generalized curvature

. For any semi-Riemannian manifold ( M, g ) we deﬁne some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces. 1


Introduction
Let (M, g) be a semi-Riemannian manifold. We denote by g, R, S, κ and C, the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature and the Weyl conformal curvature tensor of (M, g), respectively. Further, let A ∧ B be the Kulkarni-Nomizu product of symmetric (0, 2)-tensors A and B. Now we can define the (0, 2)-tensors S 2 and S 3 , the (0, 4)-tensors R · S, C · S and Q(A, B), and the (0, 6)-tensors R · R, R · C, C · R, C · C and Q(A, T ), where T is a generalized curvature tensor. For precise definitions of the symbols used, we refer to Section 2 of this paper, as well as to [32,Section 1], [35,Section 1], [36,Chapter 6] and [43,Sections 1 and 2].
A semi-Riemannian manifold (M, g), dim M = n ≥ 2, is said to be an Einstein manifold [2], or an Einstein space, if at every point of M its Ricci tensor S is proportional to g, i.e., S = κ n g (1.1) on M, assuming that the scalar curvature κ is constant when n = 2. According to [2, p. 432] this condition is called the Einstein metric condition.
Let (M, g) be a semi-Riemannian manifold of dimension dim M = n ≥ 3. We set E = g ∧ S 2 + n − 2 2 S ∧ S − κ g ∧ S + κ 2 − tr g (S 2 ) 2(n − 1) g ∧ g. (1.2) It is easy to check that the tensor E is a generalized curvature tensor. Further, we define the subsets U R and U S of M by U R = {x ∈ M | R− κ (n−1)n G = 0 at x} and U S = {x ∈ M | S − κ n g = 0 at x}, respectively, where G = 1 2 g ∧ g. If n ≥ 4 then we define the set U C ⊂ M as the set of all points of (M, g) at which which C = 0. We note that if n ≥ 4 then (see, e.g., [23]) An extension of the class of Einstein manifolds form quasi-Einstein, 2-quasi-Einstein and partially Einstein manifolds.
A semi-Riemannian manifold (M, g), dim M = n ≥ 3, is said to be a quasi-Einstein manifold, or a quasi-Einstein space, if rank (S − α g) = 1 (1.4) on U S ⊂ M, where α is some function on U S . It is known that every non-Einstein warped product manifold M × F N with a 1-dimensional (M, g) base manifold and a 2-dimensional manifold ( N , g) or an (n − 1)-dimensional Einstein manifold ( N, g), dim M = n ≥ 4, and a warping function F , is a quasi-Einstein manifold (see, e.g., [7,32]). A Riemannian manifold (M, g), dim M = n ≥ 3, whose Ricci tensor has an eigenvalue of multiplicity n − 1 is a non-Einstein quasi-Einstein manifold (cf. [22,Introduction]). We mention that quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations and the investigation on quasi-umbilical hypersurfaces of conformally flat spaces (see, e.g., [26,32] and references therein). Quasi-Einstein hypersurfaces in semi-Riemannian spaces of constant curvature were studied among others in [28,38,41,57] (see also [26] and references therein). Quasi-Einstein manifolds satisfying some pseudosymmetry type curvature conditions were investigated recently in [1,7,23,30,40].
Let (M, g), dim M = n ≥ 3, be a semi-Riemannian manifold. We note that (1.4) holds at a point x ∈ U S ⊂ M if and only if (S − α g) ∧ (S − α g) = 0 at x, i.e., (1.5), by a suitable contraction, we get immediately Using (1.1) we can easily check that the following equation is satisfied on any Einstein manifold (M, g) i.e., E = 0 on M, where the tensor E is defined by (1.2).
The semi-Riemannian manifold (M, g), dim M = n ≥ 3, will be called a partially Einstein manifold, or a partially Einstein space (cf. [5,Foreword], [75, p. 20]), if at every point x ∈ U S ⊂ M its Ricci operator S satisfies S 2 = λS + µId x , or equivalently, where λ, µ ∈ R and Id x is the identity transformation of T x M. Evidently, (1.6) is a special case of (1.8). Thus every quasi-Einstein manifold is a partially Einstein manifold. The converse statement is not true. Contracting (1.8) we get tr g (S 2 ) = λ κ + n µ. This together with (1.8) yields (cf. [24, Section 5]) In particular, a Riemannian manifold (M, g), dim M = n ≥ 3, is a partially Einstein space if at every point x ∈ U S ⊂ M its Ricci operator S has exactly two distinct eigenvalues κ 1 and κ 2 with multiplicities p and n − p, respectively, where 1 ≤ p ≤ n − 1. Evidently, if p = 1, or p = n − 1, then (M, g) is a quasi-Einstein manifold.
In Section 3 we present definitions of some classes of semi-Riemannian manifolds determined by curvature conditions of pseudosymmetry type. Investigations of semi-Riemannian manifolds satisfying some particular curvature conditions of pseudosymmetry type lead to Roter spaces (see Propositon 4.1). Roter spaces form an important subclass of the class of non-conformally flat and non-quasi-Einstein partially Einstein manifolds of dimension ≥ 4. A semi-Riemannian manifold (M, g), dim M = n ≥ 4, satisfying on U S ∩ U C ⊂ M the following equation where φ, µ and η are some functions on this set, is called a Roter type manifold, or a Roter manifold, or a Roter space (see, e.g., [6,Section 15.5], [21,32,33,36]). Equation (1.10) is called a Roter equation (see, e.g., [27,Section 1]). In Section 4 we present results on such manifolds. For instance, every Roter space (M, g), dim M = n ≥ 4, satisfies Let (M, g), dim M = n ≥ 4, be a non-partially-Einstein and non-conformally flat semi-Riemannian manifold. If its Riemann-Christoffel curvature R is at every point of U S ∩ U C ⊂ M a linear combination of the Kulkarni-Nomizu products formed by the tensors S 0 = g and S 1 = S, . . . , S p−1 , S p , where p is some natural number ≥ 2, then (M, g) is called a generalized Roter type manifold, or a generalized Roter manifold, or a generalized Roter type space, or a generalized Roter space. For instance, when p = 2, we have where φ, φ 1 , φ 2 , µ 1 , µ and η are functions on U S ∩ U C . Because (M, g) is a non-partially Einstein manifold, at least one of the functions µ 1 , φ 1 and φ 2 is a non-zero function. Equation (1.12) is called a Roter type equation (see, e.g., [27,Section 1]). We refer to [27,31,32,33,40,67,68,69,70,71] for results on manifolds (hypersurfaces) satisfying (1.12).

Preliminaries.
Throughout this paper, all manifolds are assumed to be connected paracompact manifolds of class C ∞ . Let (M, g), dim M = n ≥ 3, be a semi-Riemannian manifold, and let ∇ be its Levi-Civita connection and Ξ(M) the Lie algebra of vector fields on M. We define on M the endomorphisms X ∧ A Y and R(X, Y ) of Ξ(M) by respectively, where X, Y, Z ∈ Ξ(M) and A is a symmetric (0, 2)-tensor on M. The Ricci tensor S, the Ricci operator S and the scalar curvature κ of (M, g) are defined by respectively. The endomorphism C(X, Y ) is defined by Now the (0, 4)-tensor G, the Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C of (M, g) are defined by respectively, where X 1 , X 2 , . . . ∈ Ξ(M). For a symmetric (0, 2)-tensor A we denote by A the endomorphism related to A by g(AX, Y ) = A(X, Y ). The (0, 2)-tensors A p , p = 2, 3, . . ., are defined by A p (X, Y ) = A p−1 (AX, Y ), assuming that A 1 = A. In this way, for A = S and A = S we get the tensors S p , p = 2, 3, . . ., assuming that S 1 = S. Let B be a tensor field sending any X, Y ∈ Ξ(M) to a skew-symmetric endomorphism B(X, Y ), and let B be the (0, 4)-tensor associated with B by The tensor B is said to be a generalized curvature tensor if the following two 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, assuming that it commutes with contractions and B(X, Y ) · f = 0 for any smooth function f on M. Now for a (0, k)-tensor field T , k ≥ 1, we can define the (0, k + 2)-tensor B · T by 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 in the above formulas For a symmetric (0, 2)-tensor A and a (0, k)-tensor T , k ≥ 2, we define their Kulkarni-Nomizu tensor A ∧ T by (see, e.g., [23, Section 2]) It is obvious that the following tensors are generalized curvature tensors: R, C and A ∧ B, where A and B = T are symmetric (0, 2)-tensors. We have and (see, e.g., [23, By an application of (2.4)(a) we obtain on M the identities Further, by making use of (2.2), (2.3) and (2.5), we get immediately From (2.4) (a) it follows immmediately that Q(g, g ∧ g) = 0. Thus we have where the tensor E is defined by (1.2).

From (2.8) we get easily (see also [23, Lemma 2.2(iii)] and references therein)
Let A be a symmetric (0, 2)-tensor and T a (0, k)-tensor, k = 2, 3, . . .. The tensor Q(A, T ) is called the Tachibana tensor of A and T , or the Tachibana tensor for short (see, e.g., [34]). Using the tensors g, R and S we can define the following (0, 6)-Tachibana tensors: Q(S, R), Q(g, R), Q(g, g ∧ S) and Q(S, g ∧ S). We can check, by making use of (2.4)(a) and (2.5), that other (0, 6)-Tachibana tensors constructed from g, R and S may be expressed by the four Tachibana tensors mentioned above or vanish identically on M.
Let A be a symmetric (0, 2)-tensor on a semi-Riemannian manifold (M, g), dim M = n ≥ 3. We denote by U A the set of points of M at which A = trg(A) n g.
This, by suitable contractions yields respectively. From (2.20), by symmetrization in l, j, we obtain From  .14), completing the proof of (ii).
A semi-Riemannian manifold (M, g), dim M = n ≥ 4, is said to be Weyl-pseudosymmetric if the tensors R · C and Q(g, C) are linearly dependent at every point of M [23,26]. This is equivalent on U C to R · C = L 1 Q(g, C), (3.8) where L 1 is some function on U C . We can easily check that on every Einstein manifold (M, g), dim M ≥ 4, (3.8) turns into For a presentation of results on the problem of the equivalence of pseudosymmetry, Riccipseudosymmetry and Weyl-pseudosymmetry we refer to [26,Section 4].
Warped product manifolds M × F N , of dimension ≥ 4, satisfying on U C ⊂ M × F N , the condition R · R − Q(S, R) = L Q(g, C), (3.10) where L is some function on U C , were studied among others in [10]. In that paper necessary and sufficient conditions for M × F N to be a manifold satisfying (3.10) are given. Moreover, in that paper it was proved that any 4-dimensional warped product manifold M × F N , with a 1-dimensional base (M , g), satisfies (3.10) [10, Theorem 4.1].
If we set Λ = 0 in (4.2) then we obtain the line element of the Reissner-Nordström spacetime, see, e.g., [58, Section 9.2] and references therein. It seems that the Reissner-Nordström spacetime is the oldest example of the Roter warped product space.
(iii) In [39] a particular class of Roter warped product spaces was determined such that every manifold of that class admits a non-trivial geodesic mapping onto some Roter warped product space. Moreover, both geodesically related manifolds are pseudosymmetric of constant type.
(iii) An algebraic classification of the Roter type 4-dimensional spacetimes is given [8].
(iv) Some comments on pseudosymmetric manifolds (also called Deszcz symmetric spaces), as well as Roter spaces, are given in [9, Section 1] (see also [8, Remark 2 (iii)], [39, Remark 2.1 (iii)]): "From a geometric point of view, the Deszcz symmetric spaces may well be considered to be the simplest Riemannian manifolds next to the real space forms." and "From an algebraic point of view, Roter spaces may well be considered to be the simplest Riemannian manifolds next to the real space forms." For further comments we refer to [77].
We finish this section with the following recent result on Roter spaces.

Warped product manifolds with 2-dimensional base manifold
where τ is some function on U S ∩ U C . N be the warped product manifold with a 2dimensional semi-Riemannian manifold (M , g), an (n−2)-dimensional semi-Riemannian manifold ( N, g), n ≥ 4, a warping function F , and let ( N, g) be a space of constant curvature when n ≥ 5. Then (5.1) holds on U S ∩ U C ⊂ M × F N .
It is well-known that the Cartesian product S 1 (r 1 ) × S n−1 (r 2 ) of spheres S 1 (r 1 ) and S n−1 (r 2 ), n ≥ 4, and more generally, the warped product manifold S 1 (r 1 ) × F S n−1 (r 2 ) of spheres S 1 (r 1 ) and S n−1 (r 2 ), n ≥ 4, with a warping function F , is a conformally flat manifold. (ii) As it was stated in [56, Example 3.2], the Cartesian product S p (r 1 ) × S n−p (r 2 ) of spheres S p (r 1 ) and S k (r 2 ) such that 2 ≤ p ≤ n−2 and (n−p −1)r 2 1 = (p −1)r 2 2 is a non-conformally flat and non-Einstein manifold satisfying the Roter equation (1.10) on U S ∩U C = S p (r 1 ) ×S n−p (r 2 ). (iii) [42, Example 4.1] The warped product manifold S p (r 1 ) × F S n−p (r 2 ), 2 ≤ p ≤ n − 2, with some special warping function F , satisfies on U S ∩ U C ⊂ S p (r 1 ) × S n−p (r 2 ) the Roter equation (1.10). Thus some warped product manifolds S 2 (r 1 ) × F S n−2 (r 2 ) are Roter spaces. (iv) Properties of pseudosymmetry type of warped products with 2-dimensional base manifold, a warping function F , and an (n − 2)-dimensional fibre, n ≥ 4, assumed to be of constant curvature when n ≥ 5, were determined in [32,Sections 6 and 7]. Evidently, warped product manifolds S 2 (r 1 ) × F S n−2 (r 2 ), n ≥ 4, are such manifolds. Let g, R, S, κ and C denote the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature and the Weyl conformal curvature tensor of S 2 (r 1 )× F S n−2 (r 2 ), respectively. From [32,Theorem 7.1] it follows that on set V of all points of U S ∩ U C ⊂ S 2 (r 1 ) × F S n−2 (r 2 ) at which the tensor S 2 is not a linear combination of the tensors S and g, the Weyl tensor C is expressed by where λ is some function on V . This, by (2.2), turns into Thus (1.12) is satisfied on V . Moreover, (1.10) holds at all points of (U S ∩ U C ) \ V , at which (1.4) is not satisfied. From Lemma 2.2 it follows that (5.2) holds at all points of U S ∩ U C ⊂ S 2 (r 1 ) × F S n−2 (r 2 ), n ≥ 4, at which (1.4) is not satisfied. Finally, in view of Theorem 2.4, we can state that (5.1) holds on U S ∩ U C .

Hypersurfaces in conformally flat spaces
Let M, dim M = n ≥ 4, be a hypersurface isometrically immersed in a semi-Riemannian conformally flat manifold N, dim N = n + 1. Let g ad , H ad , G abcd = g ad g bc − g ac g bd and C abcd be the local components of the metric tensor g, the second fundamental tensor H, the (0, 4)-tensor G and the Weyl conformal curvature tensor C of M, respectively. As it was stated in [46, eq. (20)] (see also [51,  where ε = ±1, tr(H) = g ad H ad , H 2 ad = g bc H ab H cd and µ is some function on M. From (7.1), by contraction we get easily µ = ε (n − 2)(n − 1) ((tr(H)) 2 − tr(H 2 )) , (7.2) where tr(H 2 ) = g ad H 2 ad . Now (7.1) and (7.2) yield If H = tr(H) n g at a point x ∈ M, i.e., M is umbilical at x, then from (7.3) it follows immediately that the tensor C vanishes at x. If at a non-umbilical point x ∈ M, we have rank(H − αg) = 1, for some α ∈ R, i.e., M is quasi-umbilical at x, then in view of Proposition 2.1 (i), the tensor C vanishes at x. Conversely, if at a non-umbilical point x ∈ M the tensor C vanishes then in view of Proposition 2.1 (ii) we have rank(H − αg) = 1, for some α ∈ R. Thus we can present [46,Theorem 4.1] in the folowing form. Remark 7.2. Let M, dim M = n ≥ 4, be a hypersurface isometrically immersed in a semi-Riemannian conformally flat manifold N, dim N = n + 1.
(ii) The above presented result, i.e., if (7.4) is satisfied at every point of U C ⊂ M then (3.9) holds on this set, was already obtained in [50, Proposition 3.1]. We mention that Proposition 3.1 of [50] was proved without application of [63, Theorem 3.1 (i)].
(iv) Recently curvature properties of pseudosymmetry type of hypersurfaces isometrically immersed in a semi-Riemannian conformally flat manifold were investigated in [53] and [64].
Let N n+1 s (c), n ≥ 4, be a semi-Riemannian space of constant curvature with signature (s, n + 1 − s), where c = κ n(n+1) and κ is its scalar curvature. Let M, dim M = n ≥ 4, be a connected hypersurface isometrically immersed in N n+1 s (c). We denote by U H ⊂ M the set of all points at which the tensor H 2 is not a linear combination of the metric tensor g and the second fundamental tensor H of M. We have U H ⊂ U S ∩ U C ⊂ M (see, e.g., [29,34,35] or [55, p. 137]). Further, we assume that the following conditions are satisfied on U H ⊂ M H 3 = tr(H) H 2 + ψ H + ρ g and C · C = Q(g, T ), (7.9) where T is a generalized curvature tensor and ψ and ρ some functions on U H . Then in view of [35,Theorem 4.5] on U H , where λ 1 is some function on this set. Using (1.2), (7.9) and (7.10) we get immediately Q(g, E) (7.11) on U H , where the tensor E is defined by (1.2) and λ is some function on this set. In addition, if we assume that (3.9) holds on U H then (3.9) and (7.11) give κ + 2εψ n − 1 − κ n + 1 − L C C = n − 3 (n − 2) 2 (n − 1) E + λ 2 2 g ∧ g on U H , where λ 2 is some function on this set. We note that the last equation, by a contraction with the metric tensor g, yields λ 2 = 0. Thus κ + 2εψ n − 1 − κ n + 1 − L C C = n − 3 (n − 2) 2 (n − 1) E on U H .