Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 84-91
ON QUASI EINSTEIN MANIFOLDS ADMITTING A RICCI
QUARTERSYMMETRIC METRIC CONNECTION
(COMMUNICATED BY UDAY CHAND DE)
S. ALTAY DEMIRBAĞ, H. BAĞDATLI YILMAZ, S. A. UYSAL, F. ÖZEN ZENGIN
Abstract. The object of the present paper is to study quasi Einstein manifolds admitting a Ricci quartersymmetric metric connection satisfiying certain
curvature conditions.
1. Introduction
It is well known that a Riemannian manifold or a semi-Riemannian manifold
M n , n = dim M > 2, is said to be an Einstein manifold if its Ricci tensor Rkj of
type (0, 2) is of the form
R
Rkj = gkj
(1.1)
n
where R is the scalar curvature of the manifold. According to [1], (1.1) is called
the Einstein metric condition.
In 2000, the notion of a quasi Einstein manifold was introduced by Chaki and
Maity [2]. A non-flat Riemannian manifold M n is called a quasi Einstein manifold
if its Ricci tensor Rkj of type (0, 2) is not identically zero and satisfies the following:
Rkj = agkj + bpk pj
(1.2)
where a and b are scalar functions and pk is a non-zero covariant vector. It is obvious
that if b = 0 , then this manifold reduces to an Einstein manifod if a = R
n , namely,
Einstein manifolds form a subclass of quasi Einstein manifolds. Quasi Einstein
manifolds arose in the study of exact solutions of the Einstein field equations and in
the study of quasi-umbilical hypersurfaces of conformally flat spaces. This manifold
has many applications in physics, especially general relativity, e.g, a quasi Einstein
manifold can be taken as a model of perfect fluid spacetime in general relativity
[7]. This manifold has also been studied by De and Ghosh [6] and De and De [5]
and many others.
It is to be noted that Chen and Yano [3] introduced the notion of a manifold of
quasi-constant curvature. A conformally flat Riemannian manifold M n is called a
2000 Mathematics Subject Classification. 53B25.
Key words and phrases. Ricci quartersymmetric metric connection, Quasi Einstein manifold,
Sectional curvature.
c
2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted: September 08, 2011. Published: October 04, 2011.
84
ON QUASI EINSTEIN MANIFOLDS
85
manifold of quasi-constant curvature if its curvature tensor Rikjm of type (0, 4) is
of the form
Rikjm = β [gkj gim − gij gkm ] + γ [gkj ηi ηm − gij ηk ηm + gim ηk ηj − gkm ηi ηj ] (1.3)
where β, γ are scalar functions, η is a 1-form. In particularly, if γ = 0, then it
reduces to the manifold of constant curvature.
In 1970, Yano [12] studied some curvature and derivational conditions for semisymmetric connections in Riemannian manifolds. In 1975, Golab [8] introduced
the idea of a quarter-symmetric connection in differentiable manifolds. In 1980,
Misra and Pandey [10] introduced a quarter-symmetric F-connection in Riemannian manifolds with F-structures; especially, they considered the case of Kaehlerian structure. In 1982, Yano and Imai [13] studied some curvature conditions for
quarter-symmetric metric connections in Riemannian, Hermitian and Kaehlerian
manifolds.
A linear connection ∇ in a Riemannian manifold M n is called a semi-symmetric
connection if its torsion tensor T satisfies the relation [12]
T (X, Y ) = π(Y )X − π(X)Y
where π is a 1- form. In [4], De and De studied some properties of this connection.
A linear connection ∇ in M n is called a quarter symmetric connection if its torsion
tensor T satisfies the relation
T (X, Y ) = π(Y )LX − π(X)LY
(1.4)
where π has the meaning already stated, L is a tensor field of type (1, 1) [13].
If there is a Riemannian metric g on M n such that
∇g = 0
then the connection ∇ is called a metric connection [12].
In (1.4), if a tensor field L is a Ricci tensor of a Riemannian manifold M n , then
the connection ∇ of the manifold M n is called a Ricci quartersymmetric connection
[10]. In local coordinates, the torsion tensor T of the connection ∇ is the following
form
l
Tik
= Ril πk − Rkl πi
(1.5)
where π is a 1-form associated with the torsion tensor of the connection ∇ and
Ril = Rit g tl . From (1.5), it is clear that ∇ is a metric connection. Some properties
of this connection has been studied by Kamilya
and De [9].
l
l
Thus, the relation between Γik and
is given by [13], [10],
ik
l
l
Γik =
+ Ril πk − Rik π l
(1.6)
ik
l
where Γlik and
are the connection coefficients of the Ricci quartersymmetric
ik
metric connection ∇ and the Levi-Civita connection ∇, respectively and π l = πt g tl ,
π l being contravariant components of πt .
The paper is organized as follows: Some theorems about a Riemannian manifold
with a Ricci quartersymmetric metric connection and whose Ricci tensor Rkj is
Codazzi type are proved. In section 4, it is obtained necessary condition that the
Ricci quartersymmetric metric connection reduces to the semi symmetric metric
86
S. A. DEMIRBAĞ, H. B. YILMAZ, S. A. UYSAL, F. Z. ÖZEN
connection and under this condition, some properties of this manifold are also
studied.
2. Preliminaries
Let
l
Rikj
and
l
Rikj
be denoted the curvature tensor of the connection ∇ and ∇
l
respectively. The curvature tensor Rikj is defined by
∂Γlkj
∂Γlij
−
+ Γlit Γtkj − Γlkt Γtij
(2.1)
∂xi
∂xk
Then substituting (1.6) in (2.1), it is obtained the following equation for the curvature tensor of the connection ∇
l
Rikj =
Rikjm
= Rikjm − Rim πkj + Rkm πij − πim Rkj + πkm Rij
(2.2)
+ (∇i Rkm − ∇k Rim ) πj − (∇i Rkj − ∇k Rij ) πm
where
1
πkj = ∇k πj − Rkt π t πj + πt π t Rkj
2
(2.3)
and πjl = πjt g tl .
Contracting for i and m in (2.2), we get
Rkj
=
Rkj − Rπkj + Rki πij − πRkj + πki Rij
+ ∇i Rki − ∇k R πj − (∇i Rkj − ∇k Rij ) π i
(2.4)
where R = Rim g im and g im πm = π i , π = πim g im . Moreover, it follows from (1.2)
that
R = an + bP
(2.5)
i
where P = pi p .
3. Some Properties of a Riemannian Manifold admitting a Ricci
Quartersymmetric Metric Connection
In this section, we give some theorems of a Riemannian manifold admitting a
Ricci quartersymmetric metric connection. We assume that the Ricci tensor Rkj is
of Codazzi type to ∇. Then we have
Rikjm = Rikjm − Rim πkj + Rkm πij − πim Rkj + πkm Rij
(3.1)
By the aid of the equation (3.1), we can write
Rikjm − Rjmik
=
(πmi − πim ) Rjk + (πij − πji ) Rmk
(3.2)
+ (πjk − πkj ) Rim + (πkm − πmk ) Rij
If the tensor πkj is symmetric, then we find
Rikjm = Rjmik
(3.3)
In view of the equation (3.1), we get
Rikjm + Rkjim + Rjikm
=
(πij − πji ) Rkm
+ (πki − πik ) Rjm + (πjk − πkj ) Rim
Considering that the tensor πkj is symmetric in (3.4), then we can obtain
Rikjm + Rkjim + Rjikm = 0
(3.4)
ON QUASI EINSTEIN MANIFOLDS
87
Now, let us find the condition to be the symmetry of the Ricci tensor with respect
to ∇. Since Rkj is of Codazzi type to ∇, it follows from (2.4) that
Rkj = Rkj − Rπkj + Rki πij − πRkj + πki Rij
(3.5)
By using (3.5), it can be written
Rjk = Rjk − Rπjk + Rji πik − πRjk + πji Rik
(3.6)
Thus, from the equations (3.5) and (3.6), it follows that
Rkj − Rjk = R (πjk − πkj ) + Rjt (πkt − πtk ) + Rkt (πtj − πjt )
(3.7)
Finally, if the tensor πkj is symmetric, then the equation (3.7) gives
Rkj = Rjk
Thus, we can state the following theorem:
Theorem 3.1. Let a Riemannian manifold M n admit a Ricci quartersymmetric
metric connection whose Ricci tensor is Codazzi type. If the tensor πkj is symmetric, then the curvature tensor and the Ricci tensor of ∇ satisfy the following
properties:
(i) Rikjm = Rjmik ,
(ii) Rikjm + Rkjim + Rjikm = 0,
(iii) Rkj = Rjk .
After that, using the equation (3.1), we can write
∇l Rikjm
= ∇l Rikjm − ∇l (Rim πkj ) + ∇l (Rkm πij )
(3.8)
−∇l (πim Rkj ) + ∇l (πkm Rij )
Considering that the Ricci tensor Rkj of ∇ is Codazzi type and using (3.8), it
follows that
∇l Rikjm + ∇i Rkljm + ∇k Rlijm
=
Rkm [∇l πij − ∇i πlj ]
+Rlm [∇i πkj − ∇k πij ]
+Rim [∇k πlj − ∇l πkj ]
(3.9)
+Rlj [∇k πim − ∇i πkm ]
+Rij [∇l πkm − ∇k πlm ]
+Rkj [∇i πlm − ∇l πim ]
If the tensor πkj is of Codazzi type to ∇, from (3.9), we get
∇l Rikjm + ∇i Rkljm + ∇k Rlijm = 0
(3.10)
In view of this, we can state the following:
Theorem 3.2. Let a Riemannian manifold M n admit a Ricci quartersymmetric
metric connection whose Ricci tensor is Codazzi type. If the tensor πkj which is
of Codazzi type to ∇ is symmetric , then the curvature tensor Rikjm satisfies the
second Bianchi Identity to the connection ∇.
88
S. A. DEMIRBAĞ, H. B. YILMAZ, S. A. UYSAL, F. Z. ÖZEN
4. Sectional Curvature of a Quasi Einstein Manifold with a Special
Ricci Quartersymmetric Metric Connection
In this section, necessary condition that the Ricci quartersymmetric metric connection reduces to the semi-symmetric metric connection is obtained by proving
the following theorem. After that, under this condition, we study some properties
of this manifold.
By the use of (1.2) and (1.6), we get
l
l
Γik =
+ aδil πk − agik π l + bpi pl πk − pk π l
(4.1)
ik
If pi and πi are collinear, we have
πk = cpk
where c is a non-zero constant. From (4.1) and (4.2), we get
l
l
Γik =
+ acδil pk − acpl gik
ik
(4.2)
(4.3)
From (2.1) and (4.3), the curvature tensor of a Riemannian manifold admitting a
Ricci quartersymmetric metric connection is the following form
Rikjm = Rikjm − gim πkj + gkm πij − πim gkj + πkm gij
(4.4)
(ac)2
gkj P
2
and a is a non-zero constant. Therefore, we can state the following:
(4.5)
where
πkj = (ac)∇k pj − (ac)2 pk pj +
Theorem 4.1. Let M n be a quasi Einstein manifold admitting a Ricci quartersymmetric metric connection. If pi and πi are collinear, then the Ricci quartersymmetric metric connection reduces to a semi-symmetric metric connection.
From (4.5), it is clear that
πkj − πjk = (ac) (∇k pj − ∇j pk )
(4.6)
Using (1.2) and (2.3), we get
bP
ac2
gkj P
πkj = c∇k pj − c2 a +
pk pj +
2
2
Thus, it follows from (4.5) and (4.7) that
bP
ac(1 − a)
2
(1 − a)∇k pj − c a − a +
pk pj +
gkj P = 0
2
2
Hence, we can state the following:
(4.7)
(4.8)
Theorem 4.2. Let a Riemannian manifold M n admit a Ricci quartersymmetric
metric connection whose Ricci tensor satisfies the condition (1.2). If pk and πk are
collinear then pk satisfies the equation (4.8)
Now, we consider the conformal curvature tensor Cikjm defined by
1
Cikjm = Rikjm −
(Rkj gim − Rij gkm + gkj Rim − gij Rkm )
(n − 2)
R
+
(gim gkj − gij gkm )
(n − 1)(n − 2)
(4.9)
ON QUASI EINSTEIN MANIFOLDS
89
Suppose that the sectional curvature of a linear connection ∇ is independent
from the orientation chosen. Using Theorem 4.1 and remembering that Riemannian
manifold with a semi-symmetric connection is conformally flat and the curvature
tensor of ∇ is zero, if the sectional curvature of this manifold is independent from
the orientation chosen [14].
Thus, from the equation (4.9), we obtain
Rikjm
=
1
(Rkj gim − Rij gkm + gkj Rim − gij Rkm )
(n − 2)
R
−
(gim gkj − gij gkm )
(n − 1)(n − 2)
Substituting the equation (1.2) in (4.10), we get
2a
R
−
Rikjm =
(gim gkj − gij gkm )
(n − 2) (n − 1)(n − 2)
b
[pk pj gim − pi pj gkm + gkj pi pm − gij pk pm ]
+
(n − 2)
(4.10)
(4.11)
Hence, by virtue of (1.3), we can state the following:
Theorem 4.3. Suppose that a quasi Einstein manifold admits a Ricci quartersymmetric metric connection whose 1-form πk is a non-zero constant multiple of a
covariant vector pk and this manifold has a sectional curvature of the connection ∇
independent from the orientation chosen, then the manifold is of a quasi constant
curvature.
In a conformally flat Riemannian manifold, we know that
1
∇i Rkj − ∇j Rki =
[(∇i R) gkj − (∇j R) gki ]
2(n − 2)
(4.12)
Multiplying (4.4) by g im , we get
Rkj = Rkj − (n − 2)πkj − πgkj
(4.13)
Since the sectional curvature of this manifold is independent from the orientation
chosen, the curvature tensor is zero. By using (4.13), we can say that πkj is symmetric. From (4.6), pj is gradient.
If b is chosen as a constant, using (1.2) in the equation (4.12), we have
b [pj (∇i pk ) − pi (∇j pk )] =
1
[(∇i R) gkj − (∇j R) gki ]
2(n − 2)
(4.14)
Transvecting (4.14) with g kj , we find
(n − 1)
b pk (∇i pk ) − pi ∇k pk =
∇i R
2(n − 2)
(4.15)
where pk = g km pm . As we know, the following property is satisfied in a Riemannian
manifold.
∇j R
∇m Rjm =
(4.16)
2
Thus, from (1.2) and (4.16), we obtain
b∇m (pj pm ) =
∇j R
2
(4.17)
90
S. A. DEMIRBAĞ, H. B. YILMAZ, S. A. UYSAL, F. Z. ÖZEN
Hence, we get
∇i R
b pi ∇k pk =
− bpk ∇k pi
(4.18)
2
In this case, substituting (4.18) in the equation (4.15) and since pk is a gradient
vector, we have
(2n − 3)
4b pk ∇i pk =
∇i R
(4.19)
(n − 2)
For a component of tangent vector field of a geodesic line, we have [11]
pk ∇k pi = 0
(4.20)
Thus, from the equations (4.19)and (4.20), we have
∇i R = 0
(4.21)
We see that the scalar curvature R is a constant. Moreover, from (4.12) and (4.21)
we get
∇i Rkj − ∇j Rki = 0
(4.22)
Therefore, the Ricci tensor Rkj is of Codazzi type to ∇ .
This leads to the following theorem:
Theorem 4.4. Let a conformally flat quasi Einstein manifold, (b≡constant), admit
a Ricci quartersymmetric metric connection whose a covariant vector πk is a non
zero constant multiple of a covariant vector pk . If pk is a component of the tangent
vector field of the geodesic line , then the Ricci tensor Rkj is of Codazzi type to ∇
and the scalar curvature R is also a constant.
Acknowledgement : The authors are thankful to the referee for his valuable
comments and recommendations towards the improvement of the paper.
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Sezgin Altay Demirbağ, Department of Mathematics, Faculty of Sciences and Arts,,
Istanbul Technical University, Istanbul, Turkey
E-mail address: saltay@itu.edu.tr
Hülya Bağdatlı Yılmaz, Department of Mathematics, Faculty of Sciences and Arts,,
Marmara University, Istanbul, Turkey
E-mail address: hbagdatli@marmara.edu.tr
S. Aynur Uysal, Department of Mathematics, Faculty of Sciences and Arts,, Doğuş
University, Istanbul, Turkey
E-mail address: auysal@dogus.edu.tr
Füsun Özen Zengin, Department of Mathematics, Faculty of Sciences and Arts,, Istanbul Technical University, Istanbul, Turkey
E-mail address: fozen@itu.edu.tr