In [12, 13, 14, 15, 16], the authors introduced some curvature tensors similar to the projective curvature tensor of [9]. They investigated their geometrical properties and physical significance. These tensors have been recently studied in different ambient spaces [1, 4, 5, 18, 17, 20, 11]. However, we have noticed that little attention has been paid to the W3⋆–curvature tensor. This tensor is a (0, 4) tensor defined asW3⋆(U,V,Z,T)=R(U,V,Z,T)-1n-1[g(V,Z) Ric (U,T)-g (V,T) Ric (U,Z)],

where R(U, V,Z, T) = g(R((U, V) Z, T), R(U, V)Z = ∇_U∇_V −∇_U∇_V −∇_{[U,V ]}Z is the Riemann curvature tensor, ∇ is the Levi-Civita connection, and Ric (U, V) is the Ricci tensor. For simplicity, we will denote W3⋆ by ; in local coordinates, it is

The –curvature tensor has neither symmetry nor cyclic properties.

A semi-Riemannian manifold M is semi-symmetric [19] if

where R(ζ, ξ) acts as a derivation on R. M is Ricci semi-symmetric [8] if

where R(ζ, ξ) acts as a derivation on Ric. A semi-symmetric manifold is known to be Ricci semi-symmetric as well. The converse does not generally hold. Along the same line of the above definitions we say that M has a semi-symmetric –curvature tensor if

where R(ζ, ξ) acts as a derivation on .

This study was designed to fill the above mentioned gap. The relativistic significance of the –curvature tensor is investigated. First, it is shown that space-times with semi-symmetric Wjk⋆=gilWijkl⋆ tensor have Ricci semi-symmetric tensor and consequently the energy-momentum tensor is semi-symmetric. The divergence of the –curvature tensor is considered and it is proved that the energy-momentum tensor T of a space-time M is of Codazzi type if M has a divergence free –curvature tensor. If M admits a parallel –curvature tensor, then T is a parallel. Finally, a –flat perfect fluid space-time performs as a cosmological constant. A dust fluid –flat space-time satisfies Einstein’s field equation is a vacuum space.

A 4–dimensional relativistic space-time M is said to have a semi-symmetric-curvature tensor if

where R(ζ, ξ) acts as a derivation on the tensor . In local coordinates, one gets

Contracting both sides with g^il yields

where Wjk⋆=gilWijkl⋆. Thus we have the following theorem.

Theorem 2.1

M is Ricci semi-symmetric if and only ifWjk⋆=gilWijkl⋆is semi-symmetric.

The following result is a direct consequence of this theorem.

Corollary 2.2

M is Ricci semi-symmetric if the-curvature is semi-symmetric.

A space-time manifold is conformally semi-symmetric if the conformal curvature tensor is semi-symmetric.

Theorem 2.3

Assume that M is a space-time admitting a semi-symmetricWjk⋆=gilWijkl⋆. Then, M is conformally semi-symmetric if and only if it is semi-symmetric i.e..

The Einstein’s field equation is

where Λ,R, k are the cosmological constant, the scalar curvature, and the gravitational constant. Then

i.e., M is Ricci semi-symmetric if and only if the energy-momentum tensor is semi-symmetric.

Theorem 2.4

The energy-momentum tensor of a space-time M is semi-symmetric if and only ifWjk⋆=gilWijkl⋆is semi-symmetric.

Remark 2.5

A space-time M with semi-symmetric energy-momentum tensor has been studied by De and Velimirovic in [2].

It is clear that ∇μWijkl⋆=0 implies (∇μ∇ν-∇ν∇μ) Wijkl⋆=0. Thus the following result rises.

Corollary 2.6

Let M be a space-time having a covariantly constant–curvature tensor. Then M is conformally semi-symmetric and the energy-momentum tensor is semi-symmetric.

A space-time is called Ricci recurrent if the Ricci curvature tensor satisfies

where b is called the associated recurrence 1–form. Assume that the Ricci tensor is recurrent, then

Corollary 2.7

The following conditions on a space-time M are equivalent

• The Ricci tensor is recurrent with closed recurrence one form,

• T is semi-symmetric, and

• Wjk⋆=gilWijkl⋆is semi-symmetric.

The tensor Wjkl⋆h of type (1, 3) is given by Wjkl⋆h=ghiWijkl⋆=Rjklh-13[gjkRlh-gilRkh].

Consequently, one defines its divergence as

It is well known that the contraction of the second Bianchi identity gives

Thus, equation (3.1) becomes

If the –curvature tensor is divergence free, then equation (3.2) turns into

Multiplying by g^jk we have

Thus, the tensor R_ij is a Codazzi tensor and R is constant. Conversely, assume that the Ricci tensor is a Codazzi tensor. Then

However, the last equation implies that ∇_lR = 0. Consequently, the –curvature tensor has zero divergence.

Theorem 3.1

The–curvature tensor has zero divergence if and only if the Ricci tensor is a Codazzi tensor. In both cases, the scalar curvature is constant.

The divergence of the Weyl curvature tensor is given by

Remark 3.2

Since divergence free of –curvature tensor implies that R_ij is a Codazzi tensor, the conformal curvature tensor has zero divergence.

Equation (2.3) yields

The above theorem now implies the following result.

Corollary 3.3

The energy-momentum tensor is a Codazzi tensor if and only if the–curvature tensor has zero divergence. In both cases, the scalar curvature is constant.

Einstein’s field equation infers

Now, it is noted that the above theorem may be proved using this identity.

A space-time M is called -symmetric if

Applying the covariant derivative on the both sides of equation (1.1), one gets

If M is a –symmetric space-time, then

Multiplying the both sides by g^il, we get

and hence

Now, the following theorem rises.

Theorem 4.1

Assume that M is a–symmetric space-time, then M is a Ricci symmetric if the scalar curvature is constant.

The second Bianchi identity for –curvature tensor is

If the Ricci tensor satisfies ∇_mR_il = ∇_lR_im, then

Conversely, if the above equation holds, then equation (4.3) implies

Multiplying the both sides with g^ik, then we have

which means that the Ricci tensor is of Codazzi type.

Theorem 4.2

The Ricci tensor satisfies ∇_mR_il = ∇_lR_im if and only if the–curvature tensor satisfiesequation (4.4).

For a purely electro-magnetic distribution, Eequation (2.3) reduces to

Its contraction with g^ij gives

In this case, it is T = R = 0. Thus equation (4.2) yields ∇_mT_jk = 0.

Theorem 4.3

The energy-momentum tensor of a–symmetric space-time obeying Einstein’s field equation for a purely electro-magnetic distribution is locally symmetric.

Now, we consider –flat space-times. Multiplying both sides of equation (1.1) by g^il yields

Thus, a Wjk⋆–curvature flat space-time is Einstein, i.e.,

Now, equation (1.1) becomes

Theorem 5.1

A space-time manifold M is Einstein if and only ifWjk⋆=0. Moreover, a–flat space-time has a constant curvature.

A vector field ξ is said to be a conformal vector field if

where ℒ_ξ denotes the Lie derivative along the flow lines of ξ and φ is a scalar. ξ is called Killing if φ = 0. Let T_ij be the energy-momentum tensor defined on M. ξ is said to be a matter inheritance collineation if

The tensor T_ij is said to have a symmetry inheritance property along the flow lines of ξ. ξ is called a matter collineation if φ = 0. A Killing vector field ξ is a matter collineation. However, a matter collineation is not generally Killing.

Theorem 5.2

Assume that M is a–flat space-time. Then, ξ is conformal if and only if ℒ_ξT = 2φT.

Corollary 5.3

Assume that M is a–flat space-time. Then, M admits a matter collineation ξ if and only if ξ is Killing.

Equations (5.1) and (2.3) imply

Taking the covariant derivative of 5.4 we get

Since a –curvature flat space-time has ∇_lR = 0, ∇_lT_ij = 0.

Theorem 5.4

The energy-momentum tensor of a–flat space-time is covariantly constant.

Let M be a space-time and Wklm⋆i=gijWjklm⋆ be a (1, 3) curvature tensor. According to [3], there exists a unique traceless tensor ℬklmi and three unique (0, 2) tensors , ℰ_kl such that

All of these tensors are given by

and

Assume that the –curvature tensor is traceless. Then

and consequently

Theorem 5.5

Assume that M is a space-time admitting a traceless–curvature tensor. Then, M is an Einstein space-time.

For a perfect fluid space-time with the energy density μ and isotropic pressure p, we have

where u_i is the velocity of the fluid flow with g_iju^j = u_i and u_iuⁱ = −1 [10, 6, 7]. In [2, Theorem 2.2], a characterization of such space-times is given. This result leads us to the following.

Theorem 5.6

Assume that the perfect fluid space-time M is–semi-symmetric. Then, M is regarded as inflation and this fluid acts as a cosmological constant. Moreover, the perfect fluid represents the quintessence barrier.

Using Equations (5.2), we have

Multiplying the both sides by g^ij we get

For –curvature flat space-times, the scalar curvature is constant and consequently

Again, a contraction of equation (5.7) with uⁱ leads to

The comparison between (5.8) and (5.10) gives

i.e., the perfect fluid performs as a cosmological constant. Then equation (5.6) implies

For a –flat space-time, the scalar curvature is constant. Thus μ = constant and consequently p = constant. Therefore, the covariant derivative of equation (5.12) implies ∇_lT_ij = 0.

Theorem 5.7

Let M be a perfect fluid–flat space-time obeyingequation (2.3), then the μ and p are constants and μ + p = 0 i.e. the perfect fluid performs as a cosmological constant. Moreover, ∇_lT_ij = 0.

The following results are two direct consequences of being –curvature flat.

Corollary 5.8

A–flat space-time M obeyingequation (4.7) is a Euclidean space.

Corollary 5.9

Let M be a dust fluid–flat space-time satisfyingequation (2.3) (i.e. T_ij = μu_iu_j). Then M is a vacuum space-time(i.e. T_ij = 0).