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PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS
CARLO ALBERTO MANTICA AND YOUNG JIN SUH
Abstract. In this paper we introduce a new kind of tensor whose trace is
the well known Ztensor 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 (P S)n
and pseudo-concircular symmetric manifolds (P˜
CS )nis defined. This is named
pseudo -Qsymmetric and denoted with (P QS)n. Various properties of such an
n-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 sence of R. Dezcz [20]. It turns out that in this
case the Ricci tensor is Weyl compatible, a concept enlarging the classical
Derdzinski-Shen theorem about Codazzi tensors. Moreover it is shown that
a conformally flat (P QS)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 (P QS)nspace-time manifolds; in particular we take into consideration
perfect fluid space times and provide a state equation. The consequences of
the Wely 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.
1. Introduction
M.C. Chaki [6] introduced and studied a type of non flat Riemannian manifold
whose curvature tensor is not identically zero and satisfies the following equation:
(1.1) ∇iRjkl m= 2AiRjkl m+AjRikl m+AkRjil m+AlRjki m+AmRjkli
Such a manifold is called Pseudo symmetric, Akis a non null covector called as-
sociated 1-form, ∇is the operator of covariant differentiation with respect to the
metric gkl and the manifold is denoted by (P S)n. Here we have defined the Ricci
tensor to be Rkl =−Rmkl m[50] and the scalar curvature R=gijRij . This no-
tion of Pseudo symmetric is different from that of R. Deszcz [20]. Several authors
studied condition (1.1)(see [7], [8], [15], [22] and [45]). We recall that the conformal
curvature tensor is given in local coordinates by:
12010 Mathematics Subject Classification : Primary 53C15; Secondary 53C25.
2Key words and phrases : pseudo symmetric manifolds, Pseudo Q symmetric, conformal cur-
vature tensor, quasi conformal curvature tensor, conformally symmetric, conformally recurrent,
Riemannian manifolds.
* The second author was supported by grant Proj. No. NRF-2011-220-C00002 and BSRP-
2011-0025687 from National Research Foundation.
1
2 C.A. MANTICA & Y.J. SUH
Cjkl m=Rjkl m+1
n−2(δm
jRkl −δm
kRjl +Rm
jgkl −Rm
kgjl )
−R
(n−1)(n−2)(δm
jgkl −δm
kgjl ).
(1.2)
An n-dimensional Riemannian manifold is said to be conformally flat if Cm
jkl = 0 (it
may be scrutinized that the conformal curvature tensor vanishes identically for n=
3 [36]). Tarafder proved that a confrmally flat (PS )nwith non-zero 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 (P S)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]:
Kjklm +Kkljm +Kljk m = 0, Kjklm =−Kkj lm =−Kjkml ,
Kjklm =Klmk j .
If a generalized curvature tensor Ksatisfies the condition:
(1.3) ∇iKjkl m= 2AiKjkl m+AjKikl m+AkKjil m+AlKjkl m+AmKjkli ,
then the manifold is named Pseudo Ksymmetric and denoted with (P KS )n[13]. If
the previous equation holds for Kjklm =Rj klm then the manifold is called Pseudo-
Symmetric, if it holds for Kjklm =Cjklm , then the manifold is called Pseudo
Conformally Symmetric [14] (or conformally quasi recurrent [5], [27] and [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 consider-
ation in paper [35]. Some properties of (P K S)nmanifolds were studied in [30] and
[32]. For the definitions of the above mentioned curvature tensors (see for example
[21], [34], [36], [41], [42] and [46]).
Recently the present authors [31] (see also [29]) defined a generalized (0,2) sym-
metric Ztensor given by:
(1.4) Zkl =Rkl +ϕgkl ,
where ϕis an arbitrary scalar function. In [31] and [29] various properties the Z
tensor were pointed out; it was used to introduce the new differential structures of
pseudo Z-symmetric and weakly Z-symetric Riemannian manifolds. The first one
is defined by the condition [31]:
(1.5) ∇kZjl = 2AkZjl +AjZkl +AlAjk .
The fundamental properties of such manifolds were investigated in [31]. The second
is defined by the condition [29]:
(1.6) ∇kZjl =AkZjl +BjZkl +DlZjk .
A complete study of (1.6) was pursued in [29]. Finally in [33] manifolds on which
aZform is recurrent were studied. This embraces both pseudo Z-symmetric and
weakly Z-symmetric Riemannian manifolds.
In this paper we introduce a new curvature tensor whose trace is the Ztensor.
The (1,3) Qtensor is defined as:
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 3
(1.7) Qjkl m=Rjkl m−ϕ
(n−1)(δm
jgkl −δm
kgjl ),
where ϕis an arbitrary scalar function. Obviously we have Zkl =−Qmklm. Ele-
mentary properties are also Qjkl m=−Qkjl m,Qjklm+Qklj m+Qlj km= 0. The
(0,4) Qtensor is defined in the natural way:
Qjklm =Rj klm −ϕ
(n−1)Gjklm
with Gjklm =gj mgkl −gk mgjl . Thus we have also Qjklm =−Qj kml,Qj klm =Qlmkj
and the tensor Qis a generalized curvature tensor.
The notion of Qtensor is also suitable to reinterpret some differential structures
on a Riemannian manifold.
1) A Qflat manifold is simply a manifold of constant curvature. In fact from
Qjlk m= 0 we have Rjklm=ϕ
(n−1) (δm
jgkl −δm
kgjl ) and transvecting Rkl =−ϕgkl
and R=−nϕ.
2) A Q-symmetric manifold results to be Ricci symmetric [42]. From ∇iQjkl m= 0
we have ∇iRkl =∇iϕgkl and thus ∇iR=∇iϕ= 0.
3) A Qrecurrent manifold ∇iQjkl m=λiQjkl mturns out to be a generalized
recurrent manifold [16]; namely we have:
(1.8) ∇jRjkl m=λiRjkl m+(∇iϕ−λiϕ)
(n−1) (δm
jgkl −δm
kgjl ).
4) A Qharmonic ∇mQj klm= 0 results to be of harmonic conformal curvature
tensor, i.e.∇mCjkl m= 0; in fact from
∇mQjkl m=∇mRjkl m−1
(n−1)(∇jϕgkl − ∇kϕgjl)
transvecting with gjl we have ∇kϕ=−1
2∇kRand reinserting back we infer:
(1.9) ∇kRjl − ∇jRkl =1
2(n−1)(∇kRgjl − ∇jRgkl).
A manifold satisfying the previous condition is named nearly Conformally Sym-
metric 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 ∇mCjkl m= 0.
Moreover if ∇kϕ= 0 we infer ∇kR= 0 and thus an harmonic curvature tensor
∇mRjkl m= 0.
Several cases accommodate in a new kind of Riemannian manifold whose non null
Qtensor satisfies the following condition:
(1.10) ∇iQjklm = 2AQj klm +AjQiklm +AkQj ilm +AlQjkim +AmQj kli.
Such an n-dimensional manifold is named Pseudo Qsymmetric and denoted by
(P QS)n. It is worth to notice that if φ= 0 we recover a (P S)nmanifold, while if
Z=gklZk l =R+nϕ = 0 one has φ=−R
nand so we recover a Pseudo Concircular
symmetric manifold (P˜
CS )n.
In the present paper we investigate the fundamental properties of (P QS)nRie-
mannian manifolds. In Section 2 we deal with elementary properties showing that
the associated form Akis closed. The case in which the associated covector takes
4 C.A. MANTICA & Y.J. SUH
the concircular form ∇iAj=AiAj+γgij is also investigated: it will be shown that
in such case a Pseudo Symmetric Riemannian manifold in the sense of Ryszard
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 a Qrecur-
rent one are taken into consideration. In Section 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] and [30].
In this case an enlarged version of the celebrated Derdzinski-Shen theorem [18] ap-
plies. Here we discuss an alternative proof. In Section 4 we investigate conformally
flat (P QS)nRiemannian manifolds: a local form of the components of the Ricci
tensor is given: this generalizes the results of [15] and [45]. In Section 5 it is shown
that a conformally flat (P QS)nmanifold admits a proper concircular vector and
the local form of the metric tensor is given. Section 6 deals with the properties
of special conformally flat (P QS )nmanifolds. Finally in Section 7 we investigate
some interesting properties of (P QS)nspace-time manifolds; in particular we take
into consideration perfect fluid space times with cosmological constant [24] and
provide a state equation. The consequences (recently obtained in [30]) of the Wely
compatibility on the electric and magnetic part of the Weyl tensor are pointed out.
Moreover a (P QS)nscalar field space-time is considered, and interesting proper-
ties are pointed out. Throughout the paper all manifolds under consideration are
assumed to be smooth connected Hausdorff manifolds; their metrics are assumed
to be positive definite in Sections 1-6. In Section 7 we consider a smooth four di-
mensional Hausdorff space-time manifold endowed with a Lorentz metric [24] (i.e.
a metric of signature +2).
2. Elementary properties of a (P QS)nmanifold
In this section elementary properties of a (P QS)nare shown. We will show
that the covector Ais closed. Moreover interesting properties arise in the case of
concircular Ak. Let Mbe a non flat n(n≥4) dimensional (P QS )nRiemannian
manifold with metric gij and Riemannian connection ∇. We can state the following
simple theorem:
Theorem 2.1. The Qtensor of a pseudo Qsymmetric manifolds satisfies the
second Bianchi indentity:
(2.1) ∇iQjklm +∇jQkilm +∇kQijlm = 0.
Proof. Write three equations like (1.10) with a cyclic indices permutation and sum
up, taking into consideration the first Bianchi identity for the Qtensor. ¤
Now we point out some useful formulas concerning (P QS)nmanifolds: transvec-
tion equation (1.10) with gmj gives immediately:
(2.2) ∇kZjl = 2AkZjl +AjZkl +AlAjk −AmQkj lm −AmQkljm .
Transvecting equation (2.2) with gjl and with gkj gives:
∇kZ= 2AkZ+ 4AlZkl,
∇lZkl =AkZ+ 2AlZkl .
(2.3)
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 5
Combining these equations and using the relation ∇lZjl =1
2∇jR+∇jϕcoming
from the contracted second Bianchi identity one obtains ∇jϕ= 0 and ∇kR=
2AkZ+ 4AlZkl.
The last equations are the generalization of the correspondent results obtained for
(P S)nand (P˜
CS )nmanifolds (see [15], [35] and [45]). In fact we can state the
following simple remarks:
Remark 2.1. If ϕ= 0, we recover a (PS )nmanifold one has ∇kR= 2AkR+
4AlRkl. Moreover if ∇kR= 0 it is AlRkl =−AkR
2. Thus Akresults a closed one
form and it is an eigenvector of the Ricci tensor with eigenvalue −R
2.
Remark 2.2. If Z= 0, then φ=−R
nand by a simple calculation ∇kR=∇kφ= 0
and then AlRkl =R
nAk. So we have obtained that the scalar curvature is a covariant
constant and that Akis an eigenvector of the Ricci tensor with eigenvalue R
n.
Now from equation (2.2) we have simply:
(2.4) ∇kZjl − ∇jZkl =AkZj l −AjZkl + 3AmQjklm ,
and, because of the relation ∇jZkl =∇jRk l and the second contracted Bianchi
identity it follows:
(2.5) ∇kRjl − ∇jRkl =∇mRj klm=AkZj l −AjZkl + 3AmQjklm.
Now we prove that the associated covector of a (P QS)nis a closed one form. We
follow the trick already used in [22] and [5]. First a covariant derivative ∇sis
applied on equation (1.10) and the commutator is evaluated obtaining:
(∇s∇i− ∇i∇s)Qjklm = 2(∇sAi− ∇iAs)Qj klm + (∇sAj−AjAs)Qiklm
+ (∇sAk−AkAs)Qjilm + (∇sAl−AlAs)Qj kim
+ (∇sAm−AmAs)Qjkli −(∇iAj−AiAj)Qsklm
−(∇iAk−AiAk)Qjslm −(∇iAl−AiAl)Qj ksm
−(∇iAm−AiAm)Qjkls .
(2.6)
In the sequel we define ωsi =∇sAi− ∇iAs. Permuting in pairs the indices (s, i),
(j, k) and (l, m) we obtain after a long calculation (see [22] and [5])
(∇s∇i− ∇i∇s)Qjklm + (∇j∇k− ∇k∇j)Qsilm + (∇l∇m− ∇m∇l)Qsij k
= 2£ωsiQj klm +ωjk Qsilm +ωlmQsijk ¤
+ωsj Qiklm +ωsk Qjilm +ωsl Qjkim +ωsm Qjkli +ωj iQsklm +ωj lQsikm
+ωjm Qsilk +ωkiQj slm +ωliQj ksm +ωmiQj kls +ωlk Qsijm +ωmk Qsilj .
(2.7)
Now we note that (∇s∇i−∇i∇s)Qjklm = (∇s∇i−∇i∇s)Rj klm and the well know
Walker Lemma [49] about the Riemann curvature tensor:
Lemma 2.1. (Walker) The Riemann curvature tensor satisfies the following iden-
tity:
(2.8) (∇s∇i−∇i∇s)Rjklm + (∇j∇k− ∇k∇j)Rsilm + (∇l∇m− ∇m∇l)Rsij k = 0.
6 C.A. MANTICA & Y.J. SUH
Thus we are able to write equation (2.7) in the algebraic form:
2£ωsiQj klm +ωjk Qsilm +ωlmQsij k ¤
+ωsj Qiklm +ωsk Qjilm +ωsl Qjkim +ωsm Qjkli +ωj iQsklm
+ωjl Qsikm +ωjm Qsilk +ωkiQj slm +ωliQj ksm +ωmiQj kls
+ωlkQsij m +ωmkQsilj = 0.
(2.9)
Now the following Lemma due to Ewert-Krzemieniewski [22] (see also [5]) is pointed
out:
Lemma 2.2. If ωlm =−ωlm and a generalized curvature tensor Ksatisfies the
algebraic equation:
2£ωsiKj klm +ωjk Ksilm +ωlmKsij k ¤
+ωsj Kiklm +ωsk Kjilm +ωsl Kjkim +ωsm Kjkli +ωj iKsikm
+ωjm Ksilk +ωkiKj slm +ωliKj kls +ωlk Ksijm +ωmk Ksilj = 0,
then either ωlm = 0 for all l, m or Khijk = 0 for all h, i, j, k .
From equation (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 Qsymmetric Riemannian
manifold is closed.
Now we point out some consequences due to equation (2.6). Let’s suppose that
the associated covector of a (P QS)nis of the concircular form ∇iAj=AiAj+γgij ,
being γan arbitrary scalar function. This condition will be of great importance in
the next section. Then we have:
(∇s∇i− ∇i∇s)Qjklm =γ£gj sQiklm +gsk Qjilm +gslQjk im +gsmQjkli
−gij Qsklm −gik Qjslm −gil Qjksm −gim Qjkls ¤.
(2.10)
Again on noting that (∇s∇i− ∇i∇s)Qjklm = (∇s∇i− ∇i∇s)Rj klm and that a
simple computation gives:
gjs Giklm +gskGjilm +gslGj kim +gsmGjkli −gij Gsklm
−gikGj slm −gilGj ksm −gimGj kls = 0,
we conclude that:
(∇s∇i− ∇i∇s)Rjklm =γ£gj sRiklm +gsk Rjilm +gslRjk im +gsmRjkli
gij Rsklm −gik Rjslm −gil Rjksm −gim Rjkls ¤.
(2.11)
Theorem 2.3. Let Mbe an n-dimensional (P QS)nRiemannian manifold. If the
associated covector of has the form ∇iAj=AiAj+γgij , then the manifold reduces
to a Pseudo Symmetric Riemannian manifold in the sense of Ryszard Deszcz [20].
If γ= 0 the manifold reduces to a semisymmetric one [19].
Now we focus on (P QS)nsatisfying some other curvature conditions [37] and
[38]. For example we consider a Pseudo Qrecurrent, i.e.:
∇iQjklm =kiQj klm
for some covector ki. From the definition of a (P QS)nwe easily infer that:
(2.12) 0 = (2Ai−ki)Qjklm +AjQiklm +AkQjilm +AlQjkim +AmQjkli .
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 7
Now a useful Lemma due to Roter [39] is stated:
Lemma 2.3. (Roter) If cj,pjand Khijk are numbers satisfying:
csKhijk +phKsijk +piKhsjk +pjKhisk +pkKhijs = 0,
Khijk =Kjkhi =−Khikj , Khijk +Khj ki +Khkij = 0,
then cj+ 2pj= 0 or Khijk = 0.
This give immediately ki= 4Aiand consequently equation (2.12) takes the form:
−2AiQjklm +AjQiklm +AkQjilm +AiQjk lm +AmQjkli = 0.
Now a cyclic permutation of indices i,j,kis performed on the previous equation,
and the resulting three relations are added to obtain:
(2.13) AiQkjlm +AjQiklm +AkQjilm = 0.
This is a curvature condition on the Qtensor that has a discrete relevance. We
state the result:
Theorem 2.4. If a Pseudo Qsymmetric Riemannian manifold is also Qrecurrent,
then the condition (2.13) is fulfilled.
Now let us suppose that (2.13) is valid: the companion equation
AiQlmjk +AlQmij k +AmQiljk = 0
is written, and summing up we finally have:
2AiQjklm +AjQkilm +AkQijlm +AjQjkmi +AmQjkil = 0.
Thus equation (1.10) reduces to ∇iQjklm = 4AiQjk lm. We have thus proved the
following result:
Theorem 2.5. Let Mbe an n-dimensional Pseudo Qsymmetric Riemannian man-
ifold. If the condition AiQkjlm +AjQiklm +AkQjilm = 0 is fulfilled, then the
manifold reduces to a Qrecurrent one.
3. (P QS)nmanifolds with concircular associated vector: Weyl
compatibily.
In this section we consider an n-dimensional Pseudo Qsymmetric 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]
and [30]. In this case an enlarged version of the celebrated Derdzinski-Shen theorem
about Codazzi tensors applies (see [18], [27] and [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] and [18]). We remember equation (2.5):
(3.1) ∇mRjkl m= 3AmQjkl m+AkZjl −AjZkl.
The covariant derivative ∇iis thus applied to the previous expression, and then
a cyclic permutation over indices i, j, k is performed. The resulting equations are
8 C.A. MANTICA & Y.J. SUH
added to obtain, using (2.4) the second Bianchi identity for the Qtensor:
∇i∇mRjkl m+∇j∇mRkil m+∇k∇mRijlm
= 3£(∇iAm−AiAm)Qjkl m+ (∇jAm−AjAm)Qkil m+ (∇kAm−AkAm)Qijlm¤
+Zjl (∇iAk− ∇kAi) + Zkl(∇jAi− ∇iAj) + Zil (∇kAj− ∇jAk).
(3.2)
Now if we consider a concircular associated covector Ai, i.e. satisfying the condition
∇iAm=AiAm+γgim , we find:
(3.3) ∇i∇mRjkl m+∇j∇mRkil m+∇k∇mRijlm= 0.
Now using Lovelock’s identity [26] and [28]:
∇i∇mRjkl m+∇j∇mRkilm+∇k∇mRijlm=−(RimRjk lm+RjmRkil m+RkmRij lm)
we obtain simply:
(3.4) RimRj klm+Rj mRkil m+Rkm Rijl m= 0.
Theorem 3.1. Let Mbe an n-dimensional (P QS)nRiemannian manifold with
∇iAm=AiAm+γgim . Then the relation (3.4) is fulfilled.
If the Ricci tensor satisfies equation (3.4) it is named R-compatible [27] and [30]. If
we insert in the previous relation the local form of the Weyl tensor we obtain:
(3.5) RimCj klm+Rj mCkil m+Rkm Cijl m= 0.
The Ricci tensor is thus Wely-compatible. More generally a (0,2) symmetric tensor
is said to be K-compatible [30] if an analogous algebraic relation with a generalized
curvature tensor is fulfilled:
(3.6) bimKj klm+bj mKkil m+bkm Kijlm= 0.
In recent works [27] and [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 Mbe an n-dimensional Riemannian manifold with a general-
ized curvature tensor Kand a K-compatible tensor b. If X,Yand Zare three
eigenvectors of the matrix bs
rat a point xof the manifold, with eigenvalues λ,µ
and ν, then
(3.7) XiYjZkKijkl = 0
provided that λand µare different from ν.
Derdzinski and Shen [18] proved this result in the case in which bis a Codazzi
tensor. We notice that if bkl is a Codazzi tensor, then it is Riemann-compatible
[30]. Here we provide an alternative proof of the previous Theorem based only on
equation (3.6). Write the equation for the eigenvectors of the tensor bkl:
(3.8) Xibim =λXm, Y jbjm =µYm, X kbkm =vZm, W lblm =ηWm
where λ,µ,ν,ηare the eigenvalues of the matrix bs
rand Xietc belong to the tangent
bundle of the manifold. Now equation bimKj klm+bj mKkil m+bkm Kijlm= 0 is
multiplied by XiYjZkand after an indices rearrangement (m→j,i→min the
second term, i→m,m→kin the third term) one easily gets:
(3.9) XmYjZk£λKkjml +µKmkj l +νKjmkl ¤= 0.
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 9
Now for example if µ=νfrom the previous equation and the first Bianchi identity
we obtain:
(3.10) XmYjZk(λ−µ)Kkjlm = 0.
Thus if λ6=µwe have:
(3.11) 1
2(YjZk−YjZj)XmKkjlm = 0.
In this way we have proven Theorem 3.1 in the case λ6=µ. Now equation (3.9) is
multiplied by Wlto obtain:
(3.12) XmYjZkWl£λKkjml +µKmkj l +νKjmkl ¤= 0.
At this point in equation (3.4) we make the index change i→j,j→k,k→l,
l→ito obtain easily:
(3.13) bjm Kklim+bkmKlji m+blmKj kl m= 0.
Again the previous equation is multiplied by YjZkWland after an indices re-
arrangement (m→k,j→min the second term, m→l,j→min the third term)
one easily gets:
(3.14) YmZkWl£µKlkmi +νKmlki +ηKkmli ¤= 0.
This las equation is multiplied by Xiand after an indices rearrangement (i→m,
m→j) the following relation holds:
(3.15) XmYjZkWl£ηKkj ml +νKmkj l +µKjmkl ¤= 0.
At this point in equation (3.13) we make the index change i→j,j→k,k→l,
l→ito obtain straightforwardly :
(3.16) bkmKlij m+blm Kikj m+bim Kklj m= 0.
The previous equation is then multiplied by XiZkWlto obtain after an indices
rearrangement (k→m,m→lin the second term, k→m,m→iin the third
term):
(3.17) XiZmWl£νKilmj +ηKmilj +λKlmij ¤= 0.
This last equation is multiplied by Yjand after an indices rearrangement (i→m,
m→k) the following relation holds:
(3.18) XmYjZkWl£νKkjml +ηKmkj l +λKmkjl ¤= 0.
Finally in equation (3.16) we make the index change i→j,j→k,k→l,l→ito
obtain easily:
(3.19) blmKij km+bim Kjlk m+bjm Klikm= 0.
As in the previous cases the last equation is multiplied by XiYjWland after an
indices rearrangement (m→i,l→min the second term, m→j,l→min the
third term) one gets:
(3.20) XiYjWm£ηKjimk +λKmjik +µKimjk ¤= 0.
This last equation is multiplied by Zkand after an indices rearrangement i→m,
m→lthe following relation holds:
(3.21) XmYjZkWl£µKkjml +λKmkj l +ηKj mkl¤= 0.
10 C.A. MANTICA & Y.J. SUH
Now equations (3.12), (3.21) and (3.18) are rewritten in matrix form to get:
(3.22)
λ µ ν
µ λ η
ν η λ
1 1 1
XmYjZkWlKkjml
XmYjZkWlKmkjl
XmYjZkWlKjmkl
= 0
In writing the last equation we have used the obvious (Bianchi) identity:
(3.23) XmYjZkWl[Kkjml +Kmkj l +Kjmkl ] = 0.
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 λ6=η,λ6=µ,λ6=νfrom the previous relations we get
λ=η=ν=µ. So we are restricted to suppose that λis equal to one of µ,η,ν.
As in Derdzinski and Shen’s paper [18] this implies that XmYjZkWlKkj ml = 0.
In fact the symmetries of the curvature tensor imply that XmYjZkWlKkjml 6= 0
only if X,Y,Z,Wbelong to at most two distinct eigenspaces. This completes the
proof in the case of distinct eigenvalues.
Thus on a pseudo Qsymmetric Riemannian manifold with concircular associated
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 tensor
hwe define a (0,2) symmetric tensor Kas in Bourguignon’s notation [4]:
(3.25)
◦
K(h)jm =hkl Kkjml .
In [30], the authors proved the following statement:
Theorem 3.3. If bis K-compatible, and bcommutes with a symmetric (0,2) tensor
h, the symmetric tensor
◦
K(h)jm =hkl Kkjml commutes with b.
We give here again a proof for completeness.
Proof. Multiply equation (3.6) with hkl: the last term bkm hklKij lmvanishes for
symmetry reasons. The remaining terms gives the commutation relation. ¤
Taking h=band Kthe Weyl tensor, from the previous results we have:
Corollary 3.1. Let Mbe an n-dimensional (P QS)nRiemannian manifold with
∇iAm=AiAm+γgim : then the symmetric tensor
◦
C(b)jm =bkl Ckjml commutes
with b.
This result will be of great importance in Section 7 where we deal with General
Relativity.
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 11
4. Conformally flat Pseudo Qsymmetric manifolds: local form of
the Ricci tensor
In this section a (P QS)n, (n > 3) Riemannian manifold with the property
∇mCjkl 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 spe-
cialize to a conformally flat (P QS)nmanifold. It is well know that the divergence
of the conformal tensor satisfies the relation:
(4.1) ∇mCjkl m=n−3
n−2£∇mRjkl m+1
2(n−1)(∇jRgkl − ∇kRgj l)¤.
So if we consider ∇mCjkl m= 0 one immediately obtains:
(4.2) ∇mRjkl m=∇kRjl − ∇jRkl =1
2(n−1)(∇kRgjl − ∇jRgk l).
From equation (2.5) we get easily:
(4.3) 3AmQjkl m+AkZjl −AjZkl =1
2(n−1)£(∇kR)gjl −(∇jR)gkl ¤.
Inserting in the previous relation ∇kR= 2AkZ+ 4AlZkl we have simply:
(4.4)
3AmQjkl m+AkZjl −AjZkl =1
(n−1)£(AkZ+2AmZkm)gj l −(AjZ+2AmZj m)gkl ¤.
On multiplying equation (4.4) by Alwith the symmetry condition AmAlQjklm = 0
we have
AkAlZjl =AjAlZkl .
Again we multiply the previous equation by Ajto obtain:
AkAjAlZjl =AjAjAlZkl .
This last can be rewritten as
(4.5) AlZkl =AjAlZjl
AjAjAk=tAk,
where t=AjAlZjl
AjAjis a scalar function. We have proved the following Theorem:
Theorem 4.1. Let Mbe an n(n > 3)-dimensional (P QS)nRiemannian manifold
with the property ∇mCjkl m= 0. Then the vector Alis an eigenvector of the Zkl
tensor with eigenvalue t.
Inserting result (4.5) in equation (2.3) one easily obtains:
(4.6) 2(2t+Z)Ak=∇kZ.
On multiplying the previous result by Ajand interchanging the indices jand kwe
get:
(4.7) Aj∇kZ=Ak∇jZ.
Again the operation of covariant derivation is applied on equation (4.6) and the
following relation is obtained:
∇j∇kZ= 2(∇jAk)Z+ 2Ak(∇jZ) + 4(∇jt)Ak+ 4t(∇jAk).
12 C.A. MANTICA & Y.J. SUH
Now a similar equation with indices kand jexchanged is written and then sub-
tracted from the previous result to obtain finally, taking account of (4.7):
(4.8) 0 = (∇jAk− ∇kAj)(2Z+ 4t) + 4£Ak(∇jt)−Aj(∇kt)¤.
From the previous section we know that in a (P QS)nthe associated covector is
closed; we have thus:
(4.9) Ak(∇jt)−Aj(∇kt) = 0.
The following statement resumes such results:
Theorem 4.2. Let Mbe an n(n > 3)-dimensional (P QS)nRiemannian manifold
with the property ∇mCjkl m= 0. Then the relations Ak(∇jt)−Aj(∇kt) = 0 and
Aj∇kZ=Ak∇jZhold.
Hereafter we specialize to a conformally flat (P QS)n: we will show that a confor-
mally flat (P QS)nis quasi Einstein [9], generalizing the result found in [15] and
[45]. This is done in few steps.
Step 1 : Transvecting equation (4.4) with Ajand using the result (4.5) one straight-
forwardly shows that the following equation holds:
(4.10) AjAjZkl= 3AmAjQjkl m+AkAlh(n−3)t−Z
n−1] + gkl
AjAj(2t+Z)
(n−1) .
Now from the definition of the Qtensor we have:
AjAmQjkl m=AjAmRjkl m−φ
(n−1)(AjAjgkl −AkAl),
and inserting in equation (4.10) it follows that:
(4.11)
AjAjZkl = 3AmAjRjkl m+AkAlh(n−3)t−Z+ 3φ
(n−1) i+gklAjAj
(n−1) £2t+Z−3φ¤.
Step 2: From the definition of the Conformal curvature tensor, in the conformally
flat case we find the form of the Riemann tensor:
Rjkl m=1
n−2(−δm
jRkl +δm
kRjl −Rm
jgkl +Rm
kgjl )+ R
(n−1)(n−2)(δm
jgkl −δm
kgjl ).
Multiplying this by AmAjand noting that form (4.5) necessarily it is AlRkl =
(t−φ)Akwe get after some calculations:
(4.12)
AmAjRjkl m=AkAl
(n−2)h2(t−ϕ)−R
(n−1)i+AjAjgkl
(n−2) hϕ−t+R
(n−1)i−AjAj
(n−2)Rkl .
Step 3: Inserting (4.12) in (4.11) with the definition of the Ztensor after long but
straightforward calculation we find:
(4.13) Rkl =AkAl
AjAjhnt −Z
n−1i+gkl
(n−1)[ϕ+R−t].
In such a case a Riemannian manifold is said to be quasi Einstein (see [9]). We
have obtained a generalization of the results given for example in [15] and [45]. The
previous expression may be written in the more compact form:
(4.14) Rkl =αgkl +βTkTl,
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 13
where α=ϕ+R−t
n−1, β =nt−Z
n−1are the associated scalars and Tk=Ak
√AjAjis naturally
a unit covector. We have proved the following:
Theorem 4.3. An n(n > 3)-dimensional (P QS )nRiemannian manifold with the
property Cjkl m= 0 is quasi Einstein.
At this point is worth to note the following geometric remark:
Remark 4.1. If ϕ= 0 being Z=Rit follows the same result achieved in [15],
namely equation (4.13) takes the form:
(4.15) Rkl =AkAl
AjAjhnt −R
n−1i+gklhR−t
n−1i.
If ϕ=−R
n, then Z= 0 and from Remark 2.2 we easily obtain that ∇kϕ=∇kR= 0
and so from 2(2t+Z)Ak=∇kZbeing Ak6= 0 we have t= 0 and thus finally
Rkl =gkl R
n, that is, the manifold is Einstein.
From Theorems 4.2 and 4.3 a (P QS)nRiemannian manifold with the property
Cjkl m= 0 is quasi Einstein, [9] that is the Ricci tensor satisfies Rkl =αgkl +βTkTl.
We have also the relations Aj(∇kt) = Ak(∇jt), Aj(∇kZ) = Ak(∇jZ); taken in
conjunction we get:
(4.16) Aj(∇k
nt −Z
n−1) = Ak(∇j
nt −Z
n−1).
Thus multiplying the previous result by 1
√AjAjand considering Theorems 4.2 and
4.3 we can state the following:
Corollary 4.1. Let Mbe an n(n > 3)-dimensional (P QS)nRiemannian manifold
with the property Cjkl m= 0. Then the manifold is quasi Einstein and the following
is true:
(4.17) Tj(∇kβ) = Tk(∇jβ).
The notion of manifold of quasi constant curvature was introduced by Chen and
Yano [11] and generalizes a space of constant curvature. A Riemannian manifold
(n > 3) is said to be a manifold of quasi constant curvature if it is conformally flat
and the Riemann curvature tensor may be written in the form:
(4.18) Rjklm =p£gmj gkl −gmkgj l¤+q£gmj TkTl−gmk TjTl+gklTmTj−gjlTmTk¤,
where pand qare scalars (with q6= 0) and Tiis a unit covector. Now in a
conformally flat manifold the curvature tensor may be written as:
Rjklm =1
(n−2)£gmk Rjl −gjm Rkl −gkl Rjm +gjlRkm ¤
+R
(n−1)(n−2)£gmj gkl −gmk gjl ¤.
(4.19)
Now if we consider an n(n > 3)-dimensional (P QS)nRiemannian manifold with
the property ∇mCjkl m= 0 we get Rkl =αgkl +β TkTl, and inserting this in
equation (4.19) we obtain a manifold of quasi constant curvature with q=−β
(n−2) ,
p=R−2(n−1)α
(n−1)(n−2) . We may assert the following:
Theorem 4.4. Let Mbe an n(n > 3)-dimensional conformally flat (P QS)nRie-
mannian manifold: then the manifold is of quasi constant curvature.
14 C.A. MANTICA & Y.J. SUH
5. Conformally Flat Pseudo Q Symmetric manifolds: local form of
the metric tensor
In this section we study in deep conformally flat (P QS)n; in particular we point
out the existence of a proper concircular vector in such a manifold and give the
local form of the metric tensor. First we recall the following [31], [33] and [29]:
Theorem 5.1. Let Mbe an n(n > 3)-dimensional manifold whose Ricci tensor is
given by Rkl =αgkl +βTkTlwhere Tkis a unit vector: if the manifold is confor-
mally flat and the condition Tj(∇kβ) = Tk(∇jβ)is satisfied, then Tkis a proper
concircular vector.
Proof. If the manifold is conformally flat then the following naturally holds:
(5.1) ∇kRjl − ∇jRkl =1
2(n−1)(∇kRgjl − ∇jRgkl).
Equation (4.14) is then substituted in previous relation and the operations of co-
variant differentiation are performed to give straightforwardly:
(∇kβ)TjTl+β(∇kTj)Tl+βTj(∇kTl)−(∇jβ)TkTl−β(∇jTk)Tl
−βTk(∇jTl) = 1
2(n−1)(∇k˜
Rgjl − ∇j˜
Rgkl)
(5.2)
where ˜
R=R−2(n−1)α.
Recalling that Tkis a unit vector and so (∇kTl)Tl= 0, equation (5.2) is then
transvected with gjl to obtain:
(5.3) (∇kβ)−(∇lβ)TkTl−β(∇lTk)Tl−βTk(∇lTl) = 1
2(∇k˜
R).
Transvecting again equation (5.2) with TjTlgives:
(5.4) (∇kβ)−(∇lβ)TkTl−βT l(∇lTk) = 1
2(n−1)(∇k˜
R)−1
2(n−1)(∇l˜
R)TkTl.
Comparing the last two equations gives immediately:
(5.5) βTk(∇lTl) = 2−n
2(n−1)(∇k˜
R)−1
2(n−1)(∇l˜
R)TkTl.
The last result is then transvected with Tkso that the following holds:
(5.6) β(∇lTl) = −1
2(∇l˜
R)Tl.
Now equation (5.6) is substituted in (5.5) to give:
(5.7) (∇l˜
R)TkTl= (∇k˜
R).
If the last result is substituted in (5.4) one can easily obtain:
(5.8) (∇kβ)−(∇lβ)TkTl=βT l(∇lTk).
It is worth to notice that, by (5.7) one easily has (∇k˜
R)Tj= (∇j˜
R)Tk: thus
transvecting equation (5.2) with Tlgives immediately:
(5.9) β[∇kTj− ∇jTk] + Tj∇kβ−Tk∇jβ= 0.
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 15
Following the hypothesis of the theorem we immediately conclude that Tkis a closed
one form, that is:
(5.10) ∇kTj− ∇jTk= 0.
Now with this condition in mind we can transvect again equation (5.2) with Tj
recalling that (∇jTl)Tj= (∇lTj)Tj= 0 being Tjclosed to obtain the following
relation:
(5.11) ∇kTl=Tm(∇m˜
R)
2β(n−1) [TlTk−gkl].
So we conclude that Tkis a concircular vector. ¤
Now we can state the following remarks:
Remark 5.1. From (∇m˜
R)TmTk= (∇k˜
R)we easily obtain by a covariant deriv-
ative that the following is true:
(5.12) ∇j∇k˜
R= (∇jTk)(∇m˜
R)Tm+Tk∇j((∇m˜
R)Tm).
A similar relation is written with indices kand jexchanged and the resulting equa-
tions are then subtracted recalling that Tkis a closed one form to obtain finally:
(5.13) ∇j((∇m˜
R)Tm) = Tj(Tk∇k(Tm∇m˜
R)).
Remark 5.2. From (∇kβ)−(∇lβ)TkTl=βT l(∇lTk)recalling that Tkis a closed
one form one easily writes:
(5.14) (∇kβ) = (∇lβ)TkTl.
Now if the scalar function F=Tm(∇m˜
R
2β(n−1) is considered by the previous remarks one
can write ∇jf=µTj: thus the one form ωk=f Tkis closed and Tkis a proper
concircular vector.
Now taking account of Corollary 4.1 and Theorem 5.1 one can state the following:
Theorem 5.2. Let Mbe an n(n > 3)-dimensional (P QS)nconformally flat: then
the manifold admits a proper concircular vector.
Now it is well know [1] that if a conformally flat space admits a proper concircular
vector, then this space is subprojective in the sense of Kagan. In this way the
following holds:
Theorem 5.3. Let Mbe an n(n > 3)-dimensional conformally flat (P QS)n. If
the tensor Zkl is non singular, then the manifold is a subprojective space.
In [47] K. Yano proved that a necessary and sufficient condition for a Riemannian
manifold admits a concircular vector that there is a coordinate system in which the
first fundamental form may be written as:
(5.15) ds2= (dx1)2+eq(x1)g∗
αβ dxαdxβ,
where g∗
αβ =g∗
αβ (xγ) are function of xγonly (α, β, γ = 2,3,··· , n) and qis a
function of x1only. Since a conformally flat (P QS)nadmits a proper concircular
vector field, this space is the warped product 1 ×eqM∗where (M∗, g ∗) is a (n−1)-
dimensional Riemannian manifold. Gebarosky [23] proved that the warped product
1×eqM∗satisfies the condition (5.1) if and only if M∗is Einstein. Thus the
following theorem holds:
16 C.A. MANTICA & Y.J. SUH
Theorem 5.4. Let Mbe an n(>3)-dimensional conformally flat (P QS)n: then
the manifold is the warped product 1×eqM∗where M∗is Einstein.
6. Special conformally flat (PQS )nmanifolds
In this section we prove that a conformally flat (P QS)nis a special conformally
flat manifold. In [12] Chen and Yano introduced the notion of special conformally
flat manifold that is a generalization of a subprojective space: a conformally flat
Riemannian manifold is said to be special conformally flat if the (0,2) tensor defined
as:
(6.1) Hij =−1
(n−2)Rij +R
2(n−1)(n−2)gij
may be written in the form:
(6.2) Hij =−γ2
2gij +δ(∇iγ)(∇jγ)
where γ, δ are scalar functions and γ > 0. In the previous sections we have proved
that a conformally flat (P QS)nis quasi Einstein [9] and the Ricci tensor is written
in the form Rij =αgij +βTiTj: inserting this in (6.1) one easily gets:
(6.3) Hij =−γ2
2gij −βTiTj
(n−2)
where γ2=−˜
R
(n−1)(n−2) and ˜
R=R−2(n−1)αas previously defined. Now recalling
that ∇k˜
R=TkTl(∇l˜
R) = λTkwe have:
Ti=−2(n−1)(n−2)
λγ(∇iγ),
and consequently that:
(6.4) TiTj=−4(n−1)(n−2) ˜
R
λ2(∇iγ)(∇jγ).
We have thus proved the result in (6.2) with the choice δ=4β˜
R(n−1)
λ2. Thus a
conformally flat (P QS)nmanifold is a special conformally flat manifold. We state
the following:
Theorem 6.1. An n(n > 3)-dimensional conformally flat (P QS)nmanifold is a
special conformally flat manifold.
In [12] Chen and Yano proved that every simply connected special conformally flat
manifold can be isometrically immersed in an Euclidean space En+1 as a hypersur-
face. Thus from Theorem 6.1 we may assert that:
Theorem 6.2. An n(n > 3)-dimensional simply connected conformally flat (P QS )n
manifold can be isometrically immersed in an Euclidean manifold En+1 as a hy-
persurface.
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 17
7. (P QS)nspace-time manifolds
In this section we study general properties of (P QS)nspace-time manifolds.
We consider a smooth connected four dimensional Hausdorff manifold endowed
with a Lorentz metric (see [17], [24], [43] and [48]). From the results in [32] and
in [30] the Ztensor may be used to write the Einstein field equations of general
Relativity. In fact the equation Zkl =kTkl being ka suitable gravitational constant
(light speed is assumed to be normalized) with the condition ∇lZkl = 0 coming
form the stress energy tensor give ∇k(R
2+ϕ) = 0 that is ϕ=−R
2+ Λ. The
term Λ is thus the cosmological constant and Einstein’s equations take the form
Rkl −R
2gkl + Λgkl =kTkl.
If a (P QS)nspace-time is considered, i.e. Zkl =kTkl, then the condition ∇lZkl = 0
coming from the stress-energy tensor must be satisfied and so from (2.3) it follows
that AlZkl =−Ak
2Z, together with ∇k(R
2+ϕ) = 0, and moreover that ∇kZ= 0
that is ∇kR= 0. Combining this facts we can thus state the following:
Theorem 7.1. For a (P QS)nspace-time manifold, Akis a closed one form and
it is an eigenvector of the Ztensor with eigenvalue −Z
2. Moreover ∇kR= 0.
An example of this situation is a perfect fluid space-time with non null stress-energy
tensor given by the following equation:
(7.1) Tkl = (µ+p)ukul+pgkl ,
where µis the energy density, pis the isotropic pressure and uithe fluid flow velocity
with the condition uiui=−1. The fluid is called perfect because of the absence
of heat conduction terms and stress terms corresponding to viscosity. In addition
pand µare related by an equation of state governing the particular sort of perfect
fluid under consideration. In general this is an equation of the form p=p(µ, T )
where Tis the absolute temperature. However we shall only be concerned with
situations in which Tis effectively constant so that the equation of state reduces
to p=p(µ). In this case the perfect fluid is called isentropic [24].
If the condition AlZkl =−Ak
2Zis applied on Einstein’s equation Zkl =kTkl , one
can obtain the following relation:
(7.2) k(µ+p)ukAlul+kpAk=−Z
2Ak.
When this result is transvected with ukbeing uiui=−1 satisfied we can easily
obtain:
(7.3) (kµ −Z
2)Akuk= 0.
If the relation Akuk6= 0 is fulfilled, then we have kµ =Z
2. Now Zkl =kTkl
gives rise to Z=kT and from T= (3p−µ) we have simply Z=k(3p−µ). It
follows immediately that kp =Z
2and that p=µ: this is the equation of state of
the stiff-matter model [24]. Inserting kµ =Z
2in equation (7.2) one easily obtain
k(µ+p)ukAlul=−k(µ+p)Akand thus Ak=−ukAlul. Now from Z
2+φ= Λ
and Z=R+ 4φit follows that Z= 4Λ −R. Inserting these relations in Einstein’s
Equations it follows after a straightforward calculation that:
(7.4) Rkl = (4Λ −R)ukul+ Λgk l.
18 C.A. MANTICA & Y.J. SUH
We have obtained a quasi Einstein manifold and so the following theorem is true:
Theorem 7.2. A(P QS )nperfect fluid space-time manifold is quasi Einstein and
the one form Akis proportional to the fluid flow velocity.
In the sequel we study the properties of a (P QS)nspace-time with concircular as-
sociated covector, i.e. with ∇iAj=AiAj+γgij . It is well known that is such a case
the Ricci tensor is Weyl compatible, that is, RimCj klm+Rj mCkilm+Rkm Cijl m= 0.
From Einstein equations is follows that the stress-energy tensor Tkl is Weyl com-
patible:
(7.5) TimCj klm+Tj mCkil m+Tkm Cijl m= 0.
Taking account of Corollary 3.1 we state the following:
Theorem 7.3. Let Mbe an (P QS)4perfect fluid space-time manifold with ∇iAm=
AiAm+γgim . Then the symmetric tensor
◦
C(T)jm =Tkl Ckjml commutes with T.
The tensor
◦
C(T)jm =Tkl Ckjml has a deep physical significance. In General Rela-
tivity it is customary to define the Electric and Magnetic part of the Weyl tensor
(see [2] and [43]). Precisely given a normalized velocity vector uithe following (0,2)
tensors are defined:
Ekl =ujumCjklm
Hkl =1
4ujum(εαβjk Cαβ lm +εαβjlCαβkm)
(7.6)
where εijkl is the completely skew-symmetric Levi Civita symbol. The tensor Ekl
is named Electric part of the Weyl tensor, while the tensor Hkl is named Magnetic
part of the Weyl tensor; elementary properties are found to be:
gklEk l =gklHkl = 0,
ukEkl =ukHkl = 0.
The notions of generalized electric and generalized Magnetic part of the Weyl tensor
has been recently introduced in [30]. The authors defined a generalized Electric and
ageneralized Magnetic part of the Weyl tensor substituting the tensor ujumwith
an arbitrary stress energy tensor Tjm :
¯
Ekl =Tjm Cjklm
¯
Hkl =1
4Tjm (εαβjk Cαβ lm +εαβj lCαβ k m).
(7.7)
Obviously the tensor
◦
C(T)jm is nothing but the generalized electric part of the Weyl
tensor. Thus Theorem 7.3 simply asserts that for a (P QS)nperfect fluid space-
time manifold with ∇iAm=AiAm+γgim the generalized electric part of the Weyl
tensor commutes with Tkl [30]. It should be noted that if the stress-energy tensor
is of the form Tkl =αukuj+βgk l then ¯
Ekl =αEkl and ¯
Hkl =αHkl .
A further interesting property of the generalized magnetic part is pointed out in
[30]: we reproduce it here for completeness. Equation (7.5) may be written in the
form:
TimCj klm +Tj mCki lm +Tkm Cij lm = 0.
PSEUDO Q SYMMETRIC RIEMANNIAN MANIFOLDS 19
The previous equation is thus multiplied by εijkp to get:
εijkp TimCj klm +εij kpTj mCkilm +εijk pTkm Cij lm = 0.
Recalling the skew-symmetric properties of the Levi-Civita symbol we simply have:
εijkp TimCj klm =εk ijpTk mCij lm =εij kpTkm Cij lm
εijkp Tjm Ckilm =εj kipTj mCkilm =εijk pTkm Cij lm.
Thus we infer that 3εijkpTkm Cij lm = 0 and the generalized Magnetic part of
the Weyl tensor vanishes.
Theorem 7.4. [30] Let Mbe an any space-time manifold having a Weyl compatible
stress energy tensor Tkl. Thus the generalized Magnetic part of the Weyl tensor
vanishes.
We specially focus on stress-energy tensors of the form Tkl =αuiul+βgkl with
normalized covector uj. From the above discussion in this case we refer to the
standard electric and magnetic parts of the Weyl tensor Ekl and Hkl . Space-times
in which Hkl = 0 are named purely electric [43] space-times, while the condition
Ekl = 0 defines purely magnetic space-times [43]. Moreover it is well known that
purely electric space-times are of Petrov type I,Dor O(conformally flat) [43]. We
have thus:
Corollary 7.1. Let Mbe an any non conformally flat space-time manifold having
a Weyl compatible stress energy tensor Tkl =αukul+βgkl . Then Ekl = 0 and the
Petrov types are Ior D.
Finally we consider (P QS)nscalar field space-times. The Lagrangian (density) of
a real spin-0 field ψis defined as [24]:
(7.8) L=−1
2(∇kψ)(∇lψ)gkl −1
2
m2
~2ψ2.
In the previous expression mis the field mass and ~is the Planck constant divided
by 2π. The Euler-Lagrange equations are:
(7.9) ∇l∇lψ−m2
~2ψ= 0.
These are known as the Klein-Gordon equation. The stress-energy tensor of a scalar
field ψspace-time is defined as [24]:
(7.10) Tkl = (∇kψ)(∇lψ)−1
2gkl£(∇lψ)(∇lψ) + m2
~2ψ2¤.
This defines a scalar field minimally coupled with matter.
Again if the condition AlZkl =−Ak
2Zis applied on Einstein’s equation Zkl =kTkl
one can obtain the following relation:
(7.11) k[(∇kψ)Al(∇lψ) + βAk] = −Ak
2Z,
being β=−1
2£(∇lψ)(∇lψ)+ m2
~2ψ2¤. The previous equation is trasvected with ∇kψ
to obtain easily:
(7.12) ³k£(∇kψ)(∇kψ) + β¤+Z
2´Al(∇lψ) = 0.
20 C.A. MANTICA & Y.J. SUH
If we suppose that Al(∇lψ)6= 0, then we have k[(∇kψ)(∇kψ) + β] = −Z
2. Now
Zkl =kTkl gives rise to Z=kT and thus from T= (∇kψ)(∇kψ)+4βwe easily have
Z=k[(∇kψ)(∇kψ)+4β]. It follows immediately that kβ =Z
2and k(∇kψ)(∇kψ) =
−Zfrom which (∇kψ)(∇kψ) = −2β. Inserting back in equation (7.12) we obtain
Ak=∇kψhAl(∇lψ)
(∇lψ)(∇lψ)i. Moreover from the definition of the scalar βwe have simply
m= 0, i.e. the scalar field is mass-less and the Klein-Gordon equation reduces to
∇l∇lψ= 0. We are able to state the following:
Theorem 7.5. A(P QS )nscalar field space-time manifold is mass-less and the
one form Akis proportional to the field gradient.
Now from R
2+φ= Λ and Z=R+ 4φit follows that Z= 4Λ −R. We easily obtain
the interesting relation:
(7.13) (∇lψ)(∇lψ) = R−4Λ
k.
We thus state:
Theorem 7.6. On a (P QS )nscalar field space-time manifold, the field square
gradient is proportional to the scalar curvature.
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Carlo Alberto Mantica
Universit`
a degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy,
Physics Department,
I.I.S. Lagrange, Via L. Modignani 65, 20161, Milano, Italy
E-mail address:carloalberto.mantica@libero.it
Young Jin Suh
Kyungpook National University,
Department of Mathematics,
Taegu 702-701, Korea
E-mail address:yjsuh@knu.ac.kr
