The basic difference between the Riemannian and semi-Riemannian geometry is the existence of a null vector, that is, a vector v satisfying g(v, v) = 0, where g is the metric tensor. The signature of the metric g of a Riemannian manifold is (+,+,+, ...+,+, +) and of a semi-Riemannian manifold is (−,−,−, ...+,+, +). Lorentzian manifold is a special case of semi-Riemannian manifold. The signature of the metric of a Lorentzian manifold is (−,+,+, ...+,+, +). In a Lorentzian manifold three types of vectors exist such as timelike, spacelike and null vector. In general, a Lorentzian manifold (M,g) may not have a globally timelike vector field. If (M,g) admits a globally timelike vector field, it is called time orientable Lorentzian manifold, physically known as spacetime. The foundations of general relativity are based on a 4-dimensional spacetime manifold.

Let (M,g) be an n-dimensional Lorentzian manifold with the Lorentzian metric g. A Lorentzian manifold is said to be recurrent [41] if at a point x ∈ M, there exists a 1-form A on some neighbourhood of x such that ∇_XR = A(X)R, where R denotes the curvature tensor of type (1, 3) and ∇ denotes the covariant differentiation with respect to X. In 1952, Patterson [33] introduced the notion of Ricci recurrent manifolds. A Lorentzian manifold (M,g) of dimension n is said to be Ricci recurrent if its Ricci tensor S satisfies the condition

where B is a non-zero 1-form. He denotes such a manifold by R_n. Ricci recurrent manifolds have been studied by several authors.

In 1995, De, Guha and Kamilya [11] introduced the notion of generalized Ricci recurrent manifolds. A non-flat Riemannian manifold is called generalized Ricci recurrent realizing the following relation

where A and B are two non-zero 1-forms, called associated 1-forms. Such a manifold is denoted by GR_n. If the 1-form B vanishes, then the manifold reduces to a Ricci recurrent manifold R_n. This justifies the name generalized Ricci recurrent manifolds and the symbol GR_n for it. A Lorentzian manifold (M,g) of dimension n ≥ 4 is named generalized Ricci recurrent spacetimes if the relation (1.1) holds.

To characterize generalized Ricci recurrent (GR₄) spacetimes we assume that the associated vector fields U, V corresponding to the 1-forms A, B respectively are timelike vector fields. In a recent paper [21] Mallick, De and De studied generalized Ricci recurrent manifolds with applications to relativity. In the same paper the authors constructed two examples of GR_n. Also Generalized Ricci recurrent manifolds have been studied by several authors.

A Riemannian or a semi-Riemannian manifold is said to be semisymmetric [40] if its curvature tensor R satisfies the condition

for all X, Y ∈ χ(M), where R(X, Y ) acts as a derivation on the curvature tensor R. Trivial examples of semisymmetric spaces are locally symmetric spaces and all two-dimensional Riemannian spaces. But a semisymmetric space is not necessarily locally symmetric. A fundamental study on such manifolds was made by Szabo [40]. In this connection we can mention the book of Boeckx, Kowalski and Vanhecke [3] and the References there in.

Also, a Riemannian or a semi-Riemannian manifold is said to be Ricci semisymmetric [31] if the Ricci tensor S of type (0,2) satisfies the condition

for all X, Y ∈ χ(M).

On the other hand, generalized Robertson-Walker (GRW) spacetimes were introduced in 1995 by Alias, Romero and Sánchez [1, 2].

A Lorentzian manifold M of dimension n ≥ 3 is named generalized Robertson-Walker (GRW) spacetime if it is the warped product M = I ×_q²M^* with base (I,−dt²), warping function q and the fibre (M^*, g^*) is an (n-1)-dimensional Riemannian manifold [1, 2, 8, 35, 36].

If M^* is a 3-dimensional Riemannian manifold of constant curvature, the spacetime is called a Robertson-Walker (RW) spacetime. Therefore, GRW spacetimes are a wide generalization of RW spacetimes on which standard cosmology is modelled. They include the Einstein-de Sitter spacetime, the static Einstein spacetime, the Friedman cosmological models, the de Sitter spacetime and hence applications as inhomogeneous spacetimes admitting an isotropic radiation [8, 35].

Lorentzian manifolds with Ricci tensor of the form

where α, β are scalar fields and U is a unit timelike vector field corresponding to the 1-form A(that is, g(U,U) = −1), are called perfect fluid spacetimes and are of interest in general relativity. In differential geometry they are named quasi Einstein. Semi-Riemannian quasi Einstein spaces arose in the study of exact solutions of Einstein’s field equations and in the investigation of quasi-umbilical hypersurfaces of Pseudo-Euclidean spaces [12, 13]. Form (1.2) of the Ricci tensor is implied by Einstein’s equation if the energymomentum tensor of the spacetime is perfect fluid with velocity vector field U. A spacetime is called perfect fluid if the energymomentum tensor is of the form

where μ is the energy density, p is the isotropic pressure, U is a unit timelike vector field (g(U,U) = −1) metrically equivalent to the 1-form A. The fluid is called perfect because of the absence of heat conduction terms and stress terms corresponding to viscosity [19].

In addition, p and μ are related by an equation of state governing the particular sort of perfect fluid under consideration. In general, this is an equation of the form p = p(μ, T₀), where T₀ is the absolute temparature. However, we shall only be concerned with situations in which T₀ is effectively constant so that the equation of state reduces to p = p(μ). In this case, the perfect fluid is called isentropic[19]. Moreover, if p = μ, then the perfect fluid is termed as stiff matter(see [39] , page 66). Einstein’s field equation is given by S(X,Y)=r2g(X,Y)=κT(X,Y), κ is the gravitational constant. Einstein’s equation implies that matter determines the geometry of spacetime and conversly, the motion of matter is determined by the metric of the space which is non-flat.

The conformal curvature tensor is defined by [34]:

where R is the curvature tensor of type (1, 3), S is the Ricci tensor of type (0, 2), Q is the Ricci operator given by g(QX, Y) = S(X, Y) and r denotes the scalar curvature.

Perfect fluid spacetimes in four dimensions with divergence free conformal curvature tensor (that is, divC = 0) were firstly investigated by Shepley and Taub [38] and successively by Sharma [37] and Coley [9].

Recently in [27] Mantica, Molinari and De extended some results to n-dimensional perfect fluids and proved the following:

Theorem 1.1.([27])

Let (M,g) be a perfect fluid spacetime, that is, the Ricci tensor is of the form R_ij = αg_ij + βu_iu_j. If ∇_ku_j = ∇_ju_kand∇hCijkh=0 , then

• (i) u_iis a concircular vector field and it is rescalable to a timelike vector X_jsuch that ∇_kX_j = ρg_jk,

• (ii) (M,g) is a GRW spacetime with Einstein fiber,

• (iii) The velocity vector annihilates the Weyl tensor, that is, uhCijkh=0.

Recently, De et al. [14, 15] studied conformally flat almost pseudo-Ricci symmetric spacetimes and spacetimes with semisymmetric energy momentum tensor respectively. Also in [23] Mallick, Suh and De studied spacetime with pseudo-projective curvature tensor. Also several authors studied spacetimes in different way such as [16, 22, 29] and many others. In [7] Chaki and Ray studied spacetimes with covariant constant energy momentum tensor.

Motivated by the above studies in the present paper we characterize generalized Ricci recurrent spacetimes GR₄. At first we determine the nature of the associated 1-forms of GR₄ spacetimes. Next in Section 3, we consider conformally flat GR₄ spacetimes with an additional restriction and prove that such a spacetime is a perfect fluid spacetime. As a consequence we obtain several corollaries. Section 4 is devoted to study GR₄ spacetimes with Codazzi type of Ricci tensor. In this section we prove that a GR₄ spacetime with Codazzi type of Ricci tensor is a GRW spacetime with Einstein fibre. Also state equation is obtained. In Section 5, it is shown that in a GR₄ spacetime with closed associated 1-forms the energymomentum tensor is semisymmetric and Weyl compatible. Finally, we give some examples of generalized Ricci recurrent spacetimes.

Putting Y = Z = e_i in (1.1), where {e_i} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, 1 ≤ i ≤ 4, we get

where r=∑i=14ɛiS(ei,ei), ɛ = g(e_i, e_i), from which one obtains

Interchanging X and Y in (2.2) and then subtracting we infer that

Suppose the vector fields U and V corresponding to the 1-forms A and B respectively are collinear. Then it follows at once from (2.3) that the 1-form A is closed if and only if B is closed.

However if r = constant, then from (2.1) we obtain that the 1-form A is closed if and only if B is closed. If r = 0, then from (2.1) we arrive at a contradiction. Therefore in a GR₄ the scalar curvature is non-zero.

From the above discussions we can state the following:

Proposition 2.1

If the associated vector fields of a GR₄spacetime are collinear or the scalar curvature is constant, then the associated 1-form A is closed if and only if B is closed. Also the scalar curvature is non-zero in a GR₄spacetime.

In this section we characterize conformally flat GR₄ spacetimes. In the paper [21] the authors proved the following:

Theorem 3.1.([21])

A conformally flat GR_nis a quasi Einstein manifold provided the associated vector fields are collinear.

It is to be noted that the basic geometric features of GR₄ spacetimes are also being maintained in the Lorentzian manifold which is necessarily a semi-Riemannian manifold. Hence all the results of GR_n Riemannian manifold are true in GR₄ spacetime. Only the form of the Ricci tensor will be changed, because in the spacetime the associated vector field corresponding to the 1-form A is assumed to be a unit timelike vector, that is, g(U,U) = −1. In the paper [21] the authors obtained the form of the Ricci tensor as

where λ is a scalar defined by B(X) = λA(X).

In the GR₄ spacetime we take U as a unit timelike vector field, that is, g(U,U) = −1. Hence (3.1) reduces to

which implies that the spacetime is a perfect fluid spacetime.

Thus we obtain the following:

Proposition 3.2

A conformally flat generalized Ricci recurrent spacetime is a perfect fluid spacetime, provided the associated vector fields are collinear.

Einstein’s field equation without cosmological constant is given by

being κ the Einstein’s gravitational constant, T is the energymomentum tensor ([39], [32]) describing the matter content of the spacetime.

From (3.2) and (3.3) we obtain

where A is a non-zero 1-form such that A(X) = g(X,U), for all X and U is a unit timelike vector field.

Equation (3.4) is of the form of a perfect fluid spacetime

where κp=-r+λ3 and κ(p+μ)=-r+4λ3 from which it follows that p=-r+λ3κ and μ=-λκ, p being the isotropic pressure and μ the energy density.

Therefore the state equation is p=13(μ-rκ). But in a generalized Ricci recurrent spacetime the scalar curvature is non-zero. Thus the state equation indicates that the fluid spacetime is not radiative due to the presence of r and κ. Moreover, the values of p and μ are in accordance with the present day observations.

The Ricci tensor is said to be Codazzi type [17] if (∇_XS)(Y,Z) = (∇_ZS)(X, Y ). Codazzi type of Ricci tensor implies by Bianchi’s 2nd identity that the scalar curvature r is constant. Mallick, De and De [21] proved

Theorem 4.1.([21])

A GR_nwith Codazzi type of Ricci tensor is a quasi Einstein manifold whose Ricci tensor is of the form

where t = g(U,U) and s = g(U, V ).

Since we consider the associated vector field U as a unit timelike vector, the equation (4.1) can be rewritten as

which implies that the spacetime is a perfect fluid. Thus we get the following:

Proposition 4.2

A GR₄spacetime with Codazzi type of Ricci tensor is a perfect fluid spacetime.

In an n-dimensional Lorentzian manifold, we know [24] that,

Therefore if the Ricci tensor is of Codazzi type, then from (4.3) it follows that divC = 0, that is, the Weyl conformal curvature tensor is divergence free.

Lemma 4.3

Let (M,g) be a perfect fluid spacetime, that is, S(X, Y ) = αg(X, Y )+ βA(X)A(Y ), where the vector field U metrically equivalent to the 1-form A is a unit timelike vector field and β ≠ 0. If div C = 0 and dr(X) = 0, then the 1-form A is closed.

Theorem 4.4

A generalized Ricci recurrent spacetime with Codazzi type of Ricci tensor is a GRW spacetime with Einstein fiber. Also the velocity vector field U satisfies the condition C(X, Y )U = 0.

Remark 4.5

For dimension n = 4, the condition C(X, Y )U = 0 means

where the vector field U is metrically equivalent to the 1-form A. The above equation implies that [20]

Now replacing W by U in the above expression and using C(X, Y )U = 0 yields C(X, Y )Z = 0. It is known [4] that a GRW spacetime M is conformally flat if and only if M is a RW spacetime. Thus in n = 4 dimension GRW spacetime reduces to RW spacetime. Therefore a GR₄ spacetime with Codazzi type of Ricci tensor is a RW spacetime.

Since a 4-dimensional spacetime with divC = 0 and dr(X) = 0 is a Yang Pure space [18], Theorem 4.4 can be restated as:

Corollary 4.6

Any 4-dimensional perfect fluid Yang Pure space with β ≠ 0 is a GRW spacetime with Einstein fiber.

Remark 4.7

Extensions and modifications of General Relativity have a prominent role in addressing the problem of dark energy and dark matter (the so-called dark side). A generalization of Einstein’s theory is the so-called f(R) theory of gravitation. It was introduced by Buchdahl [5] in 1970. In [6], Capozziello et al. proved that an n-dimensional GRW spacetime with divergence free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f(R) gravity model. In Theorem 4.4 of our paper, we prove that a generalized Ricci recurrent spacetime with Codazzi type of Ricci tensor is a GRW spacetime with Einstein fiber. Hence, from the result of Capozziello et al., we conclude that the spacetime under consideration in our paper proclaims a perfect fluid stress-energy tensor for any f(R) gravity model.

Now we consider Einstein’s equation without cosmological constant, that is,

being κ the Einstein’s gravitational constant, T is the energymomentum tensor [32, 39] describing the matter content of the spacetime.

From (4.2) and (4.9) we obtain

where A is a non-zero 1-form such that A(X) = g(X,U), for all X and U is a unit timelike vector field.

The above equation is of the form of a perfect fluid spacetime

where κp=s-r2 and κ(p+μ)=-r4 from which it follows that p=1κ(s-r2) and μ=1κ[r4-s], p being the isotropic pressure and μ the energy density.

Therefore the state equation is p=-μ-r4κ. Since by hypothesis the Ricci tensor is of Codazzi type, therefore the scalar curvature is constant. Thus the state equation reduces to p = −μ + constant.

In this section we characterize generalized Ricci recurrent spacetimes with constant scalar curvature. Taking covariant derivative of (1.1) we get

Interchanging X and Y in (5.1) we obtain

Also from (1.1) we have

Now subtracting (5.2) and (5.3) from (5.1) and using Ricci identity we get

From (2.3) it follows that if the scalar curvature is constant, then A(X)B(Y ) − A(Y )B(X) = 0, since the 1-forms A and B are closed. Hence from (5.4) we infer that

The foregoing equation implies that in such a spacetime under consideration the Ricci tensor is semisymmetric. Therefore from Einstein’s field equations we can conclude that the energy momentum tensor is semisymmetric, that is,

In a recent paper [15] De et al. characterize spacetimes with semisymmetric energy momentum tensor. Thus all the results of [15] also hold for GR₄ spacetimes with constant scalar curvature.

Equation (5.5) gives

Now summing cyclically the above equation and applying Bianchi’s first identity we get

Any semi-Riemannian manifold satisfying (5.6) is called Riemannian compatible [25]. Thus we have

Proposition 5.1

A GR₄spacetime with constant scalar curvature is Riemann compatible.

Any semi-Riemannian manifold satisfying

is called Weyl-compatible. Weyl-compatibility have been studied in the Riemannian case by Mantica et al [26].

It is known that both conditions (5.6) and (5.7) are equivalent. Now if we use Einstein’s equation in (5.7), we get

Therefore we can state the following:

Proposition 5.2

In a GR₄spacetime with constant scalar curvature the energymomentum tensor is Weyl-compatible.

Example 6.1.([10])

A generalized concircularly recurrent manifold with constant scalar curvature is a GR_n.

Example 6.2.([28])

A quasi-conformally recurrent manifold with divergence free quasi-conformal curvature tensor is a GR_n, provided the manifold is neither conformally flat nor conformally symmetric.

Example 6.3

The so called tensor is defined by

It may be noted that the vanishing of the tensor implies that the manifold to be an Einstein manifold and hence the tensor is a measure of the deviation from an Einstein manifold [30].

–recurrent manifold is defined by

where A is a non-zero 1-form. From (6.1) and (6.2) we obtain

which implies that the –recurrent manifold is a generalized Ricci recurrent manifold.

Conversely, if the manifold is generalized Ricci recurrent, then

and using (1.1) and (2.1) we get

Hence we conclude that a –recurrent manifold is a GR_n and conversely.

Remark 6.4

The above examples also hold for GR₄ spacetimes.