On warped product generalized Roter type manifolds

Transcription

1 On warped product generalized Roter type maniolds A. A. Shaikh, H. Kundu Dedicated to Proessor ajos Tamássy on his ninety-irst birthday Abstract. Generalized Roter type maniolds orm an extended class o Roter type maniolds, which gives rise the orm o the curvature tensor in terms o algebraic combinations o the undamental metric tensor and Ricci tensors upto level 2. The object o the present paper is to investigate the characterization o a warped product maniold to be a generalized Roter-type and hence as a special case or Roter type and conormally lat maniold. We also present an example by a metric which ensures the existence o a warped product generalized Roter type maniold but is not Roter type maniold. M.S.C. 2010: 53C15, 53C25, 53C35. Key words: Roter type maniold; generalized Roter type maniold; conormally lat maniold; Ricci tensors o higher levels; warped product maniold 1 Introduction et M be an n 3-dimensional connected semi-riemannian smooth maniold equipped with a semi-riemannian metric g. We denote the evi-civita connection, the Riemann-Christoel curvature tensor, Ricci tensor, scalar curvature and the space o all smooth unctions on M by, R, S, κ and C M respectively. The maniold M is lat i R = 0 and M is o constant curvature i R is a constant multiple o the Gaussian curvature tensor. For a conormally lat maniold M, R can be expressed as R = J 1 g g + J 2 g S, where J 1, J 2 C M. Especially, M is lat resp., constant curvature and conormally lat i J 1 = J 2 = 0 resp., J 1 = κ nn 1, J κ 2 = 0 and J 1 = 2n 1n 2, J 2 = 1 n 2. The maniold M is Roter type briely, RT n; see [, 17] i R can be expressed as a linear combination o g g, g S and S S. Very recently, Shaikh et al. [25] introduced the notion o generalized Roter type maniold. A maniold is said to be generalized Roter type briely, GRT n i its curvature tensor can be expressed as a linear combination o g g, g S, S S, g S 2, S S 2 and S 2 S 2. We note Balkan Journal o Geometry and Its Applications, Vol.21, No.2, 2016, pp c Balkan Society o Geometers, Geometry Balkan Press 2016.

2 On warped product generalized Roter type maniolds 83 that the name generalized Roter type was irst used in [25]. For general properties o GRT n and its proper existence we reer the readers to see [28] and also reerences therein. The paper is organized as ollows. Section 2 is concerned with preliminaries. Section 3 deals with warped product maniolds and various curvature relations. Section is devoted to the study o warped products GRT n and obtained the characterization o such maniolds see Theorem.1. We obtain the characterization o a warped product maniold to be RT n and conormally lat. The last section deals with the proper existence o such notion by an example with a suitable metric. 2 Preliminaries et M be an n 3-dimensional semi-riemannian maniold and S 2 be its level 2 Ricci tensor o type 0,2. In terms o local coordinates, the tensor S 2 can be expressed as S 2 ij = g kl S ik S jl. Similarly the Ricci tensors o level 3 and, S 3 and S are respectively deined as S 3 ij = g kl S 2 iks jl and S ij = g kl S 3 iks jl. Now or two 0, 2 tensors A and E, their Kulkarni-Nomizu product [5], [7], [11], [18] A E is given by A E ijkl = A il E jk + A jk E il A ik E jl A jl E ik. In particular, we can deine g g, g S, S S, g S 2, S S 2 and S 2 S 2 as ollows: g g ijkl = 2g il g jk g ik g jl, g S ijkl = g il S jk + S il g jk g ik S jl S ik g jl, S S ijkl = 2S il S jk S ik S jl, g S 2 ijkl = g il S 2 jk + S 2 ilg jk g ik S 2 jl S 2 ikg jl, S S 2 ijkl = S il S 2 jk + S 2 ils jk S ik S 2 jl S 2 iks jl, S 2 S 2 ijkl = 2S 2 ils 2 jk S 2 iks 2 jl. We note that the tensor 1 2 g g is known as Gaussian curvature tensor and is denoted by G. A tensor D o type 0, on M is said to be generalized curvature tensor [5], [7], [11], i id ijkl + D jikl = 0, iid ijkl = D klij, iiid ijkl + D jkil + D kijl = 0. Moreover i D satisies the second Bianchi identity, i.e., D ijkl,m + D jmkl,i + D mikl,j = 0, then D is called a proper generalized curvature tensor, where coma denotes the covariant derivative. I A and B are two symmetric 0, 2 tensors, then A B is obviously a generalized curvature tensor. We mention that there are various generalized curvature tensors which are linear combination o Riemann-Christoel curvature tensor with Kulkarni-Nomizu products

3 8 A. A. Shaikh, H. Kundu o some tensors. One such important curvature tensor is the conormal curvature tensor C, and is given by C = R 1 n 2 g S + κ 2n 1n 2 g g. We reer the readers to see [27] or details about the various curvature tensors and geometric structures along with their equivalency. Deinition 2.1. et M be a semi-riemannian maniold satisying the ollowing condition 2.1 R = N 1 g g + N 2 g S + N 3 S S, or some N 1, N 2 and N 3 C M. The above condition is called a Roter type condition and M is called a Roter type maniold [?, 6, 13, 15, 19, 21] with N 1, N 2 and N 3 as the associated scalars. It may be mentioned that every conormally lat maniold o dimension, as well as every 3-dimensional maniold are Roter type. Deinition 2.2. et M be a semi-riemannian maniold satisying the ollowing condition 2.2 R = 1 g g + 2 g S + 3 S S + g S S S S 2 S 2, or some i C M, 1 i 6. The above condition is called a generalized Roter type condition and M is called a generalized Roter type maniold [25], [28] with i s as the associated scalars. For details about the geometric properties o generalized Roter type maniold we reer the readers to see [28]. We mention that such decompositions o R were already investigated in [8], [12], [23] and very recently in [9], [10], [2]. Throughout this paper by a proper GRT n we mean a GRT n which is not a RT n, and by a proper RT n we mean a RT n which is not conormally lat. A GRT n or a RT n is said to be special i one or more o their associated scalars are identically zero or assume some particular values. Again contracting the Roter type and generalized Roter type conditions Shaikh and Kundu [28] presented some generalizations o Einstein metric conditions. Deinition 2.3. [1] I in a semi-riemannian maniold M, S and g resp., S 2, S and g; S 3, S 2, S and g; S, S 3, S 2, S and g are linearly dependent then it is called Ein1 resp., Ein2; Ein3; Ein maniold. The Ein1 maniold is known as Einstein maniold and in this case we have S = κ n g. We note that every Eini maniold is Eini + 1 or i = 1, 2, 3 but not conversely [25]. It is well known that every maniold o constant curvature is always Einstein. κ Again a RT n is Ein2 except N 1 = 2n 1n 2, N 2 = 1 n 2, N 3 = 0; and a GRT n is Ein except 1 = 1 κ 2 κ 2 κ 2 n 1 n 1n 2, 2 = 1 n 2 κ, 3 = 1 2 n 2, 5 = 0, 6 = 0, where κ 2 = trs 2.

4 On warped product generalized Roter type maniolds 85 3 Warped product maniolds et M, g and M, g be two semi-riemannian maniolds o dimension p and n p respectively 1 p n 1. The product metric g on M = M M is deined as g = π g + σ g, where π : M M and σ : M M are the natural projections. Generalizing this notion o product metric, Kru ckovi c [22] introduced the notion o semi-decomposable metric g on M as g = π g + πσ g, where is a positive smooth unction on M. Again to construct a large class o complete maniolds o negative curvature, Bishop and O Neill [2] obtained the same notion and named as warped product maniold. We mention that in the literature o dierential geometry the name warped product is more widely used and here we also use the name warped product maniold. et M be the warped product maniold equipped with the warped product metric g. I we consider a product chart U V ; x 1, x 2,..., x p, x p+1 = y 1, x p+2 = y 2,..., x n = y n p on M, then in terms o local coordinates, g can be expressed as g ij or i = a and j = b, 3.1 g ij = g ij or i = α and j = β, 0 otherwise, where a, b {1, 2,..., p} and α, β {p + 1, p + 2,..., n}. We note that throughout the paper we consider a, b, c,... {1, 2,..., p} and α, β, γ,... {p + 1, p + 2,..., n} and i, j, k,... {1, 2,..., n}. Here M is called the base, M is called the iber and is called the warping unction o M. I = 1, then the warped product reduces to the product maniold. Moreover, when Ω is a quantity ormed with respect to g, we denote by Ω and Ω, the similar quantities ormed with respect to g and g respectively. The non-zero local components R hijk o the Riemann-Christoel curvature tensor R, S jk o the Ricci tensor S and the scalar curvature κ o M are given by R abcd = R abcd, R aαbβ = T ab g αβ, R αβγδ = R αβγδ 2 P G αβγδ, S ab = S ab n pt ab, S αβ = S αβ + Q g αβ, and 3. κ = κ + κ n p[n p 1P 2 trt ], where G ijkl = g il g jk g ik g jl are the components o Gaussian curvature and T ab = 1 2 a,b 1 2 a b, trt = g ab T ab, P = 1 2 gab a b, Q = n p 1P + trt, a = a = x a.

5 86 A. A. Shaikh, H. Kundu For more detail about warped product components o basic tensors we reer the readers to see [20], [26] and also reerences therein. Now rom above results we can easily calculate the local components o various necessary tensors. The non-zero local components o S 2, g g, g S, S S, g S 2, S S 2 and S 2 S 2 are given as ollows: 3.5 { is 2 ab = S2 ab + n ps T ab + n p 2 T 2 ab, iis 2 αβ = 1 [ S 2 αβ + 2Q S αβ + Q 2 g αβ ]. 3.6 ig g abcd = g g abcd, iig g aαbβ = 2g ab g αβ, iiig g αβγδ = 2 g g αβγδ. 3.7 ig S abcd = g S abcd n pg T abcd, iig S aαbβ = g ab S αβ + Q g αβ g αβ S ab n pt ab, iiig S αβγδ = g S αβγδ + 2Q G αβγδ. 3.8 is S abcd = S S abcd 2n ps T abcd +n p 2 T T abcd, iis S aαbβ = 2 S αβ + Q g αβ S ab n pt ab, iiis S αβγδ = S S αβγδ + 2Q S g αβγδ + Q 2 g g αβγδ. 3.9 ig S 2 abcd = g S 2 abcd + n pg S T abcd +n p 2 g T 2 abcd, iig S 2 aαbβ = 1 g ab S 2 αβ + 2Q S αβ + Q 2 g αβ g αβ S 2 ab + n ps T ab + n p 2 T 2 ab, iiig S 2 αβγδ = g S 2 αβγδ + 2Q g S αβγδ + Q 2 g g αβγδ is S 2 abcd = S S 2 abcd + n ps S T abcd +n p 2 S T 2 abcd n ps 2 T abcd n p 2 T S T abcd +n p 3 T T 2 abcd, iis S 2 aαbβ = 1 S ab n pt ab S 2 αβ + 2Q S αβ + Q 2 g αβ S αβ + Q g αβ S 2 ab + n ps T ab + n p 2 T 2 ab, iiis S 2 αβγδ = 1 [ S S 2 αβγδ + Q S S αβγδ +Q 2 S g αβγδ + Q g S 2 αβγδ +2Q 2 g S αβγδ + 2Q 3 g g αβγδ ].

6 On warped product generalized Roter type maniolds is 2 S 2 abcd = S 2 S 2 abcd + n p 2 S T S T abcd +n p 2 T 2 T 2 abcd + 2n p 3 S 2 T 2 abcd +2n p 3 S T 2 T 2 abcd +2n ps 2 S T abcd, iis 2 S 2 aαbβ = 2 S2 ab + n ps T ab + n p 2 T 2 ab S αβ 2 + 2Q S αβ + Q 2 g αβ, iiis 2 S 2 αβγδ = 1 [ S 2 S 2 2 αβγδ + Q 2 S S αβγδ +Q g g αβγδ + Q S 2 S αβγδ +2Q 2 g S 2 αβγδ + Q 3 g S αβγδ ]. From above it ollows that the components o g g, g S, S S, g S 2, S S 2 and S 2 S 2 are given in a quadratic orm o Kulkarni-Nomizu product or base and iber part, and quadratic orm o the product or the mixed part. So each o them can be expressed by a matrix. For example, g S abcd, g S aαbβ and g S αβγδ can respectively be expressed as g S S 2 T T 2 S T 1 p n g S S , p n T T S T g S S2 g Q 1 0 S 0 0 S T p n 0 0 T S T and g S S2 g Q 2 0. S S Similarly, we can get the matrix representations o the components or the other tensors g g, S S, g S 2, S S 2 and S 2 S 2. Warped product generalized Roter-type maniolds Theorem.1. I M n = M p M n p is a warped product maniold, then M satisies the generalized Roter type condition.1 R = 1 g g + 2 g S + 3 S S + g S S S S 2 S 2 i and only i i the Riemann-Christoel curvature tensor R o M can be expressed as g S S 2 T T 2 S T g p n 1 2 n p2 1 2 n p S p n 2 5n p n p S p n 6 n p 2 6 n p 1 T 2 1 2p n 3 p n 2 5p n 3 n p n p n p 2 T n p n p 2 6 n p n p 3 6 n p 6 n p 3 1 S T 2 n p 1 2 5n p 6n p 1 2 5n p2 6n p 3 6n p 2 ii R, R being the Riemann-Christoel curvature tensor o M, can be expressed as g S S2 g 6 Q Q Q Q+ 2 2Q P Q Q S 3 + Q Q Q Q Q 2 S Q Q 2 + Q Q Q Q

7 88 A. A. Shaikh, H. Kundu iii the ollowing expression vanishes identically on M: g S S 2 T T 2 S T g S S2 Q2 2 Q Q 5 Q2 2 3Q Q Q2 5 Q 5 6 Q p n Q2 p n Qp n 2n p Q 5 p n n p Q+2 6 Q 2 n p Q n p Q+2 6 Q 2 n p Q 2 6 n p2 2 6 p n. Proo. In terms o local coordinates,.1 can be expressed as.2 R ijkl = 1 g g ijkl + 2 g S ijkl + 3 S S ijkl + g S 2 ijkl + 5 S S 2 ijkl + 6 S 2 S 2 ijkl. From.2 it ollows that we can consider the ollowing three cases: I i = a, j = b, k = c, l = d; II i = α, j = β, k = γ, l = δ; III i = a, j = α, k = b, l = β. Consider the case I: i = a, j = b, k = c, l = d in.2 and using , we get R abcd = 1 g g abcd + 3 S n pt S n pt abcd + 2 g S n pt abcd + g S n ps T + n p 2 T 2 abcd + 5 S S n ps T + n p 2 T 2 abcd + 6 S n ps T + n p 2 T 2 S n ps T + n p 2 T 2 abcd. Now expressing the above in matrix orm, we get i. Similarly setting i = α, j = β, k = γ, l = δ in.2, we get ii. Again putting i = a, j = α, k = b, l = β in.2 and using , we obtain T ab g αβ = 2 1 g ab g αβ 2 [g ab S ] αβ + Q g αβ + g αβ S ab n pt ab [ S ab n pt ab 2 3 S αβ + Q g αβ + 5 S αβ 2 + 2Q S αβ + Q 2 g ] αβ g ab S 2 αβ + 2Q S αβ + Q 2 g αβ g αβ S 2 ab + n ps T ab + n p 2 T 2 ab 5 S 2 ab + n ps T ab + n p 2 T 2 ab S αβ + Q g αβ, 2 6 S2 ab + n ps T ab + n p 2 T 2 ab S 2 αβ + 2Q S αβ + Q 2 g αβ. Now simpliying above and expressing in matrix orm, we obtain iii. This completes the proo. The above theorem yields the ollowing:

8 On warped product generalized Roter type maniolds 89 Corollary.2. I M n = M p M n p is a warped product maniold with n p 3 satisying the generalized Roter-type condition.1, then i the iber M is generalized Roter type. ii the iber M is Roter type i J 1 0, where J 1 = 1 [ p + 5 trt n p + κ +2 6 trt 2 n p 2 + n ptrs T + κ ]. 2 iii the iber M is o vanishing conormal curvature tensor i J 1 0 and J J J Q 2 + 2Q Q J = 0, where J 2 = 1 [ Q n p trt 2 n p + trs T + κ 2 ] + p Q + 2trT n p Q + 2κ Q. iv the iber M is o constant curvature i J 1 = 0 and J 2 0. Corollary.3. I M n = M p M n p is a warped product maniold with p 3 satisying generalized Roter-type condition.1, then the base M is generalized Roter type i T can be expressed as a linear combination o g and S. From Theorem.1 we can easily get the necessary and suicient condition or a warped product maniold to be Roter type. Corollary.. I M n = M p M n p is a non-lat warped product maniold, then M satisies the Roter type condition.3 R = N 1 g g + N 2 g S + N 3 S S i and only i i R = g S T N g N N 2p n N S 2, 2 N 3 N 3 p n 1 T 2 N 2p n N 3 p n N 3 n p 2 g S ii R = g N N 2 Q + N 3 Q 2 P N N 3 Q, N S N 3 Q N 3 g S T iii g 2N 1 N 2 Q N 2 2N 3 Q 1 + N 2 n p 2N 3 Qn p S N 2 2N 3 2N 3 p n = 0. Proo. The result ollows rom Theorem.1, by setting 1 = N 1, 2 = N 2, 3 = N 3 and = 5 = 6 = 0.

9 90 A. A. Shaikh, H. Kundu Corollary.5. I M n = M p M n p is a non-lat warped product maniold with n p 3 satisying the Roter type condition.3, then i the iber is o Roter type. ii the iber is o vanishing conormal curvature tensor i M is conormally lat. iii the iber is o constant curvature i 2n pn 3 trt N 2 p 2N 3 κ 0. Corollary.6. I M n = M p M n p is a non-lat warped product maniold with p 3 satisying the Roter type condition.3, then the base is o Roter type i T, g and S are linearly dependent with non-zero coeicient o T. Recently, Deszcz et al. [1] studied the warped product Roter type maniold with 1-dimensional iber and showed the ollowing: Corollary.7. [1] I M n = M n 1 M 1 is a non-lat warped product maniold satisying the Roter type condition.3, then M realizes a Roter type condition at those points, where it does not satisy the Einstein metric condition. Proo. Since the dimension o M is n p = 1, rom the condition iii o Corollary., we get 2N 1 + N 2 Qg + N 2 + 2N 3 QS N 2 + 2N 3 QT = 0. I at x M, S κ n 1 g, then 1 + N 2 + 2N 3 Q 0 at x and T can be expressed as linear combination o S and g and hence by Corollary.6, M satisies a Roter type condition at x. This completes the proo. Now we can easily deduce the necessary and suicient condition or a warped product maniold to be conormally lat maniold, as ollows: Corollary.8. I M n = M p M n p, 1 p n 1 is a non-lat warped product maniold, then M is conormally lat i and only i κ i R = 2n 1n 2 g g + 1 n 2 g S n p n 2 g T, ii R [ ] κ = 2n 1n 2 + Q n P g g + 1 n 2 g S, [ ] 2κ iii 2n 1n 2 + Q n 2 g ab g αβ + 1 n 2 g S ab αβ + n 2 S ab g αβ + n p n T ab g αβ = 0. Proo. The result ollows rom Corollary. by taking N 1 = and N 3 = 0. From above we can state the ollowing: κ 2n 1n 2, N 2 = 1 n 2 Corollary.9. [3] I M n = M p M n p is a conormally lat warped product maniold, then i or n p 2, the iber is o constant curvature. ii or p 2, the base is o vanishing conormal curvature tensor. Proo. By contracting the condition iii o Corollary.8, it ollows that T is a linear combination o g and S, and S is a scalar multiple o g. Putting these in the condition i and ii o Corollary.8, we get the results.

10 On warped product generalized Roter type maniolds 91 Since or the decomposable maniold the warping unction is 1, we have T = 0, P = 0 and Q = 0. Thus applying these values in 3.2 to 3.11 we get the non-zero components o R, S, κ, S 2, g g, g S, S S, g S 2, S S 2 and S 2 S 2. Consequently, rom Theorem.1 we can state the ollowing: Corollary.10. I M n = M p M n p is a decomposable maniold, then M satisies the generalized Roter-type condition.1 i and only i i R = g S S 2 g S S , ii R = g S S2 g S S , iii g S S2 g S S = 0. Note. From the above corollary we can get the necessary and suicient conditions or a decomposable maniold to be Roter type by taking = 5 = 6 = 0, and κ conormally lat by taking 3 = = 5 = 6 = 0, 1 = 2n 1n 2 and 2 = 1 n 2. Again rom the above results we see that the decompositions o a semi-riemannian product generalized Roter type maniold are also generalized Roter type maniold but the converse is not necessarily true, in general. We also note that the same situations arise or Roter type and conormally lat maniolds. Remark.1. In this context we state the necessary and suicient conditions o a warped product maniold to be Einstein. et M n = M p M n p be a warped product maniold [1], see also [16]. Then M is Einstein i and only i i S n pt = κ n g and ii S = κ n Q g. 5 Examples Example 5.1: Consider the warped product M = M M, where M is an open interval o R with usual metric g = dx 1 2 in local coordinate x 1 and M is a - dimensional maniold equipped with a semi-riemannian metric g = dx hdx hdx 2 + hψdx 5 2 in local coordinates x 2, x 3, x, x 5, where the warping unction is a unction o x 1, and h and ψ are everywhere non-zero unctions o x 2 and x 3 respectively. We can easily evaluate the local components o necessary tensors o M. The non-zero components o the Riemann-Christoel curvature tensor R and the Ricci tensor S o M upto symmetry are h 2 2hh and ψ R 1212 = ψ R 1313 = R 11 = ψ h, ψ R 2323 = R 33 = ψ h 2, R 22 = 1 ψ h 2 2hψ + h ψ 2 ψ 3 2hh h 2 S 11 =, h 2 S22 = 1 2h + h 2 h ψ 2 2ψψ, ψ 2

11 92 A. A. Shaikh, H. Kundu S 33 = 2hh + h 2, S = 1 2 ψh + ψ + ψ h 2 h h ψ 2. ψ Then we can easily check that this maniold is generalized Roter type and Ein3 maniold. Again i i h 2 hh = 0, i.e., h = c 1 e c2x2, then it is a Ein2 maniold and hence Roter type; ii ψ 2 2ψψ = 0, i.e., ψ = c1x3 +2c 2 2 c 2, then it is a maniold o constant curvature, where c 1 and c 2 are arbitrary constants. Now by a straightorward calculation we can evaluate the components o various necessary tensors o M. The non-zero components o the Riemann-Christoel curvature tensor R and the Ricci tensor S o M upto symmetry are and hψr 1212 = ψr 1313 = ψr 11 = R 1515 = hψ 2 2, ψr 2323 = ψr 22 = R 2525 = ψ h 2 2h + h 2, h ψr 33 = R 55 = ψ h h 2, [ ] R 3535 = 1 ψ h 2 2hψ + h ψ 2 h 2 ψ 2, ψ S 11 = 2 2 2, S 22 = hh 3 h 2 h 2, ψs 33 = S 55 = 1 2hψ + 2hψ 2 + 2ψh + ψ h 2 + 2ψ ψ 2, h ψ S = 1 2 h + h + 2h 2 + h 2. h From these we can easily calculate the components o S 2, S 3, S and also the components o G, g S, S S, g S 2, S S 2 and S 2 S 2. We observe that or every, h and ψ, the maniold is Ein but not o generalized Roter type. We now discuss the results or particular value o the unctions, h and ψ step by step as ollows: Step I: I h 2 hh = 0, i.e., h = c 1 e c2x2, then M is generalized Roter type and Ein3. We note that in this case iber M is proper Roter type and hence M is a proper generalized Roter type warped product maniold with proper Roter type iber. Step II: Again consider = 0, i.e., = c 2 e c 1x 1 +c 2 e ± c 1x 1 +c 2 + c 1, 1 where c 1 and c 2 are arbitrary non-zero constants. Then the maniold is a Ein2 maniold and hence the maniold is proper Roter type. In this case iber remains also

12 On warped product generalized Roter type maniolds 93 Roter type. So M is a warped product proper Roter type maniold with proper Roter type iber. Step III: Finally, consider ψ 2 2ψψ = 0, i.e., ψ = c1x3 +2c 2 2 c 2. Then M is o constant curvature and in this case iber is also o constant curvature. I = x 1 2, h = c 2 cos 2 x 2 2c 1 and ψ = e x3, then the maniold M is a special generalized Roter type and is Ein3. In this case the iber M is proper generalized Roter type and is Ein3. Hence M is a warped product generalized Roter type maniold with proper generalized Roter type iber. 6 Conclusions The present paper is devoted to the study o warped product generalized Roter type maniolds and obtained the necessary and suicient conditions or a warped product maniold to be o generalized Roter type maniold. As a particular case we obtain the characterization o a warped product Roter type maniold. It is shown that the iber o a warped product GRT n resp., RT n, conormally lat is also generalized Roter type resp., Roter type, maniold o constant curvature. We ind out the conditions or which the iber o a warped product GRT n is Roter type, conormally lat or a maniold o constant curvature and the base is a generalized Roter type. It is also shown that the two decompositions o a GRT n product maniold are both generalized Roter type but the product o two generalized Roter type maniolds is not always generalized Roter type. Finally by suitable metric, the existence o such case is ensured by an example. Acknowledgements. The second named author grateully acknowledges to CSIR, New Delhi File No. 09/ /2010-EMR-I or the inancial assistance. All the algebraic computations o Section 5 are calculated with help o Wolram Mathematica. Reerences [1] A.. Besse, Einstein Maniolds, Springer-Verlag, Berlin-New York [2] R.. Bishop and B. O Neill, Maniolds o negative curvature, Trans. Amer. Math. Soc , 1 9. [3] M. Brozos-Vázquez, E. García-Río and R. Vázquez-orenzo, Some remarks on locally conormally lat static spacetimes, J. Math. Phys. 6, , pages. [] R. Deszcz, On some Akivis-Goldberg type metrics, Publ. Inst. Math. Beograd N.S. 7, , [5] R. Deszcz and M. G logowska, Some examples o nonsemisymmetric Riccisemisymmetric hypersuraces, Colloq. Math , [6] R. Deszcz, M. G logowska, M. Hotloś and K. Sawicz, A Survey on Generalized Einstein Metric Conditions, Advances in orentzian Geometry, Proceedings o the orentzian Geometry Conerence in Berlin, AMS/IP Studies in Advanced Mathematics 9, S.-T. Yau series ed., M. Plaue, A.D. Rendall and M. Scherner eds., 2011, 27 6.

13 9 A. A. Shaikh, H. Kundu [7] R. Deszcz, M. G logowska, M. Hotloś and Z. Ṣentürk, On certain quasi-einstein semi-symmetric hypersuraces, Ann. Univ. Sci. Budapest Eötvös Sect. Math , [8] R. Deszcz, M. G logowska, J. Je lowicki, M. Petrović-Torgašev and G. Zaindrataa, On Riemann and Weyl compatible tensors, Publ. Inst. Math. Beograd N.S. 9, , [9] R. Deszcz, M. G logowska, J. Je lowicki and G. Zaindrataa, Curvature properties o some class o warped product maniolds, Int. J. Geom. Methods Mod. Phys , pages. [10] R. Deszcz, M. G logowska, M. Petrović-Torgašev and. Verstraelen, Curvature properties o some class o minimal hypersuraces in Euclidean spaces, Filomat 29, , [11] R. Deszcz and M. Hotloś, On hypersuraces with type number two in spaces o constant curvature, Ann. Univ. Sci. Budapest Eötvös Sect. Math , [12] R. Deszcz, M. Hotloś, J. Je lowicki, H. Kundu and A.A. Shaikh, Curvature properties o Gödel metric, Int J. Geom. Method Mod. Phy. 11, 3 201, pages. [13] R. Deszcz, and D. Kowalczyk, On some class o pseudosymmetric warped products, Colloq. Math , [1] R. Deszcz, M. Plaue and M. Scherner, On Roter type warped products with 1- dimensional ibres, J. Geom. and Phys , [15] R. Deszcz and M. Scherner, On a particular class o warped products with ibres locally isometric to generalized Cartan hypersuraces, Colloquium Math , [16] A. Gȩbarowski, On Einstein warped products, Tensor, N.S , [17] M. G logowska, Curvature conditions on hypersuraces with two distinct principal curvatures, In: PDEs, submaniolds and aine dierential geometry, Banach Center Publications, , [18] M. G logowska, Semi-Riemannian maniolds whose Weyl tensor is a Kulkarni- Nomizu square, Publ. Inst. Math. Beograd N.S. 72, , [19] M. G logowska, On Roter-type identities, in: Pure and Applied Dierential Geometry - PADGE 2007, Berichte aus der Mathematik, Shaker Verlag, Aachen, 2007, [20] M. Hotloś, On conormally symmetric warped products, Ann. Academic Paedagogical Cracoviensis , [21] D. Kowalczyk, On the Reissner-Nordström-de Sitter type spacetimes, Tsukuba J. Math , [22] G.I. Kru ckovi c, On semi-reducible Riemannian spaces in Russian, Dokl. Akad. Nauk SSSR , [23] K. Sawicz, On curvature characterization o some hypersuraces in spaces o constant curvature, Publ. Inst. Math. Beograd N.S. 79, , [2] K. Sawicz, Curvature properties o some class o hypersuraces in Euclidean spaces, Publ. Inst. Math. Beograd N.S. 98, , [25] A.A. Shaikh, R. Deszcz, M. Hotlós, J. Je lowicki and H. Kundu, On pseudosymmetric maniolds, Publ. Math. Debrecen 86, ,

14 On warped product generalized Roter type maniolds 95 [26] A.A. Shaikh and H. Kundu, On weakly symmetric and weakly Ricci symmetric warped product maniolds, Publ. Math. Debrecen 81, , [27] A.A. Shaikh and H. Kundu, On quivelency o various geometric structures, J. Geom , [28] A.A. Shaikh and H. Kundu, On generlized Roter type maniolds, arxiv: v1 [math.dg] Nov 201. Author s address: Absos Ali Shaikh, Haradhan Kundu Department o Mathematics, The University o Burdwan, Golapbag, Burdwan-71310, West Bengal, India. aask2003@yahoo.co.in, aashaikh@math.buruniv.ac.in and kundu.haradhan@gmail.com