4, is a Roter type manifold with vanishing scalar curvature κ on U ⊂ M then (1.10), with L = −1, holds on this set. In Theorem 5.3 we present some converse statement. We show (Example 5.4) that under some conditions the Cartesian product of two semi-Riemannian spaces of constant curvature satisfies assumptions of Theorem 5.3.

Preliminaries
Throughout this paper, all manifolds are assumed to be connected paracompact manifolds of class C ∞ . Let (M, g) be an n-dimensional, n ≥ 3, semi-Riemannian manifold, let ∇ be its Levi-Civita connection and X(M) the Lie algebra of vector fields on M. We define on M the endomorphisms X ∧ A Y and R(X, Y ) of X(M) by respectively, where A is a symmetric (0, 2)-tensor on M and X, Y, Z ∈ X(M). The Ricci tensor S, the Ricci operator S and the scalar curvature κ of (M, g) are defined by S(X, Y ) = tr{Z → R(Z , X )Y }, g(S X, Y ) = S(X, Y ) and κ = tr S, respectively. The endomorphism C(X, Y ) is given by Finally, the (0, 4)-tensor G, the Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C of (M, g) are defined by Let B be a tensor field sending any X, Y ∈ X(M) to a skew-symmetric endomorphism B(X, Y ), and let B be a (0, 4)-tensor associated with B by It is well-known that the tensor B is said to be a generalized curvature tensor if the following conditions are fulfilled: For B as above, let B be again defined by (2.1). We extend the endomorphism B(X, Y ) to a derivation B(X, Y )· of the algebra of tensor fields on M, requiring that it commutes with contractions and If A is a symmetric (0, 2)-tensor then we define the (0, k + 2)-tensor Q(A, T ) by In this manner we obtain the (0, 6)-tensors B · B and Q(A, B). Substituting B = R or B = C, T = R or T = C or T = S, A = g or A = S in the above formulas, we get the tensors R · R, R · C, C · R, R · S, Q(g, R), Q(S, R), Q(g, C) and Q(g, S). For a symmetric (0, 2)-tensor E and a (0, k)-tensor T , k ≥ 2, we define their Kulkarni-Nomizu product E ∧ T by see [18]. The tensor E ∧ T will be called the Kulkarni-Nomizu tensor of E and T . The following tensors are generalized curvature tensors: R, C and E ∧ F, where E and F are symmetric (0, 2)-tensors. We have G = 1 2 g ∧ g and For symmetric (0, 2)-tensors E and F we have (see, e.g., [19] Sect. 3) We also have (cf. [18] eq. (3)) For a symmetric (0, 2)-tensor A we denote by A the endomorphism related to A by Further, let T be a (0, k)-tensor, k ≥ 2. We will call the tensor Q(A, T ) the Tachibana tensor of A and T , or the Tachibana tensor for short (see, e.g., [8]). By an application of (2.3) we obtain on M the identities From the tensors g, R and S we define the following (0, 6)-Tachibana tensors: Moreover, in both cases the following condition holds at x:

Some special generalized curvature tensors
Let e 1 , e 2 , . . . , e n be an orthonormal basis of T x M at a point x ∈ M of a semi-Riemannian manifold (M, g), n ≥ 3, and let g(e j , e k ) = ε j δ jk , where ε j = ±1 and h, i, j, k, l, m, r, s ∈ {1, 2, . . . , n}. For a generalized curvature tensor B on M we denote by Ric(B), κ(B) and W eyl(B) its Ricci tensor, scalar curvature and Weyl tensor, respectively. We have Let B hi jk , T hi jk and A i j be the local components of the generalized curvature tensors B and T and a symmetric (0, 2)-tensor A on M, respectively. The local components (B·T ) hi jklm and Q(A, T ) hi jklm of the tensors B · T and Q(A, T ) are the following: If we contract the last equation with g i j and g hm , then we obtain where V mi jk = g rs Ric(B) mr B si jk (see [2,11,12]). According to [9], a generalized curvature tensor B on a semi-Riemannian manifold (M, g), n ≥ 4, is called a Roter type tensor if We also have (Ric(B), G), and, equivalently, Sects. 1 and 4). respectively.
As an immediate consequence of Proposition 3.3 we have are satisfied on U S ∩ U C ⊂ M, then on this set we have    N , g). In Proposition 3.3(i) of [28] it was proved that if the conditions

Manifolds with parallel Weyl conformal curvature tensor
Let (M, g), n ≥ 4, be a semi-Riemannian manifold whose Weyl conformal curvature tensor is parallel, i.e., ∇C = 0 on M. It is obvious that the last condition implies R ·C = 0. Suppose, moreover, that the manifold (M, g) is neither conformally flat nor locally symmetric. Such manifolds are called essentially conformally symmetric manifolds, e.c.s. manifolds, in short (see, e.g., [29] and [30]). E.c.s. manifolds are semisymmetric manifolds (R · R = 0, [29] Theorem 9) satisfying κ = 0 and Q(S, C) = 0 ([29] Theorems 7 and 8). In addition, holds on M, where F is some function on M, called the fundamental function ( [30]). At every point of M we also have rank S ≤ 2 ([30] Theorem 5). We mention that the local structure of e.c.s. manifolds has already been described. We refer to [14] and [16] for the final results related to this subject. We also mention that certain e.c.s. metrics are realized on compact manifolds ( [15,17]). Suppose that F = 0 at x ∈ M. Now (4.1) implies rank S ≤ 1 at x. It is clear that if S vanishes, then (3.53) holds at x. If rank S = 1, then in view Proposition 3.9(ii) we also have (3.53) at x. Next, we assume that F is non-zero at x ∈ M. Then rank S = 2 at x. In this case (4.1) turns into (3.7) with B = R, Ric(B) = S, φ = F −1 , μ = 1 n−2 and η = 0. Therefore (3.10) and (3.11) reduce to C · R = 0 and C · C = 0, respectively. Consequently, (3.53) holds at x. Thus we have proved the following Remark 4.2 (i) E.c.s. warped product manifolds were investigated in [31], where examples of such manifolds are given. (ii) The manifolds studied in this section satisfy (1.10). They can be quasi-Einstein or not.
Moreover, the tensor C · C of such manifolds is the zero tensor.

Roter type manifolds satisfying (1.10)
We recall that if the curvature tensor R of a semi-Riemannian manifold (M, g), n ≥ 4, is a linear combination of the Kulkarni-Nomizu products S ∧ S, g ∧ S and G = 1 2 g ∧ g on U S ∩ U C ⊂ M, i.e., (1.7) holds on this set, then (M, g) is called a Roter type manifold. Such manifolds were investigated among others in [32] and [33]. We also refer to [10] for a survey on Roter type manifolds, as well as on Roter type hypersurfaces. Curvature properties of manifolds satisfying (1.7) are presented in Proposition 3.2 (for B = R).
Remark 5.1 ([11] Theorem 4.1 and Corollary 4.1) Let (M, g), n ≥ 4, be a semi-Riemannian manifold and let U = U S ∩ U C ⊂ M. If R · C − C · R = L Q(g, C) holds on U for some function L, then R · R = L Q(g, R) and C · R = 0 on this set. In particular, if R · C = C · R holds on U, then R · R = R · C = C · R = 0 on this set. Therefore we consider manifolds satisfying (1.10) and (1.7) on U with non-zero function L.
As an immediate consequece of Proposition 3.2(ii) we get Theorem 5.2 Let (M, g), n ≥ 4, be a a semi-Riemannian manifold satisfying (1.7) on U = U S ∩ U C ⊂ M. If κ = 0 on U then (1.10), with L = −1, holds on this set.
We also have a converse to this result. (M, g), n ≥ 4, be a semi-Riemannian manifold satisfying (1.10) and (1.7) on U = U S ∩ U C ⊂ M, and let U 1 ⊂ U be the set of all points at which the functions L and L C , defined by (1.10), (3.10) and (3.11) (for B = R), respectively, are nowhere zero on this set. Then on U 1 we have

Theorem 5.3 Let
Proof From (3.9)(b) and (3.10) we obtain on U This, together with (1.10), yields We restrict our considerations to the set U 1 . Now the last equation turns into We note that if we had rank (S + (L C − L R )L −1 g) = 1 at a point of U 1 then -in a standard way -we would obtain C = 0 from (1.7), which is a contradiction. Therefore on U 1 , where λ is a function on this set. Now (5.4), via (2.2), turns into The decomposition of R is unique on U 1 (see, e.g., [24] Lemma 3.1). Therefore (1.7) and (5.6) yield φ = λ and (a) μ = 1 n − 2 + φ(L C − L R )L −1 , .