# A Study of Generalized ProjectiveP−Curvature Tensor on Warped Product Manifolds

Authors: Uday Chand De; Abdallah Abdelhameed Syied; Nasser Bin Turki; Suliman Alsaeed
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main aim of this study is to investigate the effects of theP−curvature flatness, P−divergence-free characteristic, and P−symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P−curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P−curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P−divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P−symmetric warped product manifold is considered.


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## Body

## 1. Introduction

Curvature tensors play a significant role in mathematics and physics. This is why many researchers have introduced and studied many curvature tensors in various ways, as well as they have shown the importance of these curvature tensors. For instance, the deviation of a space from constant curvature is measured by the concircular curvature tensor (for more details, see [1]). The Weyl curvature tensor describes the distorting but volume-preserving tidal effects of gravitation on a material body.TheP−curvature tensor was first coined by De et al. in 2021 [2]. This curvature tensor is a good generalization of projective [3], conharmonic [4], M−projective [5], and the set of Wi−curvature tensors which was introduced by Pokhariyal and Mishra [6–10]. This curvature tensor is given by(1)Pijkl=a0Rijkl+a1gijRkl+a2gikRjl+a3gilRjk+a4gjkRil+a5gjlRik+a6gklRij,where ai are constants, Rijkl is the Riemann tensor, and Rij is the Ricci tensor [2]. The authors studied this curvature tensor on pseudo-Riemannian manifolds and space times of general relativity. It is proved that pseudo-Riemannian manifolds M will be Einstein manifold if M admits a traceless P−curvature tensor and will be of constant scalar curvature if M is of P−curvature flat. Pseudo-Riemannian manifolds with P−divergence-free characteristic were investigated in Gray’s seven subspaces. As a final point, they studied perfect fluid space times when the P−curvature tensor is flat, and in this case, many interesting results are obtained.Geometers have considered all well-known curvature tensors on the warped product manifolds. For instance,W2−curvature tensor on warped product manifolds is studied in [11]. Also, concircular curvature tensor on warped product manifold is considered in [12]. Motivated by these kinds of studies and others, this paper aims to investigate the P−curvature tensor on the warped product manifolds.This paper is organized as follows: In Section2, the basic properties of the warped product manifold are presented. In Section 3, we consider the P−curvature flat warped product manifold. We prove that the base and the fiber manifolds of the P−curvature flat warped product manifold are Einstein manifold; also in this case, the forms of the P−curvature tensor on the base and the fiber manifolds are obtained. Section 4 is devoted to study the P−divergence-free warped product manifold. It is proven that the base and the fiber manifolds of the warped product manifold with P−divergence-free characteristic are of constant scalar curvature. In addition, these factor manifolds of the P−divergence-free warped product whose Ricci tensor is of Codazzi type are Ricci symmetric manifolds. Finally, we prove that the warped product manifold is P−symmetric if and only if the base and the fiber manifolds are P−symmetric manifolds.


## 2. On Singly Warped Product Manifold

LetM¯,g¯ and M˜,g˜ be two pseudo-Riemannian manifolds with dimensions dimM¯=n¯ and dimM˜=n˜=n−n¯, where n>n¯>1. And, let F:M¯⟶0,∞ be a smooth positive function on M¯. Consider the product manifold M¯×M˜ with its natural projections π:M¯×M˜⟶M¯ and η:M¯×M˜⟶M˜. Then, the singly warped product manifold M=M¯×fM˜ is the product manifold M¯×M˜ furnished with the metric tensor(2)g=g¯⊕Fg˜.The manifoldM¯ is called the base manifold, whereas M˜ is called the fiber manifold [13, 14]. A warped product manifold M=M¯×FM˜ is called trivial if the warping function F is constant. In this case, M=M¯×FM˜ is the Riemannian product M=M¯×M˜F, where M˜F is the manifold M˜ equipped with metric Fg˜, which is homothetic to g˜.Curvatures of the warped product manifold depend on the curvatures of its fiber and base manifolds. It is noted that the curvatures of the Riemannian product manifold split as a sum of the corresponding curvatures of the first and second factor manifolds since both of the metric and the Levi–Civita connection split as a sum. It is natural now to discuss the deviation in the relation between the different curvature formulas in warped product manifolds and their factor manifolds due to the existence of a nontrivial warping function.Let∂/∂xa, ∂/∂xb,… denote the basis vector fields on a neighborhood U¯ of the base manifold M¯, where a,b,…,∈1,…,n¯, whereas ∂/∂xα, ∂/∂xβ,… denote the basis vector fields on a neighborhood U˜ of the fiber manifold M˜, where α,β,….∈n¯+1,…,n. Likewise, ∂/∂xi, ∂/∂xj,… denote the basis vector fields on a neighborhood U¯×U˜ of the warped product manifold M¯×FM˜, where i,j,…,∈1,…,n. The local components of the metric tensor g=g¯×Fg˜ of the warped product manifold M¯×FM˜ are(3)gij=g¯abfori=a,j=b,Fg˜αβfori=α,j=β,0otherwise.The local componentsΓijh of the Levi–Civita connection on the warped product M=M¯×FM˜ are as follows:(4)Γbca=Γ¯bca,Γβγα=Γ˜βγα,Γαβa=12g¯abFbG˜αβ,Γaβα=12FFaδβα,Γabα=Γαba=0,where Fa=∂aF=∂F/∂xa and G˜αβγδ=g˜αγg˜βδ−g˜αδg˜βγ.On the warped productM=M¯×FM˜, the local components of the Riemannian curvature tensor Rijkl are given by [15–18](5)Rabcd=R¯abcd,(6)Rαβγδ=FR˜αβγδ−14Δ¯FG˜αβγδ,(7)Rαabβ=−12Tabg˜αβ,where Δ¯F=g¯abFaFb and Tab is a tensor of type 0,2 with local components Tab=∇¯bFa−1/2FFaFb and Tαβ=Taα=0.The local components of the Ricci curvatureRij of the warped product M=M¯×FM˜ are the following [16, 17]:(8)Rab=R¯ab−n˜2FTab,(9)Raα=0,(10)Rαβ=R˜αβ−12trT+n˜−12FΔ¯Fg˜αβ,where trT=g¯abTab.It is well known that [16, 17](11)Rabcd;e=R¯abcd;e,Rαβγδ;ε=FR˜αβγδ;ε,Rαabβ;ε=0,where “semicolon” refers to the covariant derivative with respect to the metric.Also,(12)Rab;c=R¯ab;c−n˜2∇cTabF,Rαβ;γ=R˜αβ;γ,Rab;ε=0.


## 3.P−Curvature Flat Singly Warped Product Manifolds

In this section, we consider that the warped product manifoldM=M¯×FM˜ is a P−curvature flat manifold. The local components of the considered P−curvature tensor of the warped product manifold M¯×FM˜, which in general do not vanish identically, are the following Pabcd, Pαabβ, Pαβγδ, and Pabαβ, whereas the local components Pabcα and Paαβγ vanish.Let us calculate the first component of theP−curvature tensor of the warped product manifold M which is(13)Pαabβ=a0Rαabβ+a1gαaRbβ+a2gαbRaβ+a3gαβRab+a4gabRαβ+a5gaβRαb+a6gbβRαa.In virtue of (3) and (9), we have(14)Pαabβ=a0Rαabβ+a3gαβRab+a4gabRαβ.Utilizing equations (3), (7), (8), and (10) in equation (14), we infer(15)Pαabβ=−a02Tabg˜αβ+a3Fg˜αβR¯ab−n˜2FTab+a4g¯abR˜αβ−12trT+n˜−12FΔ¯Fg˜αβ.Suppose thatM is P−curvature flat, that is, Pαabβ=0. Thus,(16)−a02Tabg˜αβ+a3Fg˜αβR¯ab−n˜2FTab+a4g¯abR˜αβ−12trT+n˜−12FΔ¯Fg˜αβ=0.A contraction withg˜αβ implies(17)0=−a02Tabn˜+a3Fn˜R¯ab−n˜2FTab+a4g¯abR˜−12trT+n˜−12FΔ¯Fn˜,R¯ab=a02a3FTab+n˜2FTab−a4g¯aba3FR˜n˜−12trT+n˜−12FΔ¯F.One more contraction withgab gives(18)0=−a0n˜2trT+a3n˜FR¯−n˜2FtrT+a4n¯R˜−12trT+n˜−12FΔ¯Fn˜,R˜=a0n˜2a4n¯trT−a3n˜Fa4n¯R¯−n˜2FtrT+n˜2trT+n˜−12FΔ¯F.Equation (17) and (18) together imply(19)R¯ab=g¯abn¯R¯,which means that the base manifold M¯ is Einstein manifold.Contracting equation (16) with g¯ab gives(20)0=−a02trTg˜αβ+a3Fg˜αβR¯−n˜2FtrT+a4n¯R˜αβ−12trT+n˜−12FΔ¯Fg˜αβ,R˜αβ=a02a4n¯trTg˜αβ−a3a4n¯Fg˜αβR¯−n˜2FtrT+12trT+n˜−12FΔ¯Fg˜αβ.Multiplying equation (20) with g˜αβ implies(21)0=−a02trTn˜+a3n˜FR¯−n˜2FtrT+a4n¯R˜−12trT+n˜−12FΔ¯Fn˜=0,R¯=−a02a3FtrT+n˜2FtrT−a4n¯a3n˜FR˜−12trT+n˜−12FΔ¯Fn˜.Substituting equation (21) into (20), we have(22)R˜αβ=g˜αβn˜R˜,which means that the fiber manifold M˜ is Einstein manifold.The second component of theP−curvature tensor is(23)Pabαβ=a0Rabαβ+a1gabRαβ+a2gaαRbβ+a3gaβRbα+a4gbαRaβ+a5gbβRaα+a6gαβRab.In virtue of (3) and (9), one gets(24)Pabαβ=a1gabRαβ+a6gαβRab.The use of equations (3), (7), (8), and (10) implies(25)Pabαβ=a1g¯abR˜αβ−12trT+n˜−12FΔ¯Fg˜αβ+a6Fg˜αβR¯ab−n˜2FTab.Now, consider thatP−curvature tensor is flat; that is, Pabαβ=0, and hence,(26)a1g¯abR˜αβ−12trT+n˜−12FΔ¯Fg˜αβ+a6Fg˜αβR¯ab−n˜2FTab=0.A contraction withg˜αβ implies(27)0=a1g¯abR˜−12trT+n˜−12FΔ¯Fn˜+a6Fn˜R¯ab−n˜2FTab,R¯ab=n˜2FTab−a1g¯aba6Fn˜R˜−12trT+n˜−12FΔ¯Fn˜.Again, contracting equation (27) with g¯ab gives(28)0=a1n¯R˜−12trT+n˜−12FΔ¯Fn˜+a6Fn˜R¯−n˜2FtrT,R˜=12trT+n˜−12FΔ¯Fn˜−a6Fn˜a1n¯R¯−n˜2FtrT.Combining the previous two equations, we reveal that(29)R¯ab=g¯abn¯R¯.Similarly, we can obtain(30)R˜αβ=g˜αβn˜R˜.From the above discussion, we are in a position to state the following.Theorem 1.
LetM=M¯×FM˜ be a P−curvature flat singly warped product manifold furnished with the metric tensor g=g¯×Fg˜. Then, the base and the fiber manifolds of the warped product manifold M¯×FM˜ are Einstein manifold.Moving on to the next component, we have(31)Pabcd=a0Rabcd+a1gabRcd+a2gacRbd+a3gadRbc+a4gbcRad+a5gbdRac+a6gcdRab.Using (3), (8), and (5), we obtain(32)Pabcd=a0R¯abcd+a1g¯abR¯cd−n˜2FTcd+a2g¯acR¯bd−n˜2FTbd+a3g¯adR¯bc−n˜2FTbc+a4g¯bcR¯ad−n˜2FTad+a5g¯bdR¯ac−n˜2FTac+a6g¯cdR¯ab−n˜2FTab.The previous equation can be rewritten in the following form:(33)Pabcd=a0R¯abcd+a1g¯abR¯cd+a2g¯acR¯bd+a3g¯adR¯bc+a4g¯bcTad+a5g¯bdR¯ac+a6g¯cdR¯ab−n˜2Fa1g¯abTcd+a2g¯acTbd+a3g¯adTbc+a4g¯bcTad+a5g¯bdTac+a6g¯cdTab.Remember that theP−curvature tensor on the base manifold is of the form(34)P¯abcd=a0R¯abcd+a1g¯abR¯cd+a2g¯acR¯bd+a3g¯adR¯bc+a4g¯bcTad+a5g¯bdR¯ac+a6g¯cdR¯ab.Consequently, we can obtain the following:(35)Pabcd=P¯abcd−n˜2Fa1g¯abTcd+a2g¯acTbd+a3g¯adTbc+a4g¯bcTad+a5g¯bdTac+a6g¯cdTab.Suppose thatM=M¯×FM˜ is a P−curvature flat; that is, Pabcd=0. This leads to(36)P¯abcd=n˜2Fa1g¯abTcd+a2g¯acTbd+a3g¯adTbc+a4g¯bcTad+a5g¯bdTac+a6g¯cdTab,which is the form of the P−curvature tensor of the base manifold M¯. Thus, we can state the following theorem.Theorem 2.
LetM=M¯×FM˜ be a P−curvature flat singly warped product manifold equipped with the metric tensor g=g¯×Fg˜. Then, the P−curvature tensor on the base manifold M¯ is given by(37)P¯abcd=n˜2Fa1g¯abTcd+a2g¯acTbd+a3g¯adTbc+a4g¯bcTad+a5g¯bdTac+a6g¯cdTab.Assume thatTab=0; then, equation (36) implies(38)P¯abcd=0,which means that the base manifold is P−curvature flat.Corollary 1.
The base manifoldM¯ of the warped product manifold M is P−curvature flat if the warped product manifold M is P−curvature flat and Tab=0.The last component of theP−curvature tensor is(39)Pαβγδ=a0Rαβγδ+a1gαβRγδ+a2gαγRβδ+a3gαδRβγ+a4gβγRαδ+a5gβδRαγ+a6gγδRαβ.Using (6) and (10), we obtain(40)Pαβγδ=a0FR˜αβγδ−14Δ¯FG˜αβγδ+a1Fg˜αβR˜γδ−12trT+n˜−12FΔ¯Fg˜γδ+a2Fg˜αγR˜βδ−12trT+n˜−12FΔ¯Fg˜βδ+a3Fg˜αδR˜βγ−12trT+n˜−12FΔ¯Fg˜βγ+a4Fg˜βγR˜αδ−12trT+n˜−12FΔ¯Fg˜αδ+a5Fg˜βδR˜αγ−12trT+n˜−12FΔ¯Fg˜αγ+a6Fg˜γδR˜αβ−12trT+n˜−12FΔ¯Fg˜αβ.The previous equation can be rewritten in the following form:(41)Pαβγδ=Fa0R˜αβγδ+a1g˜αβR˜γδ+a2g˜αγR˜βδ+a3g˜αδR˜βγ+a4g˜βγR˜αδ+a5g˜βδR˜αγ+a6g˜γδR˜αβ−12trT+n˜−12FΔ¯F14a0Δ¯FG˜αβγδ+a1+a6Fg˜αβg˜γδ+a2+a5Fg˜αγg˜βδ+a3+a4Fg˜αδg˜βγ.TheP−curvature tensor on the fiber manifold is given by(42)P˜αβγδ=a0R˜αβγδ+a1g˜αβR˜γδ+a2g˜αγR˜βδ+a3g˜αδR˜βγ+a4g˜βγR˜αδ+a5g˜βδR˜αγ+a6g˜γδR˜αβ.Thus,(43)Pαβγδ=FP˜αβγδ−trT+n˜−12FΔ¯F14a0PG˜αβγδ+a1+a6Fg˜αβg˜γδ+a2+a5Fg˜αγg˜βδ+a3+a4Fg˜αδg˜βγ.IfM is P−curvature flat, that is, Pαβγδ=0, then(44)P˜αβγδ=12trT+n˜−12FΔ¯F14Fa0Δ¯FG˜αβγδ+a1+a6g˜αβg˜γδ+a2+a5g˜αγg˜βδ+a3+a4g˜αδg˜βγ.Thus, we have the following.Theorem 3.
LetM=M¯×FM˜ be a P−curvature flat singly warped product manifold with the metric tensor g=g¯×Fg˜. Then, the P−curvature tensor on the fiber manifold is of the form(45)P˜αβγδ=trT+n˜−12FΔ¯F14Fa0PG˜αβγδ+a1+a6g˜αβg˜γδ+a2+a5g˜αγg˜βδ+a3+a4g˜αδg˜βγ.


## 4.P−Divergence-Free Warped Product Manifold

The divergence of theP−curvature tensor is given by [2](46)∇hPjklh=a1∇jRkl+a2−a0∇kRjl+a3+a0∇lRkj+12a4gjk∇lR+12a5gjl∇kR+12a6gkl∇jR.IfP−curvature tensor is divergence-free, that is, ∇hPjklh=0, then(47)0=a1∇jRkl+a2−a0∇kRjl+a3+a0∇lRkj+12a4gjk∇lR+12a5gjl∇kR+12a6gkl∇jR.Contracting withgkl and using the relation ∇kRjk=1/2∇jR, we get(48)0=2a1+a2+a3+a4+a5+a6n∇jR.If2a1+a2+a3+a4+a5+a6n≠0, then ∇jR=0. And hence, equation (46) reduces to(49)a1∇jRkl+a2−a0∇kRjl+a3+a0∇lRkj=0.We thus have the following.Lemma 1.
A warped manifoldM¯×FM˜ with divergence-free P−curvature tensor is of constant scalar curvature and Ricci tensor satisfies equation (49), provided 2a1+a2+a3+a4+a5+a6n≠0.The divergence component∇hPαβγh of the P−curvature tensor on the warped product manifold M¯×FM˜ is(50)∇hPαβγh=a1∇αRβγ+a2−a0∇βRαγ+a3+a0∇γRβα+12a4gαβ∇γR+12a5gαγ∇βR+12a6gβγ∇αR.IfM¯×FM˜ is P−divergence-free, that is, ∇hPαβγh=0, then(51)0=a1∇αRβγ+a2−a0∇βRαγ+a3+a0∇γRβα+12a4gαβ∇γR+12a5gαγ∇βR+12a6gβγ∇αR.Using the obtained result in the previous lemma, we can have(52)a1∇αRβγ+a2−a0∇βRαγ+a3+a0∇γRβα=0.In view of equation (12), we infer(53)a1∇αR˜βγ+a2−a0∇βR˜αγ+a3+a0∇γR˜βα=0.Contracting withg˜βγ and using ∇αR˜βα=1/2∇βR˜, we get(54)2a1+a2+a3∇αR˜=0,If2a1+a2+a3≠0, then(55)∇αR˜=0,which means that the fiber manifold M˜ of the warped product manifold is of constant scalar curvature. Thus, we can state the following theorem.Theorem 4.
The fiber manifoldM˜ of P−divergence-free warped product manifold M is of constant scalar curvature.The next divergence component is(56)∇hPabch=a1∇aRbc+∇bRac+a3+a0∇cRab+12a4gab∇cR+12a5gac∇bR+12a6gbc∇aR.Assuming thatM is P−divergence-free and utilizing the obtained result in the previous lemma, we have(57)a1∇aRbc+a2−a0∇bRac+a3+a0∇cRab=0.In virtue of equation (12), we get(58)0=a1∇aR¯bc+a2−a0∇bR¯ac+a3+a0∇cR¯ab−n˜2∇aTbcF+∇bTacF+∇cTabF.IfTab=0, then(59)a1∇aR¯bc+a2−a0∇bR¯ac+a3+a0∇cR¯ab=0.Multiplying this withgbc and using ∇aR¯ba=1/2∇bR¯, we have(60)2a1+a2+a3∇aR¯=0.If2a1+a2+a3≠0, then(61)∇aR¯=0,which means that the base manifold M¯ of the warped product manifold M is of constant scalar curvature. Thus, we conclude the following.Theorem 5.
The base manifoldM¯ of P−divergence-free warped product manifold M is of constant scalar curvature, provided Tab=0.Now, consider the warped productM has a Codazzi Ricci tensor; that is, ∇lRkj=∇kRjl. And consequently, M is of constant scalar curvature. Thus, equation (46) leads to(62)∇hPjklh=a1+a2+a3∇jRkl.Proposition 1.
A warped product manifoldM with Codazzi Ricci tensor is P− divergence-free if and only if it has symmetric Ricc'i tensor, provided a1+a2+a3≠0.Now, the divergence component∇hPαβγh of the P−curvature tensor is(63)∇hPαβγh=a1+a2+a3∇αRβγ.Assume that the warped product manifoldM is P−divergence-free, and hence, one gets(64)a1+a2+a3∇αRβγ=0.Using equation (12), we get(65)a1+a2+a3∇αR˜βγ=0.Ifa1+a2+a3≠0, then(66)∇αR˜βγ=0.Thus, we have the folllowing.Theorem 6.
LetM be a P−divergence-free warped product whose Ricci tensor is of Codazzi type. Then, the Ricci tensor of the fiber manifold is symmetric.Also, the divergence component∇hPabch is(67)∇hPabch=a1+a2+a3∇aRbc.IfM¯×FM˜ is P−divergence-free, then(68)a1+a2+a3∇aRbc=0.In view of equation (12), we get(69)a1+a2+a3R¯ab;c−n˜2∇cTabF=0.IfTab=0 and a1+a2+a3≠0, then(70)R¯ab;c=0.We thus can state the following.Theorem 7.
LetM be a warped product with Codazzi Ricci tensor. Then the Ricci tensor of the base manifold M¯ is symmetric.


## 5. Semisymmetries of theP−Curvature Tensor

It is well known that a manifoldM is said to be semisymmetric if its Riemann tensor satisfies(71)∇m∇n−∇n∇mRijkl=0.A manifoldM is said to be Ricci semisymmetric if its Ricci tensor satisfies(72)∇m∇n−∇n∇mRij=0.TheP−curvature tensor is called semisymmetric if(73)∇m∇n−∇n∇mPijkl=0.Applying∇m∇n−∇n∇m on both sides of equation (1), one can have(74)∇m∇n−∇n∇mPijkl=∇m∇n−∇n∇ma0Rijkl+a1gijRkl+a2gikRjl+a3gilRjk+a4gjkRil+a5gjlRik+a6gklRij,∇m∇n−∇n∇mPijkl=a0∇m∇n−∇n∇mRijkl+a1gij∇m∇n−∇n∇mRkl+a2gik∇m∇n−∇n∇mRjl+a3gil∇m∇n−∇n∇mRjk+a4gjk∇m∇n−∇n∇mRil+a5gjl∇m∇n−∇n∇mRik+a6gkl∇m∇n−∇n∇mRij.Thus, we have the following:Proposition 2.
A pseudo-Riemannian manifoldM admits a semisymmetric P−curvature tensor if and only if M is semisymmetric.Now, assume thatM has a semisymmetric P−curvature tensor; that is,(75)∇m∇n−∇n∇mPijkl=0.Thus, we have(76)0=a0∇m∇n−∇n∇mRijkl+a1gij∇m∇n−∇n∇mRkl+a2gik∇m∇n−∇n∇mRjl+a3gil∇m∇n−∇n∇mRjk+a4gjk∇m∇n−∇n∇mRil+a5gjl∇m∇n−∇n∇mRik+a6gkl∇m∇n−∇n∇mRij.A contraction withgik implies(77)−a5gjl∇m∇n−∇n∇mR=a0+a1+a2n+a3+a4+a6∇m∇n−∇n∇mRjl.A multiplication withgjl implies(78)a0+a1+a2n+a3+a4+na5+a6∇m∇n−∇n∇mR=0.Ifa0+a1+a2n+a3+a4+na5+a6≠0, then(79)∇m∇n−∇n∇mR=0.And hence, equation (77) becomes(80)a0+a1+a2n+a3+a4+a6∇m∇n−∇n∇mRjl=0.Ifa0+a1+a2n+a3+a4+a6≠0, we have(81)∇m∇n−∇n∇mRjl=0,which means that M is Ricci semisymmetric.Proposition 3.
A pseudo-Riemannian manifoldM with semisymmetric P−curvature tensor is Ricci semisymmetric.Taking the covariant derivative of the first component of theP−curvature tensor, which is given by equation (14), we infer(82)Pαabβ;ε=a0Rαabβ;ε+a3gαβRab;ε+a4gabRαβ;ε.Using equations (11) and (12) in the previous equation, we have(83)Pαabβ;ε=a4g¯abR˜αβ;ε.Suppose that theP−curvature tensor is symmetric; that is, Pαabβ;ε=0. Thus,(84)a4g¯abR˜αβ;ε=0.A contraction withg¯ab gives(85)a4n¯R˜αβ;ε=0.Ifa4≠0, one may have(86)R˜αβ;ε=0,which means that the fiber manifold M˜ of the warped product manifold M¯×FM˜ is Ricci symmetric. We thus can state the following.Theorem 8.
LetM be a warped product manifold with symmetric P−curvature tensor. Then, the fiber manifold M˜ of M¯×FM˜ is a Ricci symmetric manifold.The covariant derivative of the componentPαβγδ is(87)Pαβγδ;ε=a0Rαβγδ;ε+a1gαβRγδ;ε+a2gαγRβδ;ε+a3gαδRβγ;ε+a4gβγRαδ;ε+a5gβδRαγ;ε+a6gγδRαβ;ε.The use of equations (3), (11), and (12) implies(88)Pαβγδ;ε=Fa0R˜αβγδ;ε+a1g˜αβR˜γδ;ε+a2g˜αγR˜βδ;ε+a3g˜αδR˜βγ;ε+a4g˜βγR˜αδ;ε+a5g˜βδR˜αγ;ε+a6g˜γδR˜αβ;ε,Pαβγδ;ε=FP˜αβγδ;ε.Thus, we can state the following.Theorem 9.
LetM¯×FM˜ be a warped product manifold with symmetric P−curvature tensor. Then, the fiber manifold M˜ has symmetric P−curvature tensor.The covariant derivative of the componentPabcd is(89)Pabcd;e=a0Rabcd;e+a1gabRcd;e+a2gacRbd;e+a3gadRbc;e+a4gbcRad;e+a5gbdRac;e+a6gcdRab;e.Utilizing (3), (11), and (12) entails that(90)Pabcd;e=a0R¯abcd;e+a1g¯abR¯cd;e+a2g¯acR¯bd;e+a3g¯adR¯bc;e+a4g¯bcR¯ad;e+a5g¯bdR¯ac;e+a6g¯cdR¯ab;e−n˜2∇ea1g¯abTcdF+a2g¯acTbdF+a3g¯adTbcF+a4g¯bcTadF+a5g¯bdTacF+a6g¯cdTabF,Pabcd;e=P¯abcd;e−n˜2∇ea1g¯abTcdF+a2g¯acTbdF+a3g¯adTbcF+a4g¯bcTadF+a5g¯bdTacF+a6g¯cdTabF.If∇eTab=0, we have(91)Pabcd;e=P¯abcd;e.We thus can state the following theorem.Theorem 10.
LetM¯×FM˜ be a warped product manifold M with symmetric P−curvature tensor. Then, the base manifold M¯ has a symmetric P−curvature tensor, provided ∇eTab=0.



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