r in
ON PSEUDO SEMICONFORMALLY
SYMMETRIC MANIFOLDS
t
Bull. Korean Math. Soc. 0 (0), No. 0, pp. 1–0
http://dx.doi.org/10.4134/BKMS.b151007
pISSN: 1015-8634 / eISSN: 2234-3016
Jaeman Kim
fP
Abstract. In this paper, a type of Riemannian manifold (namely, pseudo
semiconformally symmetric manifold) is introduced. Also the several geometric properties of such a manifold is investigated. Finally the existence
of such a manifold is ensured by a proper example.
1. Introduction
Ah
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As a special subgroup of the conformal transformation group, Ishii [10] introduced the notion of conharmonic transformation under which a harmonic
function transforms into a harmonic function. In [10] the conharmonic curvai
ture tensor Hjkl
of type (1,3) on a Riemannian manifold (M n , g) of dimension
n ≥ 4 was defined as follows:
1
i
i
(1.1)
Hjkl
= Rjkl
−
(gjk rli − δki rjl + δli rjk − gjl rki ),
n−2
which remains invariant under conharmonic transformation, where R and r are
the Riemannian curvature and Ricci curvature tensors respectively.
In [22] Shaikh and Hui showed that the conharmonic curvature tensor satisfies the symmetries and skew symmetries properties of the Riemannian curvature tensor as well as cyclic ones. The conharmonic curvature tensor has
many applications in the theory of general relativity. In [1] Abdussattar investigated its physical significance in the theory of general relativity. This tensor
has also been studied by Siddiqui and Ahsan [23]; Ghosh, De and Taleshian [9]
and many others. In [11] the author introduces a type of curvature-like tensor
called semiconformal curvature tensor such that its (1,3) components remain
invariant under conharmonic transformation. More precisely, the semiconfori
mal curvature tensor Pjkl
of type (1,3) on a Riemannian manifold (M n , g) is
Received December 8, 2015; Revised April 29, 2016.
2010 Mathematics Subject Classification. 53A55, 53B20.
Key words and phrases. pseudo semiconformally symmetric manifold, harmonic semiconformal curvature tensor, scalar curvature, closed 1-forms, recurrent tensors, Einstein manifolds, Ricci tensor, pseudo symmetric, parallel vector.
This study is supported by 2016 Research Grant from Kangwon National University (No.
520160311).
c
2016
Korean Mathematical Society
1
2
JAEMAN KIM
(1.2)
i
i
i
Pjkl
= −(n − 2)bCjkl
+ [a + (n − 2)b]Hjkl
,
t
defined as follows:
r in
i
where a, b are constants not simultaneously zero and Cjkl
is the conformal
i
curvature tensor of type (1,3). Note that the conformal curvature tensor Cjkl
of type (1,3) remains invariant under conformal transformation and that such a
i
tensor is traceless. The conformal curvature tensor Cjkl
of type (1,3) is defined
as follows:
(1.3)
1
s
i
i
Cjkl
= Rjkl
−
(gjk rli −δki rjl +δli rjk −gjl rki )+
(δ i gjk −δki gjl ),
n−2
(n − 1)(n − 2) l
fP
where s is the scalar curvature.
1
, then the semiconformal curvature
In particular, if a = 1 and b = − n−2
tensor reduces to conformal curvature tensor whereas for a = 1 and b = 0,
such a tensor turns into conharmonic curvature tensor. The semiconformal
curvature tensor Pijkl of type (0,4) possesses the several symmetric and skew
symmetric properties as well as the cyclic ones. For instance, it is easy to see
that the semiconformal curvature tensor Pijkl of type (0.4) holds
Pijkl = −Pjikl = −Pijlk = Pklij
and
Pijkl + Pkijl + Pjkil = 0.
Ah
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(1.4)
Pijkl belongs to a class of tensors named generalized curvature tensors and
denoted with Kijkl . They were introduced by Kobayashi and Nomizu [12] and
satisfy properties (1.4).
In [3] Chaki introduced a type of Riemannian manifold (M n , g) whose curvature tensor Rijkl of type (0,4) satisfies the condition
Rijkl;m = 2Am Rijkl + Ai Rmjkl + Aj Rimkl + Ak Rijml + Al Rijkm ,
where A is a nonzero 1-form and the semicolon denotes the covariant differentiation with respect to the metric tensor g. Such a manifold is called a pseudo
symmetric manifold. This manifold has received a great deal of attention and is
studied in considerable detail by many authors [3, 4, 5, 6, 8, 18]. Motivated by
the above studies, in the present paper, we introduce a pseudo semiconformally
symmetric manifold. A Riemannian manifold (M n , g) of dimension n ≥ 4 (this
condition is assumed throughout the paper as for n ≤ 3, the conformal curvature tensor vanishes) is said to be pseudo semiconformally symmetric if its
pseudo semiconformal curvature tensor Pijkl of type (0,4) satisfies the relation
(1.5)
Pijkl;m = 2Am Pijkl + Ai Pmjkl + Aj Pimkl + Ak Pijml + Al Pijkm ,
where A is an associated 1-form which is not zero.
If a generalized curvature tensor Kijkl satisfies the condition
Kijkl;m = 2Am Kijkl + Ai Kmjkl + Aj Kimkl + Ak Kijml + Al Kijkm ,
ON PSEUDO SEMICONFORMALLY SYMMETRIC MANIFOLDS
3
r in
t
then the manifold is named pseudo-K symmetric and denoted with (P KS)n
[7]. Some properties of (P KS)n manifolds were studied in [13] and [15].
The purpose of this paper is to investigate the various properties of pseudo
semiconformally symmetric manifold on which some geometric conditions are
imposed.
2. Pseudo semiconformally symmetric manifolds
Let (M n , g) be a Riemannian manifold. The semiconformal curvature tensor
of (M n , g) is said to be harmonic if the divergence of the semiconformal
i
vanishes, i.e.,
curvature tensor Pjkl
i
Pjkl
h
Pjkl;h
= 0.
fP
(2.6)
Notice that in this paper, we adopt the Einstein convention (that is, when an
index variable appears once in an upper and once in a lower position in a term,
it implies summation of that term over all the values of the index). Now we
can state the following:
Theorem 2.1. Let (M n , g) be a Riemannian manifold with harmonic semiconformal curvature tensor. If the constant [a + (n − 2)b] in (1.2) is nonzero,
then the scalar curvature s of (M n , g) is constant.
Proof. By virtue of the second Bianchi identity, we have
m
Rjkl;m
= rjk;l − rjl;k
Ah
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(2.7)
and then
1
s;l
2
Taking account of (1.1), (1.2) and (1.3) we obtain from (2.7) and (2.8)
(2.8)
k
rl;k
=
m
m
m
Pjkl;m
= − (n − 2)bCjkl;m
+ [a + (n − 2)b]Hjkl;m
s;l
s;k
n−3
)[rjk;l − rjl;k −
gjk +
gjl ]
= − (n − 2)b(
n−2
2(n − 1)
2(n − 1)
n−3
1
(2.9)
+ [a + (n − 2)b][(
)(rjk;l − rjl;k ) −
(gjk s;l − gjl s;k )].
n−2
2(n − 2)
Using the condition (2.6) and multiplying (2.9) by g jk we get from (2.8)
s;l
n−3
s;l
k
)[s;l − rl;k
n+
]
0 = − (n − 2)b(
−
n−2
2(n − 1)
2(n − 1)
n−3
1
k
+ [a + (n − 2)b][(
)(s;l − rl;k
)−
(ns;l − s;l )]
n−2
2(n − 2)
−s;l
= [a + (n − 2)b](
),
n−2
which yields from [a + (n − 2)b] 6= 0 that the scalar curvature s is constant.
This completes the proof.
4
JAEMAN KIM
m
m
Kjkl;m
= cRjkl;m
+ d[s;l gjk − s;k gjl ],
r in
t
Note that Theorem 2.1 is a particular case of the following result concerning harmonic generalized curvature tensors, i.e., generalized curvature tensors
m
with the property Kjkl;m
= 0 (see [14, Prop. 4.6], [16, Theorem 2.2] and [17,
Theorem 3.7]): Let (M n , g) be an n-dimensional Riemannian manifold having
a generalized curvature tensor with the property
c
m
where c and d are constants. If Kjkl;m
= 0 and the condition d 6= 2(n−1)
is
satisfied, then the scalar curvature is a covariant constant s;j = 0.
Concerning pseudo semiconformally symmetric manifold, we have:
fP
Theorem 2.2. Let (M n , g) be a pseudo semiconformally symmetric manifold
with harmonic semiconformal curvature tensor. If the constant [a + (n − 2)b]
in (1.2) is nonzero, then the scalar curvature s of (M n , g) is zero.
Proof. By virtue of (1.5) and (2.6), we have
(2.10)
m
m
m
m
0 = 2Am Pjkl
+ Am Pmjkl + Aj Pmkl
+ Ak Pjml
+ Al Pjkm
.
Multiplying (2.10) by g jk , we have from (1.1), (1.2) and (1.3)
−s
−s
) + Al [a + (n − 2)b](
)
n−2
n−2
−s
−s
) + Al [a + (n − 2)b](
n)
− Al [a + (n − 2)b](
n−2
n−2
−s
−s
= 2Al [a + (n − 2)b](
) + Al [a + (n − 2)b](
n)
n−2
n−2
n+2
= − Al [a + (n − 2)b](
)s,
n−2
Ah
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0 = 2Al [a + (n − 2)b](
which yields from [a + (n − 2)b] 6= 0 and A 6= 0 that the scalar curvature s of
(M n , g) vanishes. This completes the proof.
Theorem 2.3. Let (M n , g) be a pseudo semiconformally symmetric manifold.
If both the scalar curvature s of (M n , g) and the constant [a + (n − 2)b] in (1.2)
are not zero, then the associated 1-form A in (1.5) is closed.
Proof. Multiplying (1.5) by g il and then multiplying the relation obtained thus
by g jk , we have
−s;m
[a + (n − 2)b](
)n
n−2
−s
−s
)n + Am [a + (n − 2)b](
)
= 2Am [a + (n − 2)b](
n−2
n−2
−s
−s
+ Am [a + (n − 2)b](
) + Am [a + (n − 2)b](
)
n−2
n−2
−s
(2.11)
+ Am [a + (n − 2)b](
).
n−2
ON PSEUDO SEMICONFORMALLY SYMMETRIC MANIFOLDS
5
2(n + 2)
Am s.
n
Taking the covariant derivative of (2.12), we get
s;m =
2(n + 2)
[Am;t s + Am s;t ]
n
2(n + 2)
2(n + 2)
[Am;t s +
Am At s]
=
n
n
s;mt =
(2.13)
r in
(2.12)
t
By virtue of [a + (n − 2)b] 6= 0, we have from (2.11)
because of (2.12).
Therefore it follows from (2.13) that
which yields from s 6= 0 that
2(n + 2)
s[Am;t − At;m ],
n
fP
0 = s;mt − s;tm =
Am;t − At;m = 0,
showing that the associated 1-form A is closed. This completes the proof.
n
A Riemannian manifold (M , g) is said to be recurrent if its curvature tensor
Rijkl of type (0.4) satisfies the condition
(2.14)
Rijkl;m = Bm Rijkl ,
Ah
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where the associated 1-form B is nonzero. Now we can state:
Theorem 2.4. Let (M n , g) be a pseudo semiconformally symmetric manifold
with [a+(n−2)b] 6= 0 and s 6= 0. If the manifold is recurrent, then the associated
n
B.
1-forms A in (1.5) and B in (2.14) satisfy the relation A = 2(n+2)
Proof. Taking the covariant derivative of (1.2), we have from (1.1) and (1.3)
Pijkl;m = − (n − 2)bCijkl;m + [a + (n − 2)b]Hijkl;m
s;m
(gil gjk − gik gjl )]
= − (n − 2)b[
(n − 1)(n − 2)
1
(2.15)
+ a[Rijkl;m −
(gjk ril;m − gik rjl;m + gil rjk;m − gjl rik;m )].
(n − 2)
Multiplying (2.14) by g il and then multiplying the relation obtained thus by
g jk , we have
(2.16)
rjk;m = Bm rjk
and then
(2.17)
s;m = Bm s.
Taking account of (2.14), (2.16) and (2.17) we have from (2.15)
Pijkl;m = −
Bm sb
(gil gjk − gik gjl )
(n − 1)
6
JAEMAN KIM
1
(gjk Bm ril − gik Bm rjl + gil Bm rjk − gjl Bm rik )].
(n − 2)
t
(2.18) +a[Bm Rijkl −
r in
Multiplying (2.18) by g il and then multiplying the relation obtained thus by
g jk , we get
n
Bm s[a + (n − 2)b].
(2.19)
g il g jk Pijkl;m = −
n−2
On the other hand, multiplying (1.5) by g il and then multiplying the relation
obtained thus by g jk , we get
(2.20)
g il g jk Pijkl;m = Am [a + (n − 2)b]sn
−2(n + 2)
.
n(n − 2)
Bm =
This completes the proof.
fP
Taking account of [a + (n − 2)b] 6= 0 and s 6= 0, we have from (2.19) and (2.20)
2(n + 2)
Am .
n
A Riemannian manifold (M n , g) is said to be Einstein if its Ricci tensor r is
proportional to the metric tensor g (that is, r = ns g). Note that in this case,
its scalar curvature s is constant under n ≥ 3 [2].
Now we have the following.
Ah
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Lemma 2.5. Let (M n , g) be a pseudo semiconformally symmetric manifold
with [a + (n − 2)b] 6= 0. If the manifold is Einstein, then its Ricci tensor
vanishes.
Proof. By virtue of (1.1), (1.2) and (1.3), we have
(2.21)
a
bs
Pijkl = aRijkl −
(gil rjk − gik rjl + ril gjk − rik gjl ) −
(gil gjk − gik gjl ).
n−2
n−1
From rij = ns gij it follows that s;i = 0 and thus rij;k = 0 and from (2.9) it is
m
inferred that Pjkl;m
= 0. From Theorem 2.2 it is s = 0 and consequently rij =
0. Moreover it follows from (2.21) that Pijkl = aRijkl , Pijkl;m = aRijkl;m . Theorem 2.6. Let (M n , g) be a pseudo semiconformally symmetric manifold
with [a + (n − 2)b] 6= 0 and a 6= 0. If the manifold is Einstein, then the manifold
is pseudo symmetric.
Proof. Taking account of Lemma (2.5) and (2.21), the relation (1.5) leads to
aRijkl;m = 2Am aRijkl + Ai aRmjkl + Aj aRimkl + Ak aRijml + Al aRijkm .
Since a 6= 0, the last relation reduces to
Rijkl;m = 2Am Rijkl + Ai Rmjkl + Aj Rimkl + Ak Rijml + Al Rijkm ,
showing that the manifold is pseudo symmetric.
ON PSEUDO SEMICONFORMALLY SYMMETRIC MANIFOLDS
7
r in
t
Theorem 2.7. Let (M n , g) be a pseudo semiconformally symmetric manifold
with [a + (n − 2)b] 6= 0. If the manifold admits a parallel vector field V , then
we have either V m Am = 0 or s = 0.
Proof. From the Ricci identity and a parallel vector field V , it follows
t
t
t
0 = V;jk
− V;kj
= V m Rmjk
.
(2.22)
Taking the covariant derivative of (2.22), we obtain
t
V m Rmjk;l
= 0.
Multiplying the last relation by gti we have
or equivalently
fP
V m Rimjk;l = 0
V m Rjkim;l = 0.
(2.23)
Taking account of the second Bianchi identity, we get from (2.23)
V m Rjkli;m + V m Rjkml;i = 0
or equivalently
V m Rjkli;m − V m Rjklm;i = 0,
which reduces to
V m Rjkli;m = 0
(2.24)
Ah
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do
because of (2.23).
Multiplying (2.24) by g ji and then multiplying the relation obtained thus
by g kl , we have
V m rkl;m = 0
(2.25)
and then
V m s;m = 0.
(2.26)
From (2.21), (2.24), (2.25) and (2.26), it follows that
V m Pijkl;m = 0
or equivalently
(2.27)
V m [2Am Pijkl + Ai Pmjkl + Aj Pimkl + Ak Pijml + Al Pijkm ] = 0.
Multiplying (2.27) by g il and then multiplying the relation obtained thus by
g jk , we get from (2.21)
−s
)(2n + 4)][a + (n − 2)b] = 0,
V m [Am (
n−2
which leads to either V m Am = 0 or s = 0. This completes the proof.
Note that results (2.22), (2.23), (2.24), (2.25) and (2.26) are essentially contained in Lemma 1.5 of [21].
We immediately have the following:
8
JAEMAN KIM
r in
t
Corollary 2.8. Let (M n , g) be a pseudo semiconformally symmetric manifold
with [a + (n − 2)b] 6= 0. If the covariant derivative of its associated 1-form A
vanishes, then the scalar curvature s of (M n , g) vanishes.
Proof. Multiplying g im to the given condition Ai;l = 0, we have
Am
;l = 0.
Now it follows from Theorem 2.7 that we have either Am Am = 0 or s = 0,
which leads to
s=0
because of A 6= 0. This completes the proof.
fP
Now we will provide a proper example of a pseudo semiconformally symmetric manifold.
n
Example. Let (R+
, g) be a Riemannian manifold given by
n
R+
= {(x1 , x2 , . . . , xn )|xi > 0, i = 1, 2, . . . , n}
and
g = f (dx1 )2 + δαβ dxα dxβ + 2dx1 dxn ,
where Greek indices α and β run over the range 2, 3, . . . , n − 1, and
f = (Eαβ + δαβ )xα xβ e(x
1 2
)
.
Ah
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Here δαβ is the Kronecker delta, and Eαβ is constant and satisfies the relations
Pn−1
Eαβ = 0 if α 6= β; Eαβ = constant (6= 0) if α = β; |Eαβ | < 1; α=2 Eαα = 0.
This kind of metric was appeared in [19, 20]. In the metric described as above,
the only nonvanishing components of Christoffel symbols, the curvature tensors
and the Ricci tensors are, according to [19, 20]
1
1
1
Γβ11 = − E αβ f,α , Γn11 = f,1 , Γn1α = f,α ,
2
2
2
1
1
f,αβ , r11 = δ αβ f,αβ ,
2
2
where the comma denotes the partial differentiation with respect to the coordinates. It is easy to see that the relations
(2.28)
R1αβ1 =
f,αβ = 2(Eαβ + δαβ )e(x
1 2
)
and
(2.29)
δ αβ f,αβ = 2(n − 2)e(x
1 2
)
hold. From (2.28) and (2.29), it follows that the only nonzero components for
the curvature tensor Rijkl and the Ricci tensor rjk are
R1αα1 =
1 2
1
f,αα = (Eαα + 1)e(x )
2
ON PSEUDO SEMICONFORMALLY SYMMETRIC MANIFOLDS
9
t
and
1 2
1
f,αβ δ αβ = (n − 2)e(x ) .
2
From the given metric g, we obtain gni = gin = 0 for i 6= 1, which yields
g 11 = 0. Therefore the scalar curvature s of (M n , g) vanishes because s =
g ij rij = g 11 r11 = 0. Hence by considering the results mentioned above and
(2.21), we find the only nonzero components for the semiconformal curvature
tensor Pijkl as
r11 =
r in
(2.30)
1 2
1
(gαα r11 )] = aEαα e(x ) .
n−2
In this case, it follows from (2.31) that the only nonzero components of the
covariant derivative of Pijkl are
(2.32)
P1αα1 = a[R1αα1 −
fP
(2.31)
1 2
P1αα1;1 = 2x1 aEαα e(x
)
= 2x1 P1αα1 .
1
Let us consider the associated 1-form A as Ai = x2 for i = 1 and 0 otherwise.
To verify the relation (1.5), it is sufficient to prove the relation
P1αα1;1 = 4A1 P1αα1 .
Taking account of the definition of the associated 1-form A and (2.32), it is
easy to see that the last relation holds. The other components of each term of
(1.5) vanishes identically and hence the relation (1.5) holds.
Ah
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Acknowledgements. The author is thankful to the referee for his valuable
suggestions towards the improvement of this paper.
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fP
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Ah
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Jaeman Kim
Department of Mathematics Education
Kangwon National University
Chunchon 200-701, Korea
E-mail address: [email protected]