CUBO, A Mathematical Journal Vol. 23, no. 01, pp. 87–96, April 2021 DOI: 10.4067/S0719-06462021000100087 Hyper generalized pseudo Q-symmetric semi-Riemannian manifolds Adara M. Blaga1 Manoj Ray Bakshi2 Kanak Kanti Baishya3 1 Department of Mathematics, West University of Timişoara, Timişoara, România. adarablaga@yahoo.com 2,3 Department of Mathematics, Kurseong College, Kurseong, Darjeeling, India. raybakshimanoj@gmail.com; kanakkanti.kc@gmail.com ABSTRACT The object of the present paper is to study the properties of a hyper generalized pseudo Q-symmetric semi-Riemannian manifold, proving that under certain assumptions, it is a per- fect fluid spacetime. RESUMEN El objetivo del presente art́ıculo es estudiar las propiedades de una variedad semi-Riemanniana hiper generalizada pseudo Q-simétrica, probando que bajo ciertas condiciones, es un espacio-tiempo fluido perfecto. Keywords and Phrases: Q-curvature tensor, perfect fluid spacetime. 2020 AMS Mathematics Subject Classification: 53C15, 53C25. Accepted: 21 January, 2021 Received: 04 July, 2020 ©2021 A. M. Blaga et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000100087 https://orcid.org/0000-0003-0237-3866 https://orcid.org/0000-0002-5981-7715 https://orcid.org/0000-0001-9331-7960 88 A. M. Blaga, M. R. Bakshi & K. K. Baishya CUBO 23, 1 (2021) 1 Introduction Let R, S, L and r denote the curvature tensor, Ricci tensor, Ricci operator and the scalar curvature of a (semi)-Riemannian manifold, respectively. It is Mantica and Suh [5] who have introduced the notion of Q-curvature tensor. In an n-dimensional Riemannian or semi-Riemannian manifold (Mn,g) (n > 2), the Q-curvature tensor is defined as R(Y,U,V,W) = Q(Y,U,V,W) + ψ n− 1 [g(Y,W)g(U,V ) −g(Y,V )g(U,W)], (1.1) where Y,U,V,W are arbitrary vector fields on Mn and ψ is a scalar function. Semi-Riemannian manifolds with Ricci tensor S of the form S(Y,V ) = ag(Y,V ) + bT(Y )T(V ), for any vector fields Y,V , are often termed as perfect fluid spacetimes, where a and b are scalars and the vector field %, metrically equivalent to the 1-form T (that is, g(Y,%) = T(Y )), is a unit time like vector field (that is, g(%,%) = −1). An n-dimensional semi-Riemannian manifold is said to be hyper generalized pseudo Q-symme- tric (which will be abbreviated hereafter as (HGPQS)n) if it satisfies the equation (∇XQ)(Y,U,V,W) (1.2) = 2A1(X)Q(Y,U,V,W) + A1(Y )Q(X,U,V,W) +A1(U)Q(Y,X,V,W) + A1(V )Q(Y,U,X,W) +A1(W)Q(Y,U,V,X) + 2A2(X)(g ∧S)(Y,U,V,W) +A2(Y )(g ∧S)(X,U,V,W) + A2(U)(g ∧S)(Y,X,V,W) +A2(V )(g ∧S)(Y,U,X,W) + A2(W)(g ∧S)(Y,U,V,X), where (g ∧S)(Y,U,V,W) = g(Y,W)S(U,V ) + g(U,V )S(Y,W) (1.3) −g(Y,V )S(U,W) −g(U,W)S(Y,V ) and A1, A2 are non-zero 1-forms whose g-dual vector fields will be denoted by θ1 and θ2, i.e. A1(X) = g(X,θ1) and A2(X) = g(X,θ2). We organized our paper as follows: section 2 is concerned with preliminaries. After prelimi- naries, some curvature properties of (HGPQS)n manifolds are studied in section 3. It is obtained that the Q-curvature tensor in a (HGPQS)n manifold satisfies 2nd Bianchi’s identity. It is further obtained that the scalar function ψ is always constant. In section 4 we investigate properties of divergence-free (HGPQS)n manifolds and we prove that a divergence-free (HGPQS)n manifold (n > 2) under the assumption A1(Q(Y,U)V ) = 0 is a perfect fluid spacetime as well as the integral CUBO 23, 1 (2021) Hyper generalized pseudo Q-symmetric semi-Riemannian manifolds 89 curves of the vector field % are geodesics and the vector field % is irrotational, if the associated vector fields % and σ corresponding to the 1-forms T1 and T2 are related by (r − 1)% + nσ = 0. 2 Preliminaries In this section, some relations useful to the study of (HGPQS)n manifolds are obtained. Let {ei} be an orthonormal basis of the tangent space at each point of the manifold, where 1 ≤ i ≤ n. From (1.1) we can easily verify that the tensor Q satisfies the following properties: (i) Q(Y,U)V + Q(U,Y )V = 0, (ii) Q(Y,U)V + Q(U,V )Y + Q(V,Y )U = 0, (2.1) where g(Q(X,Y )U,V ) = Q(X,Y,U,V ). Also from (1.1) we have n∑ i=1 �iQ(X,Y,ei,ei) = 0 = n∑ i=1 �iQ(ei,ei,W,U) (2.2) and n∑ i=1 �iQ(ei,Y,V,ei) = n∑ i=1 �iQ(Y,ei,ei,V ) = S(Y,V ) −ψg(Y,V ) (2.3) =: Z(Y,V ), where �i = g(ei,ei) = ±1, S(X,Y ) = n∑ i=1 �ig(R(X,ei)ei,Y ), r = n∑ i=1 �iS(ei,ei). From (1.1) and (2.1) it follows that (i) Q(X,Y,U,V ) + Q(X,Y,V,U) = 0, (ii) Q(X,Y,U,V ) −Q(U,V,X,Y ) = 0. (2.4) 3 Some curvature properties of (HGPQS)n manifolds In this section we prove that in a (HGPQS)n manifold, the Q-curvature tensor satisfies 2nd Bianchi’s identity, that is, (∇XQ)(Y,U,V,W) + (∇Y Q)(U,X,V,W) + (∇UQ)(X,Y,V,W) = 0. (3.1) 90 A. M. Blaga, M. R. Bakshi & K. K. Baishya CUBO 23, 1 (2021) In view of (1.1), (1.2) and (3.1) we get (∇XQ)(Y,U,V,W) + (∇Y Q)(U,X,V,W) + (∇UQ)(X,Y,V,W) (3.2) = A1(V )[Q(Y,U,X,W) + Q(U,X,Y,W) + Q(X,Y,U,W)] +A1(W)[Q(Y,U,V,X) + Q(U,X,V,Y ) + Q(X,Y,V,U)] +A2(V )[(g ∧S)(Y,U,X,W) + (g ∧S)(U,X,Y,W) +(g ∧S)(X,Y,U,W)] + A2(W)[(g ∧S)(Y,U,V,X) +(g ∧S)(U,X,V,Y ) + (g ∧S)(X,Y,V,U)]. Using (1.3) and 1st Bianchi’s identity for the Q-curvature tensor in (3.2) and then simplifying, we obtain (3.1). Thus we can state the following: Theorem 3.1. The Q-curvature tensor in a (HGPQS)n manifold satisfies 2nd Bianchi’s identity. Using (1.1) in (3.1), we have (∇XR)(Y,U,V,W) + (∇Y R)(U,X,V,W) + (∇UR)(X,Y,V,W) (3.3) − dψ(X) (n− 1) [g(Y,W)g(U,V ) −g(Y,V )g(U,W)] − dψ(Y ) (n− 1) [g(U,W)g(X,V ) −g(U,V )g(X,W)] − dψ(U) (n− 1) [g(X,W)g(Y,V ) −g(X,V )g(Y,W)] = 0. By virtue of 2nd Bianchi’s identity for the Riemannian curvature tensor, (3.3) yields dψ(X) (n− 1) [g(Y,W)g(U,V ) −g(Y,V )g(U,W)] (3.4) + dψ(Y ) (n− 1) [g(U,W)g(X,V ) −g(U,V )g(X,W)] + dψ(U) (n− 1) [g(X,W)g(Y,V ) −g(X,V )g(Y,W)] = 0. Contracting U and V in (3.4), we have (n− 2)[dψ(X)g(Y,W) −dψ(Y )g(X,W)] = 0 (3.5) which yields after further contraction (n− 1)(n− 2)dψ(X) = 0. This implies that dψ(X) = 0, that is, ψ is constant since n > 2 and leads to the following: Theorem 3.2. In a (HGPQS)n manifold, the scalar function ψ is always constant. CUBO 23, 1 (2021) Hyper generalized pseudo Q-symmetric semi-Riemannian manifolds 91 Consequently, one can easily bring out the following: Theorem 3.3. In a (HGPQS)n manifold, (divQ)(X,Y )Z and (divR)(X,Y )Z are equivalent. In view of (1.1), (1.2) and Theorem 3.2 we have (∇XR)(Y,U,V,W) (3.6) = 2A1(X)Q(Y,U,V,W) + A1(Y )Q(X,U,V,W) +A1(U)Q(Y,X,V,W) + A1(V )Q(Y,U,X,W) +A1(W)Q(Y,U,V,X) + 2A2(X)(g ∧S)(Y,U,V,W) +A2(Y )(g ∧S)(X,U,V,W) + A2(U)(g ∧S)(Y,X,V,W) +A2(V )(g ∧S)(Y,U,X,W) + A2(W)(g ∧S)(Y,U,V,X) which yields (∇XS)(U,V ) (3.7) = [F1(X) + F2(X)]S(U,V ) + F2(U)S(X,V ) + F2(V )S(U,X) +[F3(X) + F4(X)]g(U,V ) + F4(U)g(X,V ) + F4(V )g(U,X) +A1(Q(X,U)V ) −A1(Q(V,X)U) after contraction over Y and W , where F1(X) = A1(X) + (n + 1)A2(X), F2(X) = A1(X) + (n− 3)A2(X), F3(X) = rA2(X) −ψA1(X) + 3A2(LX), F4(X) = rA2(X) −ψA1(X) −A2(LX), where L is the Ricci operator defined by g(LX,Y ) = S(X,Y ). Definition 3.4. An n-dimensional semi-Riemannian manifold is called almost generalized pseudo Ricci symmetric if the non-flat Ricci curvature tensor satisfies the equation (∇XS)(U,V ) = [A(X) + B(X)]S(U,V ) + A(U)S(X,V ) + A(V )S(U,X) +[C(X) + D(X)]g(U,V ) + C(U)g(X,V ) + C(V )g(U,X), where A,B,C and D are non-zero 1-forms whose g-dual vector fields will be denoted by γ1, γ2, δ1 and δ2, i.e. A(X) = g(X,γ1), B(X) = g(X,γ2), C(X) = g(X,δ1) and D(X) = g(X,δ2). Thus we can state the following: 92 A. M. Blaga, M. R. Bakshi & K. K. Baishya CUBO 23, 1 (2021) Theorem 3.5. A (HGPQS)n manifold (n > 2) under the assumption A1(Q(X,U)V ) = A1(Q(V,X)U) is necessarily almost generalized pseudo Ricci symmetric. Making use of (2.3) in (3.7), we get (∇XZ)(U,V ) (3.8) = [F1(X) + F2(X)]Z(U,V ) + F2(U)Z(X,V ) + F2(V )Z(U,X) +[F3(X) + ψF1(X) + F4(X) + ψF2(X)]g(U,V ) +[F4(U) + ψF2(U)]g(X,V ) + [F4(V ) + ψF2(V )]g(U,X), where Z = S −ψg is the tensor considered in ([4], [6], [7]). This leads to the following: Theorem 3.6. A (HGPQS)n manifold (n > 2) under the assumption A1(Q(X,U)V ) = A1(Q(V,X)U) is necessarily almost generalized pseudo Z-symmetric. 4 (HGPQS)n manifolds (n > 2) with divQ = 0 Let (Mn,g) be a semi-Riemannian manifold of dimension n and let {ei} be an orthonormal basis of the tangent space TpM at any point p ∈ M and �i = ±1. Then the divergence of a vector field U is defined as divU = n∑ i=1 �ig(∇eiU,ei), and the divergence of a tensor field of type (1, 3), which is a tensor field of type (0, 3), is defined as (divK)(X,Y )Z = n∑ i=1 �ig((∇eiK)(X,Y )Z,ei). Now (divQ)(Y,U)V = n∑ i=1 �ig((∇eiQ)(Y,U)V,ei) = n∑ i=1 �i[2A1(ei)Q(Y,U,V,ei) + A1(Y )Q(ei,U,V,ei) +A1(U)Q(Y,ei,V,ei) + A1(V )Q(Y,U,ei,ei) +A1(ei)Q(Y,U,V,ei) + 2A2(ei)(g ∧S)(Y,U,V,ei) +A2(Y )(g ∧S)(ei,U,V,ei) + A2(U)(g ∧S)(Y,ei,V,ei) +A2(V )(g ∧S)(Y,U,ei,ei) + A2(ei)(g ∧S)(Y,U,V,ei)] CUBO 23, 1 (2021) Hyper generalized pseudo Q-symmetric semi-Riemannian manifolds 93 = 3A1(Q(Y,U)V ) + A1(Y )[S(U,V ) −ψg(U,V )] −A1(U)[S(Y,V ) −ψg(Y,V )] + 3A2(Y )S(U,V ) +3A2(LY )g(U,V ) − 3A2(LU)g(Y,V ) − 3A2(U)S(Y,V ) +A2(Y )[(n− 2)S(U,V ) + rg(U,V )] −A2(U)[(n− 2)S(Y,V ) + rg(Y,V )] = 3A1(Q(Y,U)V ) + S(U,V )[A1(Y ) + (n + 1)A2(Y )] −S(Y,V )[A1(U) + (n + 1)A2(U)] +g(U,V )[3A2(LY ) + rA2(Y ) −ψA1(Y )] −g(Y,V )[3A2(LU) + rA2(U) −ψA1(U)] = 3A1(Q(Y,U)V ) + T1(Y )S(U,V ) −T1(U)S(Y,V ) +T2(Y )g(U,V ) −T2(U)g(Y,V ), hence (divQ)(Y,U)V = 3A1(Q(Y,U)V ) + T1(Y )S(U,V ) −T1(U)S(Y,V ) (4.1) +T2(Y )g(U,V ) −T2(U)g(Y,V ), where T1(Y ) = A1(Y ) + (n + 1)A2(Y ) =: g(Y,%), for % = θ1 + (n + 1)θ2, T2(Y ) = 3A2(LY ) + rA2(Y ) −ψA1(Y ) =: g(Y,σ), for σ = 3Lθ2 + rθ2 −ψθ1. Assuming (divQ)(Y,U)V = 0 and A1(Q(Y,U)V ) = 0, we get from the above equation T1(Y )S(U,V ) + T2(Y )g(U,V ) = T1(U)S(Y,V ) + T2(U)g(Y,V ). (4.2) Now contracting (4.2) over U and V we get S(Y,%) = rT1(Y ) + (n− 1)T2(Y ). (4.3) Again putting V = % in (4.2) we get (n− 2)[T1(Y )T2(U) −T1(U)T2(Y )] = 0, (4.4) which under the assumption n > 2 implies T1(Y )T2(U) = T1(U)T2(Y ). Now putting U = % in (4.2) and using (4.3) and (4.4) we get T1(%)S(Y,V ) + T2(%)g(Y,V ) = T1(Y )[rT1(V ) + nT2(V )] (4.5) and we can state: 94 A. M. Blaga, M. R. Bakshi & K. K. Baishya CUBO 23, 1 (2021) Theorem 4.1. A divergence-free (HGPQS)n manifold (n > 2) under the assumption A1(Q(Y,U)V ) = 0 is a perfect fluid spacetime with unit timelike vector field %, provided the associ- ated vector fields % and σ corresponding to the 1-forms T1 and T2 are related by (r−1)% + nσ = 0. In this case, (4.5) becomes S(Y,V ) = ag(Y,V ) −T1(Y )T1(V ), (4.6) where a =: T2(%). Again, (divQ)(Y,U)V = 0 gives (∇Y S)(U,V ) − (∇US)(Y,V ) = 0. (4.7) Now using (4.6) in (4.7) we find da(Y )g(U,V ) −da(U)g(Y,V ) (4.8) −[T1(V )(∇Y T1)(U) + T1(U)(∇Y T1)(V )] +[T1(V )(∇UT1)(Y ) + T1(Y )(∇UT1)(V )] = 0. Taking a frame field and contracting Y and V we get (n− 1)da(U) + [T1(U)(δT1) + (∇%T1)(U)] = 0, (4.9) where δT1 = n∑ i=1 �i(∇eiT1)(ei). Setting V = Y = % in (4.8) we find (∇%T1)(U) = −da(U) −da(%)T1(U). (4.10) Substituting (4.10) in (4.9) we get (n− 2)da(U) + T1(U)(δT1) −da(%)T1(U) = 0 (4.11) which yields δT1 = (n− 1)da(%) (4.12) for U = %. Using (4.12) in (4.11) we obtain da(U) = −T1(U)da(%), (4.13) provided n > 2. CUBO 23, 1 (2021) Hyper generalized pseudo Q-symmetric semi-Riemannian manifolds 95 Putting V = % in (4.8) and using (4.13) we get (∇Y T1)(U) − (∇UT1)(Y ) = 0. This means that the 1-form T1 is closed, that is, dT1(Y,U) = 0. Hence g(∇U%,Y ) = g(∇Y %,U) for all U,Y, (4.14) which yields g(∇%%,Y ) = g(∇Y %,%), (4.15) for U = %. Since g(∇Y %,%) = 0, from (4.15) it follows that g(∇%%,Y ) = 0 for all Y . Hence ∇%% = 0. This implies that the integral curves of the vector field % are geodesics. Therefore we can state the following: Theorem 4.2. In a divergence-free (HGPQS)n manifold (n > 2) under the assumption A1(Q(Y,U)V ) = 0, the integral curves of the unit timelike vector field % are geodesics, provided the associated vector fields % and σ corresponding to the 1-forms T1 and T2 are related by (r − 1)% + nσ = 0. Taking into account that the divergence of the conformal curvature tensor of a Riemannian manifold (Mn,g) is ([3], [6]): (divC)(X,Y )Z = n− 3 n− 2 [(∇XS)(Y,Z) − (∇Y S)(X,Z)] (4.16) = n− 3 n− 2 (divQ)(X,Y )Z, for any vector fields X,Y,Z on Mn, from the Lemma 2.1 of [2] we infer Theorem 4.3. Let (M,g) be a (HGPQS)n perfect fluid spacetime (n > 2). If (divQ)(X,Y )Z = 0, for any vector fields X,Y,Z on M, then the unit timelike vector field % is irrotational. Also, in [2] was proved the following result: Theorem 4.4. [2] Let (M,g) be a (HGPQS)n perfect fluid spacetime (n > 2). If (divQ)(X,Y )Z = 0, for any vector fields X,Y,Z on M, then (M,g) is a GRW spacetime whose fiber is Einstein. Acknowledgements. The authors are grateful to the referees for the valuable suggestions and remarks that definitely improved the paper. 96 A. M. Blaga, M. R. Bakshi & K. K. Baishya CUBO 23, 1 (2021) References [1] K. K. Baishya, F. Ozen Zengin and J. Mikeš, “On hyper generalised weakly symmetric mani- folds”, Nineteenth International Conference on Geometry, Integrability and Quantization, 02– 07, June 2017, Varna, Bulgaria Iväılo M. Mladenov and Akira Yoshioka, Editors Avangard Prima, Sofia 2018, pp. 1–10. [2] C. A. Mantica, U. C. De, Y. J. Suh, and L. G. Molinari, “Perfect fluid spacetimes with harmonic generalized curvature tensor”, Osaka J. Math., vol. 56, pp. 173–182, 2019. [3] C. A. Mantica, and L. G. Molinari, “A second-order identity for the Riemann tensor and applications”, Colloq. Math., vol. 122, no. 1, pp. 69–82, 2011. [4] C. A. Mantica, and L. G. Molinari, “Weakly Z-symmetric manifolds”, Acta Math. Hung., vol. 135, no. 1-2, pp. 80–96, 2012. [5] C. A. Mantica, and Y. J. Suh, “Pseudo Q-symmetric semi-Riemannian manifolds”, Int. J. Geom. Meth. Mod. Phys., vol. 10, no. 5, 2013. [6] C. A. Mantica, and Y. J. Suh, “Pseudo Z-symmetric Riemannian manifolds with harmonic curvature tensors”, Int. J. Geom. Meth. Mod. Phys., vol. 9, no. 1, 2012, 1250004. [7] C. A. Mantica, and Y. J. Suh, “Recurrent Z-forms on Riemannian and Kaehler manifolds”, Int. J. Geom. Meth. Mod. Phys., vol. 9, no. 7, 2012, 1250059. Introduction Preliminaries Some curvature properties of (HGPQS)n manifolds (HGPQS)n manifolds (n>2) with divQ=0