CURVATURE PROPERTIES OF GENERALIZED PP-WAVE METRICS

The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metrics. It is shown that a generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisfies the pseudosymmetric type condition P · P = − 1 3Q(S, P ). Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent. Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. Again the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally, we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.


Introduction
The class of pp-wave metrics (see [29,71]) arose during the study of exact solutions of Einstein's field equations. The term "pp-wave" is given by Ehlers and Kundt [29], where "pp" abbreviates the term "plane-fronted gravitational waves with parallel rays". The term "plane-fronted gravitational waves" means there admit a geodesic null vector field whose twist, expansion and shear are zero. The term "plane rays" implies that the rotation of the vector field vanishes. For vacuum type N, this ensures the existence of a covariantly constant vector field which is parallel to the null vector field. There are various forms of generalized pp-wave metrics in different coordinates. The pp-wave belongs to the class of solutions admitting a non-expanding, shear-free and twist-free null congruence and it admits a null Killing vector.
The family of pp-wave space-times was first discussed by Brinkmann [4] and interpreted in terms of gravitational waves by Peres [46]. According to Brinkmann, a pp-wave spacetime is any Lorentzian manifold whose metric tensor can be described, with respect to Brinkmann coordinates, in the form where H is any nowhere vanishing smooth function. Again it is well known that a Lorentzian manifold with parallel lightlike (null) vector field is called Brinkmann-wave ( [4,30]). A Brinkmann-wave is called pp-wave if its curvature tensor R satisfies the condition R ij pq R pqkl = 0 ( [30,44,69]). In 1984, Radhakrishna and Singh [47] presented a class of solutions to Einstein-Maxwell equation for the null electrovac Petrov type N gravitational field. They presented a metric of the form where U = U (u, x 3 , x 4 ) and P = P (x 3 , x 4 ) are two nowhere vanishing smooth functions. For the simplicity of notation, we write the variable u as x, and the function U as h and P as f . Then the aforesaid metric can be written as In Section 3 we show that the metric (1.2) admits a covariantly constant null vector field and it satisfies the condition R ij pq R pqkl = 0 if (1.3) f 2 3 + f 2 4 − f (f 33 + f 44 ) = 0. Thus we can say that the metric (1.2) is a Brinkmann-wave and it becomes a pp-wave if (1.3) holds. Hence we can say the metric (1.2) as "generalized pp-wave metric". We note that for f ≡ −2 and h = − 1 2 H(u, x 3 , x 4 ), the solution (1.2) reduces to the pp-wave metric (1.1).
The main object of the present paper is to investigate the geometric structures admitted by the generalized pp-wave metric (1.2). It is interesting to note that the metric (1.2) without any other condition admits several geometric structures, such as Ricci generalized pseudosymmetry, 2-quasi Einstein and generalized quasi-Einstein in the sense of Chaki [8]. Again it is shown that the pp-wave metric (i.e., (1.2) with condition (1.3)) is Ricci recurrent but not recurrent, semisymmetric, R-space by Venzi, conformal curvature 2-forms are recurrent, Ricci tensor is Riemann compatible and fulfills a pseudosymmetric type condition due to the projective curvature tensor P . For the study of pseudosymmetric type conditions with projective curvature tensor we refer the reader to see the recent papers of Shaikh and Kundu ( [58,59]). It is interesting to note that for such a metric P · R = 0 but P · R = 0.
It is also shown that the metric is weakly Ricci symmetric and weakly cyclic Ricci symmetric for different associated 1-forms, which ensures the existence of infinitely many solutions of associated 1-forms of such structures. Again we investigate the condition for which such a spacetime is locally symmetric and recurrent.
The paper is organized as follows. Section 2 deals with defining conditions of different curvature restricted geometric structures, such as recurrent, semisymmetry, pseudosymmetry, weakly symmetry etc. as preliminaries. Section 3 is devoted to the investigation of curvature restricted geometric structures admitted by the generalized pp-wave metric (1.2). Section 4 is mainly concerned with the geometric structures admitted by pp-wave metric and plane wave metric. Section 5 deals with the investigation of the conditions under which the energy-momentum tensor of such spacetimes are parallel, Codazzi type and cyclic parallel. Finally, the last section is devoted to make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric as well as pp-wave metric.

Preliminaries
Let M be a connected smooth semi-Riemannian manifold of dimension n (≥ 3) equipped with the semi-Riemannian metric g. Let R, R, S, S and κ be respectively the Riemann-Christoffel curvature tensor of type (0, 4), the Riemann-Christoffel curvature tensor of type (1,3), the Ricci tensor of type (0, 2), the Ricci tensor of type (1, 1) and the scalar curvature of M .
In terms of Kulkarni-Nomizu product, the conformal curvature tensor C, the concircular curvature tensor W , the conharmonic curvature tensor K ( [36,83]) and the Gaussian curvature tensor G can be expressed as Again the projective curvature tensor P of type (0, 4) is given by For a symmetric (0, 2)-tensor A, we get an endomorphism A defined by g(AX, Y ) = A(X, Y ). Then its k-th level tensor A k of type (0, 2) is given by where A k−1 is the endomorphism corresponding to A k−1 , k = 2, 3, . . ., and A 1 = A. Definition 2.1 ([3, 60]). A semi-Riemannian manifold M is said to be Ein(2), Ein (3) and Ein(4) respectively if Definition 2.2 ([3, 60]). A semi-Riemannian manifold M is said to be generalized Roter type ( [56,60]) if its Riemann-Christoffel curvature tensor R can be expressed as a linear combination of g ∧ g, g ∧ S, S ∧ S, g ∧ S 2 , S ∧ S 2 and S 2 ∧ S 2 . Again M is said to be Roter type (see [16,17]) if R can be expressed as a linear combination of g ∧ g, g ∧ S and S ∧ S.
Again for a (0, 4)-tensor D, an endomorphism D(X, Y ) and the corresponding (1, 3)-tensor D can be defined as Again for a symmetric (0, 2)-tensor A, another endomorphism X ∧ A Y (see [13,22]) can be defined as By operating D(X, Y ) and X ∧ A Y on a (0, k)-tensor B, k ≥ 1, we can obtain two (0, k + 2)-tensors D · B and Q(A, B) respectively given by (see [18,19,25,54,55] and also references therein): In terms of local coordinates system, D · B and Q(A, B) can be written as
are (0,4) curvature tensors. In particular, if c i 's are all constants, then it is called B-pseudosymmetric type manifold of constant type or otherwise non-constant type.

Definition 2.5.
A semi-Riemannian manifold M is said to be quasi-Einstein (resp. 2-quasi-Einstein) if at each point of M , rank(S − αg) ≤ 1 (resp., ≤ 2) for a scalar α. Also M is said to be generalized quasi-Einstein in the sense of Chaki [8] if for some 1-forms Π and Ω.
Quasi-Einstein, as well as 2-quasi-Einstein manifolds were investigated among others in [2, 9-11, 19, 21, 22] and [25]. Definition 2.6. Let D be a (0, 4)-tensor and E be a symmetric (0, 2)-tensor on a semi-Riemannian manifold M . Then E is said to be D-compatible ([20, 39, 40] Definition 2.7. A semi-Riemannian manifold M is said to be weakly cyclic Ricci symmetric [64] if its Ricci tensor satisfies the condition for three 1-forms Π, Ω and Θ on M . Such a manifold is called weakly cyclic Ricci symmetric manifold with solution (Π, Ω, Θ). Moreover if the first term of left hand side is equal to the right hand side, then it is called weakly Ricci symmetric manifold [77]. Definition 2.8. Let D be a (0, 4) tensor and Z be a (0, 2)-tensor on a semi-Riemannian manifold M . Then the corresponding curvature 2-forms Ω m (D)l (see [3,38]) are called recurrent if and only if ( [41][42][43]) and 1-forms Λ (Z)l [72] are called recurrent if and only if for an 1-form Π. Definition 2.9 ([48, 59, 81]). Let L(M ) be the vector space formed by all 1-forms Θ on a semi-Riemannian manifold M satisfying

Curvature Properties of Generalized pp-Wave Metric
We can now write the metric tensor g of the generalized pp-wave metric (1.2) as follows: Then the non-zero components of its Riemann-Christoffel curvature tensor R, Ricci tensor S and scalar curvature κ of (1.2) are given by Again the non-zero components of the conformal curvature tensor C and the projective curvature tensor P are given by Now the non-zero components (upto symmetry) of the energy momentum tensor where c = speed of light in vacuum, G = gravitational constant and Λ = cosmological constant, are given by Then the non-zero components (upto symmetry) of covariant derivative ∇T of the energy momentum tensor T are given by From above we see that the Ricci tensor S of (1.2) is of the form where α = Therefore, the metric (1.2) is generalized quasi-Einstein in the sense of Chaki. More- and ∇η = 0. So there exists a null covariantly constant vector field ζ, where ζ is the corresponding vector field of η (i.e., g(ζ, X) = η(X), for all X). Hence we can conclude that the spacetime with the metric (1.2) is a generalized pp-wave metric. Now from the value of the components of various tensors related to (1.2), we can state the following.
Now from the components of R, we see that the only non-zero component (upto . Hence from definition, we can state the following.
Hence by above theorem the metric is a pp-wave metric.

Curvature Properties of pp-Wave and Plane Wave Metric
In this section we investigate the curvature restricted geometric structures admitted by the pp-wave metric. Since under the condition (1.3) the generalized pp-wave metric (1.2) becomes a pp-wave metric, putting this condition we get the non-zero components of R, S, C and P of the pp-wave metric given as follows: Using the values of the components of g, R, S and C we get (i) κ = 0, (ii) R · R = 0, (iii) R · S = 0, (iv) Q(S, R) = 0, (v) R · C = 0, (vi) C · R = 0, (vii) C · C = 0 and (viii) Q(S, C) = 0. The energy momentum tensor T is given by Then the non-zero components of covariant derivative of T are given by From the above calculations, we can state the following. , where Π 1 and Ω 1 are arbitrary scalars. (ix) C · R = 0 and hence C · S = 0, C · C = 0 and C · P = 0. (x) P · R = 0 but P · R = 0. Also but P · S = P · S = 0. (xi) Ricci tensor is Riemann compatible as well as Weyl compatible. (xii) P · P = − 1 3 Q(S, P ). Remark 4.1. From the value of the local components (presented in Section 4) of various tensors of the pp-wave metric, we can easily conclude that the metric is (i) not conformally symmetric and hence not locally symmetric or projectively symmetric; (ii) not conformally recurrent and hence not recurrent or not projectively recurrent; (iii) not super generalized recurrent [67] and hence not hyper generalized recurrent [65], weakly generalized recurrent [66]; (iv) not weakly symmetric [76] for R, C, P, W and K and hence not Chaki pseudosymmetric [7] for R, C, P, W and K; (v) neither cyclic Ricci parallel [31] nor of Codazzi type Ricci tensor although its scalar curvature is constant; (vi) not harmonic, i.e., div R = 0 and moreover div C = 0, div P = 0.
Remark 4.2. In [12] it was shown that if Q(S, R) = 0, then R = LS ∧ S for some scalar L if S is not of rank 1. Recently, in Example 1 of [59] a metric with Q(S, R) = 0, on which S is not of rank 1 and R = e x 1 S ∧ S is presented. It is interesting to mention that the rank of the Ricci tensor of the pp-wave metric is 1 and here R = 0 but S ∧ S = 0. Remark 4.3. It is well-known that every Ricci recurrent space with Π as the 1-form of recurrency, is weakly Ricci symmetric with solution (Π, 0, 0). It is interesting to mention that there may infinitely many solutions for a weakly Ricci symmetric manifold. The pp-wave metric ((1.2) with condition (1.3)) is weakly Ricci symmetric with solution , where Π 1 and Ω 1 are arbitrary scalars.
Again it is clear that the pp-wave metric (1.1) in Brinkmann coordinates, is a special case of (1.   and hence S ∧ S = 0 and S 2 = 0. Here ||η|| = 0 and ∇η = 0. (ix) Q(S, R) = Q(S, C) = 0 but R or C is not a scalar multiple of S ∧ S as S is of rank 1. (x) C · R = 0 and hence C · S = 0, C · C = 0 and C · P = 0. (xi) P · R = 0 but P · R = 0. Also but P · S = P · S = 0. (xii) Ricci tensor is Riemann compatible as well as Weyl compatible. (xiii) P · P = − 1 3 Q(S, P ). Again the non-vacuum pp-wave solution presented in [69] is a special case of (1.1) for H(x, x 3 , x 4 ) = 2a 1 e a 2 x 3 −a 3 x 4 . Hence the line element is explicitly given by: Now the geometric properties of the metric (4.3) can be stated as follows. Again the generalized plane wave metric [71] is given by Hence, it is a special case of (1.1) and we can state the following. Remark 4.4. From Corollary 4.2 we see that the metric (4.3) is recurrent but not locally symmetric and from Corollary 4.1 we see that the metric (1.1) is Ricci recurrent but not recurrent. These results support the well-known facts that every locally symmetric manifold is recurrent but not conversely, and every recurrent manifold is Ricci recurrent but not conversely.

Energy-Momentum Tensor of Generalized pp-Wave Metric
In this section we discuss about the energy-momentum tensor of the generalized pp-wave metric (1.2) and also the other special forms, such as (1.1), (4.3) and (4.4).
From the values of the energy momentum tensor T of the generalized pp-wave metric (1.2), we can conclude that T is of rank 1 if the cosmological constant is zero and (1.3) holds. In this case Again it is easy to check that η is null. Thus we can state the following. We note that recently Shaikh et al. [62] studied the curvature properties of pure radiation metric. Again from the values of the components of ∇T of the metric (1.2), we get Now putting the condition (1.3) to above, we get Now, from (4.2), (5.1) and (5.2), we can state the following.
It is easy to check that the scalar curvature of this metric is zero and it is conformally flat. Now the non-zero components of its energy momentum tensor T and its derivative ∇T are given by Thus we can easily check that the Ricci tensor and the energy momentum tensor of (5.3) are codazzi type but not cyclic parallel.

Robinson-Trautman Metric and Generalized pp-Wave Metric
Recently, Shaikh et al. [53] studied the curvature properties of Robinson-Trautman metric. The line element of Robinson-Trautman metric in {t, r, x 3 , x 4 }-coordinate is given by where a, b, q are constants and f is a function of the real variables x 3 and x 4 . In this section we make a comparison between the curvature properties of the Robinson-Trautman metric (6.1) and generalized pp-wave metric (1.2) as well as the pp-wave metric (1.1). (v) (6.1) is pseudosymmetric due to conformal curvature tensor whereas (1.1) is semisymmetric due to conformal curvature tensor; (vi) the metric (6.1) realizes S ∧ S = 0 but (1.1) satisfies S ∧ S = 0; (vii) the metric (6.1) is Roter type and hence Ein(2) but (1.1) is not Roter type but Ein(3).