PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS

c World Scientific Publishing Company DOI:10.1142/S0219887813500138

PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS

CARLO ALBERTO MANTICA

Physics Department, Universit`a Degli Studi di Milano Via Celoria 16, 20133, Milano, Italy I.I.S. Lagrange, Via L. Modignani 65

20161 Milano, Italy carloalberto.mantica@libero.it

YOUNG JIN SUH Kyungpook National University

Department of Mathematics Taegu 702-701, Korea

yjsuh@knu.ac.kr

Received 13 June 2012 Accepted 8 October 2012 Published 29 January 2013

In this paper, we introduce a new kind of tensor whose trace is the well-knownZtensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)_nand pseudo-concircular symmetric manifolds (P ˜CS)_n is defined. This is named pseudo-Q-symmetric and denoted with (PQS)_n. Various prop- erties of such ann-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo- symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc.

Math. Belgian Ser. A44(1992) 1–34]. It turns out that in this case the Ricci tensor isWeyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)_nmanifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)_n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The conse- quences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)_nscalar field space-time is considered, and interesting properties are pointed out.

Keywords: Pseudo-symmetric manifolds; pseudo-Qsymmetric; conformal curvature ten- sor; quasi-conformal curvature tensor; conformally symmetric; conformally recurrent;

Riemannian manifolds.

Mathematics Subject Classification 2010: Primary 53C15, Secondary 53C25

1. Introduction

Chaki [6] introduced and studied a type of nonflat Riemannian manifold whose curvature tensor is not identically zero and satisfies the following equation:

∇iR_jkl^m= 2A_iR_jkl^m+A_jR_ikl^m+A_kR_jil^m+A_lR_jki^m+A^mR_jkli. (1.1) Such a manifold is called pseudo-symmetric,A_k is a non-null covector called asso- ciated one-form,∇ is the operator of covariant differentiation with respect to the metricg_kl and the manifold is denoted by (PS)_n. Here we have defined the Ricci tensor to beR_kl =−R_mkl^m[50] and the scalar curvatureR=g^ijR_ij. This notion of pseudo-symmetric is different from that of Deszcz [20]. Several authors studied condition (1.1) (see [7,8,15,22,45]). We recall that the conformal curvature tensor is given in local coordinates by:

C_jkl^m=R_jkl^m+ 1

n−2(δ_j^mR_kl−δ^m_k R_jl+R_j^mg_kl−R^m_k g_jl)

− R

(n−1)(n−2)(δ^m_j g_kl−δ_k^mg_jl). (1.2) Ann-dimensional Riemannian manifold is said to be conformally flat ifC_jkl^m= 0 (it may be scrutinized that the conformal curvature tensor vanishes identically for n= 3 [36]). Tarafder proved that a conformally flat (PS)_n with nonzero constant scalar curvature is a subprojective space in the sense of Kagan [45]. In [15] the authors obtained the same results without assuming any restriction on the scalar curvature. In [22] Ewert–Krzemieniewski proved the existence of a (PS)_n. Condition (1.1) was then extended to other curvature tensors. We recall that a (0, 4) tensor Kis a generalized curvature tensor if [25]:

K_jklm+K_kljm+K_ljkm = 0, K_jklm =−K_kjlm=−K_jkml, K_jklm =K_lmkj. If a generalized curvature tensorK satisfies the condition:

∇iK_jkl^m= 2A_iK_jkl^m+A_jK_ikl^m+A_kK_jil^m+A_lK_jkl^m+A^mK_jkli, (1.3) then the manifold is named pseudo-K-symmetric and denoted with (PKS)_n [13]. If the previous equation holds forK_jklm=R_jklm then the manifold is called pseudo- symmetric, if it holds forK_jklm=C_jklm, then the manifold is called pseudo-confor- mally symmetric [14] (or conformally quasi-recurrent [5,27,38]), if it holds for the Weyl projective tensor then it is called pseudo-projective symmetric [10]. Moreover a pseudo-concircular symmetric manifold was taken into consideration in paper [35].

Some properties of (PKS)_n manifolds were studied in [30, 32]. For the definitions of the above-mentioned curvature tensors (see, for example [21,34,36,41,42,46]).

Recently the present authors [31] (see also [29]) defined a generalized (0, 2) symmetricZ tensor given by:

Z_kl =R_kl+ϕg_kl, (1.4)

where ϕ is an arbitrary scalar function. In [31, 29] various properties the Z ten- sor were pointed out; it was used to introduce the new differential structures of

pseudo-Z-symmetric and weakly Z-symmetricRiemannian manifolds. The first one is defined by the condition [31]:

∇kZ_jl = 2A_kZ_jl+A_jZ_kl+A_lZ_jk. (1.5) The fundamental properties of such manifolds were investigated in [31]. The second is defined by the condition [29]:

∇kZ_jl =A_kZ_jl+B_jZ_kl+D_lZ_jk. (1.6) A complete study of (1.6) was pursued in [29]. Finally in [33] manifolds on which a Z form is recurrent were studied. This embraces both pseudo-Z-symmetricand weakly Z-symmetricRiemannian manifolds.

In this paper, we introduce a new curvature tensor whose trace is theZ tensor.

The (1, 3) Qtensor is defined as:

Q_jkl^m=R_jkl^m− ϕ

(n−1)(δ_j^mg_kl−δ^m_k g_jl), (1.7) whereϕis an arbitrary scalar function. Obviously we haveZ_kl=−Q_mkl^m. Elemen- tary properties are alsoQ_jkl^m=−Q_kjl^m,Q_jkl^m+Q_klj^m+Q_ljk^m= 0. The (0, 4)Q tensor is defined in the natural way:

Q_jklm=R_jklm− ϕ

(n−1)G_jklm,

with G_jklm =g_jmg_kl −g_kmg_jl. Thus we have alsoQ_jklm =−Q_jkml,Q_jklm =Q_lmkj and the tensorQis a generalized curvature tensor.

The notion ofQtensor is also suitable to reinterpret some differential structures on a Riemannian manifold.

(1) A Q flat manifold is simply a manifold of constant curvature. In fact from Q_jlk^m= 0 we haveR_jkl^m= _(n−1)^ϕ (δ^m_j g_kl−δ_k^mg_jl) and transvectingR_kl =−ϕg_kl andR=−nϕ.

(2) AQ-symmetric manifold results to be Ricci symmetric [42]. From∇iQ_jkl^m= 0 we have∇iR_kl =∇iϕg_kl and thus∇iR=∇iϕ= 0.

(3) A Q recurrent manifold ∇_iQ_jkl^m = λ_iQ_jkl^m turns out to be a generalized recurrent manifold [16]; namely we have:

∇_jR_jkl^m=λ_iR_jkl^m+(∇_iϕ−λ_iϕ)

(n−1) (δ^m_j g_kl−δ_k^mg_jl). (1.8) (4) A Q harmonic ∇_mQ_jkl^m = 0 results to be of harmonic conformal curvature

tensor, i.e.∇mC_jkl^m= 0 (see [3]); in fact from

∇_mQ_jkl^m=∇_mR_jkl^m− 1

(n−1)(∇_jϕg_kl− ∇_kϕg_jl)

transvecting withg^jl we have∇_kϕ=−₂¹∇_kRand reinserting back we infer:

∇kR_jl − ∇jR_kl= 1

2(n−1)(∇kRg_jl− ∇jRg_kl). (1.9)

A manifold satisfying the previous condition is namednearly conformally symmetric and denoted with (NCS)_n: they were introduced by Roter [40] and studied also in [44]. It is easily seen that the condition (1.9) is equivalent to ∇_mC_jkl^m = 0.

Moreover, if ∇kϕ = 0 we infer ∇kR = 0 and thus a harmonic curvature tensor

∇_mR_jkl^m= 0.

Several cases accommodate in a new kind of Riemannian manifold whose non- nullQtensor satisfies the following condition:

∇iQ_jklm= 2AQ_jklm+A_jQ_iklm+A_kQ_jilm+A_lQ_jkim +A_mQ_jkli. (1.10) Such an n-dimensional manifold is named pseudo-Q-symmetric and denoted by (PQS)_n. It is worth to notice that if φ= 0 we recover a (PS)_n manifold, while if Z=g^klZ_kl =R+nϕ= 0 one hasφ=−^R_n and so we recover a pseudo-concircular symmetric manifold (P ˜CS)_n.

In the present paper we investigate the fundamental properties of (PQS)_n Rie- mannian manifolds. In Sec.2, we deal with elementary properties showing that the associated formA_k is closed. The case in which the associated covector takes the concircular form∇iA_j =A_iA_j+γg_ij is also investigated: it will be shown that in such case a pseudo-symmetric Riemannian manifold in the sense of Deszcz [20] is recovered; if γ = 0 the manifold reduces to a semisymmetric one [19]. Moreover, some other curvature conditions that reduce the manifold to aQrecurrent one are taken into consideration. In Sec. 3, we will show that in the case of concircular associated covector, the Ricci tensor results to be Weyl compatible. This notion was recently introduced by one of the present authors in [27, 30]. In this case an enlarged version of the celebrated Derdzinski–Shen theorem [18] applies. Here, we discuss an alternative proof. In Sec.4, we investigate conformally flat (PQS)_n Rie- mannian manifolds: a local form of the components of the Ricci tensor is given:

this generalizes the results of [15, 45]. In Sec. 5, it is shown that a conformally flat (PQS)_n manifold admits a proper concircular vector and the local form of the metric tensor is given. Section6deals with the properties of special conformally flat (PQS)_n manifolds. Finally, in Sec.7, we investigate some interesting properties of (PQS)_n space-time manifolds; in particular we take into consideration perfect fluid space-times with cosmological constant [24] and provide a state equation. The con- sequences (recently obtained in [30]) of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Moreover a (PQS)_n scalar field space-time is considered, and interesting properties are pointed out. Throughout the paper all manifolds under consideration are assumed to be smooth by connected Hausdorff manifolds; their metrics are assumed to be positive definite in Secs.1–6.

In Sec. 7, we consider a smooth four-dimensional Hausdorff space-time manifold endowed with a Lorentz metric [24] (i.e. a metric of signature +2).

2. Elementary Properties of a (PQS)_n Manifold

In this section elementary properties of a (PQS)_nare shown. We will show that the covectorAis closed. Moreover interesting properties arise in the case of concircular

A_k. Let M be a nonflat n(n ≥ 4)-dimensional (PQS)_n Riemannian manifold with metric g_ij and Riemannian connection ∇. We can state the following sim- ple theorem.

Theorem 2.1. TheQtensor of a pseudo-Q-symmetric manifolds satisfies the sec- ond Bianchi identity:

∇_iQ_jklm+∇_jQ_kilm+∇_kQ_ijlm = 0. (2.1) Proof. Write three equations like (1.10) with a cyclic indices permutation and sum up, taking into consideration the first Bianchi identity for theQtensor.

Now we point out some useful formulas concerning (PQS)_nmanifolds: transvec- tion equation (1.10) withg^mj gives immediately:

∇_kZ_jl = 2A_kZ_jl+A_jZ_kl+A_lZ_jk −A^mQ_kjlm−A^mQ_kljm. (2.2) Transvecting equation (2.2) withg^jl and withg^kj gives:

∇kZ= 2A_kZ+ 4A^lZ_kl,

∇^lZ_kl=A_kZ+ 2A^lZ_kl. (2.3) Combining these equations and using the relation ∇^lZ_jl = ¹₂∇jR+∇jϕ coming from the contracted second Bianchi identity one obtains ∇jϕ = 0 and ∇kR = 2A_kZ+ 4A^lZ_kl.

The last equations are the generalization of the correspondent results obtained for (PS)_nand (P ˜CS)_nmanifolds (see [15,35,45]). In fact, we can state the following simple remarks:

Remark 2.1. If ϕ = 0, we recover a (PS)_n manifold one has ∇kR = 2A_kR+ 4A^lR_kl. Moreover if ∇kR = 0 it is A^lR_kl = −A_k^R₂. Thus A_k results a closed one-form and it is an eigenvector of the Ricci tensor with eigenvalue−^R₂.

Remark 2.2. IfZ= 0, thenφ=−^R_n and by a simple calculation∇_kR=∇_kφ= 0 and thenA^lR_kl =^R_nA_k. So we have obtained that the scalar curvature is a covariant constant and thatA_k is an eigenvector of the Ricci tensor with eigenvalue ^R_n.

Now from Eq. (2.2) we have simply:

∇kZ_jl− ∇jZ_kl =A_kZ_jl−A_jZ_kl+ 3A^mQ_jklm, (2.4) and, because of the relation ∇_jZ_kl = ∇_jR_kl and the second contracted Bianchi identity it follows:

∇kR_jl− ∇jR_kl =∇mR_jkl^m=A_kZ_jl−A_jZ_kl+ 3A^mQ_jklm. (2.5) Now we prove that the associated covector of a (PQS)_nis a closed one-form. We follow the trick already used in [22,5]. First a covariant derivative∇sis applied on

Eq. (1.10) and the commutator is evaluated obtaining:

(∇s∇i− ∇i∇s)Q_jklm = 2(∇sA_i− ∇iA_s)Q_jklm+ (∇sA_j−A_jA_s)Q_iklm + (∇sA_k−A_kA_s)Q_jilm+ (∇sA_l−A_lA_s)Q_jkim + (∇sA_m−A_mA_s)Q_jkli−(∇iA_j−A_iA_j)Q_sklm

−(∇iA_k−A_iA_k)Q_jslm−(∇iA_l−A_iA_l)Q_jksm

−(∇_iA_m−A_iA_m)Q_jkls. (2.6) In the sequel we defineω_si =∇sA_i− ∇iA_s. Permuting in pairs the indices (s, i), (j, k) and (l, m) we obtain after a long calculation (see [22,5])

(∇_s∇_i− ∇_i∇_s)Q_jklm+ (∇_j∇_k− ∇_k∇_j)Q_silm+ (∇_l∇_m− ∇_m∇_l)Q_sijk

= 2[ω_siQ_jklm+ω_jkQ_silm+ω_lmQ_sijk]

+ω_sjQ_iklm+ω_skQ_jilm+ω_slQ_jkim+ω_smQ_jkli+ω_jiQ_sklm+ω_jlQ_sikm +ω_jmQ_silk+ω_kiQ_jslm+ω_liQ_jksm+ω_miQ_jkls+ω_lkQ_sijm+ω_mkQ_silj.

(2.7) Now we note that (∇s∇i−∇i∇s)Q_jklm= (∇s∇i−∇i∇s)R_jklm and the well-known Walker lemma [49] about the Riemann curvature tensor:

Lemma 2.1 (Walker). The Riemann curvature tensor satisfies the following identity:

(∇s∇i− ∇i∇s)R_jklm+ (∇j∇k− ∇k∇j)R_silm + (∇l∇m− ∇m∇l)R_sijk = 0. (2.8) Thus we are able to write Eq. (2.7) in the algebraic form:

2[ω_siQ_jklm+ω_jkQ_silm +ω_lmQ_sijk]

+ω_sjQ_iklm+ω_skQ_jilm+ω_slQ_jkim+ω_smQ_jkli+ω_jiQ_sklm +ω_jlQ_sikm+ω_jmQ_silk+ω_kiQ_jslm+ω_liQ_jksm+ω_miQ_jkls

+ω_lkQ_sijm+ω_mkQ_silj = 0. (2.9)

Now the following lemma due to Ewert-Krzemieniewski [22] (see also [5]) is pointed out:

Lemma 2.2. If ω_lm = −ω_ml and a generalized curvature tensor K satisfies the algebraic equation:

2[ω_siK_jklm+ω_jkK_silm+ω_lmK_sijk]

+ ω_sjK_iklm+ω_skK_jilm+ω_slK_jkim+ω_smK_jkli+ω_jiK_sklm+ω_jlK_sikm + ω_jmK_silk+ω_kiK_jslm+ω_liK_jksm+ω_miK_jkls+ω_lkK_sijm+ω_mkK_silj = 0, then eitherω_lm= 0 for alll, morK_hijk= 0 for allh, i, j, k.

From Eq. (2.7), Lemmas 2.1 and 2.2, it is clear that the following statement holds.

Theorem 2.2. The associated covector of a pseudo-Q-symmetric Riemannian manifold is closed.

Now we point out some consequences due to Eq. (2.6). Let us suppose that the associated covector of a (PQS)_n is of the concircular form ∇iA_j = A_iA_j +γg_ij, beingγ an arbitrary scalar function. This condition will be of great importance in Sec. 3. Then we have:

(∇_s∇_i− ∇_i∇_s)Q_jklm =γ[g_jsQ_iklm+g_skQ_jilm+g_slQ_jkim+g_smQ_jkli

−g_ijQ_sklm−g_ikQ_jslm−g_ilQ_jksm−g_imQ_jkls]. (2.10) Again on noting that (∇s∇i− ∇i∇s)Q_jklm = (∇s∇i − ∇i∇s)R_jklm and that a simple computation gives:

g_jsG_iklm+g_skG_jilm+g_slG_jkim+g_smG_jkli−g_ijG_sklm

−g_ikG_jslm−g_ilG_jksm−g_imG_jkls= 0, we conclude that:

(∇s∇i− ∇i∇s)R_jklm =γ[g_jsR_iklm+g_skR_jilm+g_slR_jkim +g_smR_jkli−g_ijR_sklm

−g_ikR_jslm−g_ilR_jksm−g_imR_jkls]. (2.11) Theorem 2.3. Let M be an n-dimensional(PQS)_n Riemannian manifold. If the associated covector has the form∇_iA_j=A_iA_j+γg_ij,then the manifold reduces to a pseudo-symmetric Riemannian manifold in the sense of Deszcz[20]. Ifγ= 0 the manifold reduces to a semisymmetric one [19].

Now we focus on (PQS)_n satisfying some other curvature conditions [37, 38].

For example we consider aQ-recurrent, i.e.:

∇iQ_jklm =k_iQ_jklm

for some covectork_i. From the definition of a (PQS)_n we easily infer that:

0 = (2A_i−k_i)Q_jklm+A_jQ_iklm+A_kQ_jilm+A_lQ_jkim+A_mQ_jkli. (2.12) Now a useful lemma due to Roter [39] is stated:

Lemma 2.3 (Roter). Ifc_j,p_j andK_hijk are numbers satisfying:

c_sK_hijk+p_hK_sijk+p_iK_hsjk+p_jK_hisk+p_kK_hijs = 0, K_hijk =K_jkhi =−K_hikj, K_hijk+K_hjki+K_hkij = 0, thenc_j+ 2p_j= 0or K_hijk = 0.

This gives immediatelyk_i = 4A_i and consequently Eq. (2.12) takes the form:

−2A_iQ_jklm+A_jQ_iklm+A_kQ_jilm+A_iQ_jklm+A_mQ_jkli= 0.

Now a cyclic permutation of indicesi,j, kis performed on the previous equation, and the resulting three relations are added to obtain:

A_iQ_kjlm+A_jQ_iklm+A_kQ_jilm = 0. (2.13) This is a curvature condition on the Q tensor that has a discrete relevance. We state the result:

Theorem 2.4. If a pseudo-Q-symmetric Riemannian manifold is alsoQ-recurrent, then the condition(2.13)is fulfilled.

Now let us suppose that (2.13) is valid: the companion equation A_iQ_lmjk+A_lQ_mijk+A_mQ_iljk = 0

is written, and summing up we finally have:

2A_iQ_jklm+A_jQ_kilm+A_kQ_ijlm+A_lQ_jkmi +A_mQ_jkil = 0.

Thus Eq. (1.10) reduces to∇iQ_jklm = 4A_iQ_jklm. We have thus proved the following result:

Theorem 2.5. LetM be ann-dimensional pseudo-Q-symmetric Riemannian man- ifold. If the conditionA_iQ_kjlm+A_jQ_iklm+A_kQ_jilm = 0is fulfilled,then the manifold reduces to aQ-recurrent one.

3. (PQS)_n Manifolds with Concircular Associated Vector: Weyl Compatibility

In this section, we consider ann-dimensional pseudo-Q-symmetric manifold with associated covector of concircular form. We will show that this condition implies that the Ricci tensor is Weyl compatible. This notion was recently introduced in [27,30]. In this case an enlarged version of the celebrated Derdzinski–Shen theorem about Codazzi tensors applies (see [18, 27, 30] for a detailed discussion). We will prove this theorem here in an alternative way. As a consequence, strong restrictions on the structure of the Weyl tensor are imposed, with geometric and topological implications (see [30,18]). We remember Eq. (2.5):

∇_mR_jkl^m= 3A_mQ_jkl^m+A_kZ_jl−A_jZ_kl. (3.1) The covariant derivative∇_iis thus applied to the previous expression, and then a cyclic permutation over indicesi, j, k is performed. The resulting equations are

added to obtain, using (2.4), the second Bianchi identity for theQtensor:

∇i∇mR_jkl^m+∇j∇mR_kil^m+∇k∇mR_ijl^m

= 3[(∇_iA_m−A_iA_m)Q_jkl^m+ (∇_jA_m−A_jA_m)Q_kil^m + (∇kA_m−A_kA_m)Q_ijl^m] +Z_jl(∇iA_k− ∇kA_i)

+Z_kl(∇jA_i− ∇iA_j) +Z_il(∇kA_j− ∇jA_k). (3.2) Now if we consider a concircular associated covectorA_i, i.e. satisfying the con- dition∇iA_m=A_iA_m+γg_im, we find:

∇_i∇_mR_jkl^m+∇_j∇_mR_kil^m+∇_k∇_mR_ijl^m= 0. (3.3) Now using Lovelock’s identity [26,28]:

∇i∇mR_jkl^m+∇j∇mR_kil^m+∇k∇mR_ijl^m

=−(R_imR_jkl^m+R_jmR_kil^m+R_kmR_ijl^m), we obtain simply:

R_imR_jkl^m+R_jmR_kil^m+R_kmR_ijl^m= 0. (3.4) Theorem 3.1. Let M be an n-dimensional (P QS)_n Riemannian manifold with

∇iA_m=A_iA_m+γg_im. Then the relation (3.4) is fulfilled.

If the Ricci tensor satisfies Eq. (3.4) it is named R-compatible [27, 30]. If we insert in the previous relation the local form of the Weyl tensor we obtain:

R_imC_jkl^m+R_jmC_kil^m+R_kmC_ijl^m= 0. (3.5) The Ricci tensor is thusWeyl-compatible. More generally a (0, 2) symmetric tensor is said to beK-compatible[30] if an analogous algebraic relation with a generalized curvature tensor is fulfilled:

b_imK_jkl^m+b_jmK_kil^m+b_kmK_ijl^m= 0. (3.6) In recent works [27–30] it is shown that an extended version of Derdinski–Shen theorem in [18] is valid when the previous equation is verified. The following result was obtained.

Theorem 3.2. Let M be an n-dimensional Riemannian manifold with a gener- alized curvature tensor K and aK-compatible tensor b. If X, Y andZ are three eigenvectors of the matrix b^s_r at a point x of the manifold, with eigenvalues λ, µ andν,then

XⁱY^jZ^kK_ijkl= 0, (3.7)

provided that λandµare different fromν.

Derdzinski and Shen [18] proved this result in the case in whichbis a Codazzi tensor. We notice that ifb_klis a Codazzi tensor, then it is Riemann-compatible [30].

Here we provide an alternative proof of the previous theorem based only on Eq. (3.6). Write the equation for the eigenvectors of the tensorb_kl:

Xⁱb_im =λX_m, Y^jb_jm =µY_m, X^kb_km=vZ_m, W^lb_lm =ηW_m, (3.8) whereλ, µ,ν,η are the eigenvalues of the matrixb^s_randXⁱ, etc. belonging to the tangent bundle of the manifold. Now equationb_imK_jkl^m+b_jmK_kil^m+b_kmK_ijl^m= 0 is multiplied byXⁱY^jZ^k and after an indices rearrangement (m→j,i→min the second term,i→m,m→kin the third term) one easily gets:

X^mY^jZ^k[λK_kjml+µK_mkjl+νK_jmkl] = 0. (3.9) Now for example if µ = ν from the previous equation and the first Bianchi identity we obtain:

X^mY^jZ^k(λ−µ)K_kjlm= 0. (3.10) Thus ifλ=µwe have:

1

2(Y^jZ^k−Y^jZ^j)X^mK_kjlm = 0. (3.11) In this way we have proven Theorem 3.1 in the case λ = µ. Now Eq. (3.9) is multiplied byW^lto obtain:

X^mY^jZ^kW^l[λK_kjml+µK_mkjl+νK_jmkl] = 0. (3.12) At this point in Eq. (3.4) we make the index changei→j,j→k,k→l,l→ito obtain easily:

b_jmK_kli^m+b_kmK_lji^m+b_lmK_jkl^m= 0. (3.13) Again the previous equation is multiplied by Y^jZ^kW^l and after an indices rearrangement (m → k, j → m in the second term, m → l, j → m in the third term) one easily gets:

Y^mZ^kW^l[µK_lkmi+νK_mlki+ηK_kmli] = 0. (3.14) This last equation is multiplied byXⁱ and after an indices rearrangement (i→m, m→j) the following relation holds:

X^mY^jZ^kW^l[ηK_kjml+νK_mkjl+µK_jmkl] = 0. (3.15) At this point in Eq. (3.13) we make the index changei→j,j→k,k→l,l→i to obtain straightforwardly:

b_kmK_lij^m+b_lmK_ikj^m+b_imK_klj^m= 0. (3.16) The previous equation is then multiplied by XⁱZ^kW^l to obtain after an indices rearrangement (k → m, m → l in the second term,k →m, m → i in the third term):

XⁱZ^mW^l[νK_ilmj +ηK_milj +λK_lmij] = 0. (3.17)

This last equation is multiplied byY^j and after an indices rearrangement (i→m, m→k) the following relation holds:

X^mY^jZ^kW^l[νK_kjml+ηK_mkjl+λK_mkjl] = 0. (3.18) Finally in Eq. (3.16) we make the index change i→j, j →k, k→l, l→i to obtain easily:

b_lmK_ijk^m+b_imK_jlk^m+b_jmK_lik^m= 0. (3.19) As in the previous cases the last equation is multiplied by XⁱY^jW^l and after an indices rearrangement (m →i, l →m in the second term, m→ j, l →m in the third term) one gets:

XⁱY^jW^m[ηK_jimk +λK_mjik +µK_imjk] = 0. (3.20) This last equation is multiplied byZ^k and after an indices rearrangementi→m, m→l the following relation holds:

X^mY^jZ^kW^l[µK_kjml+λK_mkjl+ηK_jmkl] = 0. (3.21) Now Eqs. (3.12), (3.21) and (3.18) are rewritten in matrix form to get:







λ µ ν µ λ η ν η λ

1 1 1









X^mY^jZ^kW^lK_kjml X^mY^jZ^kW^lK_mkjl X^mY^jZ^kW^lK_jmkl



= 0. (3.22)

In writing the last equation we have used the obvious (Bianchi) identity:

X^mY^jZ^kW^l[K_kjml+K_mkjl+K_jmkl] = 0. (3.23) These are the same equations (in form) that we may find in the paper of Derdzinski and Shen [18]. Now in order to obtain a non-null curvature tensor the rank of the matrix should be at most two. This fact leads to the following restrictions:

(η−λ)(µ+ν−λ−η) = 0, (λ−ν)(η+µ−ν−λ) = 0, (µ−λ)(η+ν−λ−µ) = 0.

(3.24)

If we suppose that λ = η, λ = µ, λ = ν from the previous relations we get λ=η =ν =µ. So we are restricted to suppose that λis equal to one ofµ, η, ν.

As in Derdzinski and Shen’s paper [18] this implies thatX^mY^jZ^kW^lK_kjml= 0. In fact the symmetries of the curvature tensor imply thatX^mY^jZ^kW^lK_kjml = 0 only ifX,Y,Z,W belong to at most two distinct eigenspaces. This completes the proof in the case of distinct eigenvalues.

Thus on a pseudo-Q-symmetric Riemannian manifold with concircular associ- ated vector the Ricci tensor is Weyl compatible and the eigenvectors of the Ricci tensor pose strong restrictions on the structure of the Weyl tensor.

Equation (3.6) has another important consequence. Given a (0, 2) symmetric tensorhwe define a (0, 2) symmetric tensorK as in Bourguignon’s notation [4]:

K◦_(h)jm=h^klK_kjml. (3.25)

In [30], the authors proved the following statement:

Theorem 3.3. IfbisK-compatible,andbcommutes with a symmetric(0,2)tensor h,the symmetric tensorK^◦_(h)jm=h^klK_kjml commutes withb.

We give here again a proof for completeness.

Proof. Multiply Eq. (3.6) with h^kl: the last termb_kmh^klK_ijl^m vanishes for sym- metry reasons. The remaining terms give the commutation relation.

Taking h = b and K the Weyl tensor, from the previous results we have the following corollary:

Corollary 3.1. Let M be an n-dimensional (P QS)_n Riemannian manifold with

∇iA_m = A_iA_m+γg_im: then the symmetric tensor C^◦_(b)jm = b^klC_kjml commutes withb.

This result will be of great importance in Sec. 7 where we deal with general relativity.

4. Conformally Flat Pseudo-Q-Symmetric Manifolds:

Local Form of the Ricci Tensor

In this section a (PQS)_n, (n > 3) Riemannian manifold with the property

∇mC_jkl^m = 0 [42], i.e. with harmonic conformal curvature tensor is considered.

Some interesting properties are derived. In the sequel of the section we will special- ize to a conformally flat (PQS)_n manifold. It is well known that the divergence of the conformal tensor satisfies the relation:

∇mC_jkl^m= n−3 n−2

∇mR_jkl^m+ 1

2(n−1)(∇jRg_kl− ∇kRg_jl)

. (4.1)

So if we consider∇mC_jkl^m= 0 one immediately obtains:

∇_mR_jkl^m=∇_kR_jl− ∇_jR_kl = 1

2(n−1)(∇_kRg_jl− ∇_jRg_kl). (4.2) From Eq. (2.5) we get easily:

3A_mQ_jkl^m+A_kZ_jl−A_jZ_kl= 1

2(n−1)[(∇kR)g_jl−(∇jR)g_kl]. (4.3) Inserting in the previous relation∇kR= 2A_kZ+ 4A^lZ_kl we have simply:

3A_mQ_jkl^m+A_kZ_jl−A_jZ_kl

= 1

(n−1)[(A_kZ+ 2A^mZ_km)g_jl−(A_jZ+ 2A^mZ_jm)g_kl]. (4.4)

On multiplying Eq. (4.4) by A^l with the symmetry conditionA_mA_lQ_jk^lm = 0 we have

A_kA^lZ_jl =A_jA^lZ_kl. Again we multiply the previous equation byA^j to obtain:

A_kA^jA^lZ_jl =A_jA^jA^lZ_kl. This last can be rewritten as

A^lZ_kl = A^jA^lZ_jl

A_jA^j A_k=tA_k, (4.5)

wheret= ^A_A^jAlZjl

jA^j is a scalar function. We have proved the following theorem.

Theorem 4.1. Let M be ann(n >3)-dimensional (P QS)_n Riemannian manifold with the property ∇mC_jkl^m = 0. Then the vector A^l is an eigenvector of the Z_kl tensor with eigenvalue t.

Inserting result (4.5) in Eq. (2.3) one easily obtains:

2(2t+Z)A_k =∇_kZ. (4.6)

On multiplying the previous result byA_j and interchanging the indicesjandkwe get:

A_j∇kZ=A_k∇jZ. (4.7)

Again the operation of covariant derivation is applied on Eq. (4.6) and the following relation is obtained:

∇j∇kZ= 2(∇jA_k)Z+ 2A_k(∇jZ) + 4(∇jt)A_k+ 4t(∇jA_k).

Now a similar equation with indices k and j exchanged is written and then subtracted from the previous result to obtain finally, taking account of (4.7):

0 = (∇_jA_k− ∇_kA_j)(2Z+ 4t) + 4[A_k(∇_jt)−A_j(∇_kt)]. (4.8) From the previous section we know that in a (PQS)_n the associated covector is closed; we have thus:

A_k(∇_jt)−A_j(∇_kt) = 0. (4.9) The following statement resumes such results:

Theorem 4.2. Let M be ann(n >3)-dimensional (P QS)_n Riemannian manifold with the property ∇mC_jkl^m = 0. Then the relations A_k(∇jt)−A_j(∇kt) = 0 and A_j∇_kZ =A_k∇_jZ hold.

Hereafter, we specialize to a conformally flat (PQS)_n: we will show that a con- formally flat (PQS)_n is quasi-Einstein [9], generalizing the result found in [15,45].

This is done in few steps.