Research Article Some Curvature Properties of -Manifolds LCS
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Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 380657, 6 pages
http://dx.doi.org/10.1155/2013/380657
Research Article
Some Curvature Properties of (LCS)π-Manifolds
Mehmet Atçeken
Department of Mathematics, Faculty of Arts and Science, Gaziosmanpasa University, 60100 Tokat, Turkey
Correspondence should be addressed to Mehmet Atçeken; mehmet.atceken382@gmail.com
Received 14 January 2013; Revised 4 March 2013; Accepted 6 March 2013
Academic Editor: Narcisa C. Apreutesei
Copyright © 2013 Mehmet AtcΜ§eken. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The object of the present paper is to study (LCS)π -manifolds with vanishing quasi-conformal curvature tensor. (LCS)π -manifolds
satisfying Ricci-symmetric condition are also characterized.
1. Introduction
Recently, in [1], Shaikh introduced and studied Lorentzian
concircular structure manifolds (briefly (LCS)-manifold)
which generalizes the notion of LP-Sasakian manifolds,
introduced by Matsumoto [2].
Generalizing the notion of LP-Sasakian manifold in 2003
[1], Shaikh introduced the notion of (LCS)π -manifolds along
with their existence and applications to the general theory
of relativity and cosmology. Also, Shaikh and his coauthors
studied various types of (LCS)π -manifolds by imposing the
curvature restrictions (see [3–6]). In [7, 8], the authors also
studied (LCS)2π+1 -manifolds.
The submanifold of an (LCS)π -manifold is studied by
Atceken and Hui [9, 10] and Shukla et al. [11]. In [12], Yano
and Sawaki introduced the quasi-conformal curvature tensor,
and later it was studied by many authors with curvature
restrictions on various structures [13].
After then, the same author studied weakly symmetric
(LCS)π -manifolds by several examples and obtain various
results in such manifolds. In [7], authors shown that a pseudo
projectively flat and pseudo projectively recurrent (LCS)π
manifolds are π-Einstein manifold.
On the other hand, in [5], authors proved the existence
of π-recurrent (LCS)3 manifold which is neither locally
symmetric nor locally π-symmetric by nontrivial examples.
Furthermore, they also give the necessary and sufficient
conditions for a (LCS)π -manifold to be locally π-recurrent.
In this study, we have investigated the quasi-conformal
flat (LCS)π -manifolds satisfying properties such as Riccisymmetric, locally symmetric, and π-Einstein. Finally, we
give an example for π-Einstein manifolds.
2. Preliminaries
An π-dimensional Lorentzian manifold π is a smooth connected paracompact Hausdorff manifold with a Lorentzian
metric tensor π, that is, π admits a smooth symmetric tensor
field π of the type (2, 0) such that, for each π ∈ π,
ππ : ππ (π) × ππ (π) σ³¨→ R
(1)
is a nondegenerate inner product of signature (−, +, +, . . . , +).
In such a manifold, a nonzero vector ππ ∈ ππ(π) is said to be
timelike (resp., nonspacelike, null, and spacelike) if it satisfies
the condition ππ (ππ , ππ ) < 0 (resp., ≤0, =0, >0). These cases
are called casual character of the vectors.
Definition 1. In a Lorentzian manifold (π, π), a vector field π
defined by
π (π, π) = π΄ (π)
(2)
for any π ∈ Γ(ππ) is said to be a concircular vector field if
(∇π π΄) π = πΌ {π (π, π) + π€ (π) π΄ (π)}
(3)
for π ∈ Γ(ππ), where πΌ is a nonzero scalar function, π΄ is a
1-form, π€ is also closed 1-form, and ∇ denotes the Levi-Civita
connection on π [7].
2
Abstract and Applied Analysis
Let π be a Lorentzian manifold admitting a unit timelike
concircular vector field π, called the characteristic vector field
of the manifold. Then we have
π (π, π) = −1.
(4)
Since π is a unit concircular unit vector field, there exists a
nonzero 1-form π such that
π (π, π) = π (π) .
(5)
Μ (π, π) π = π
(π, π) π
πΆ
The equation of the following form holds:
(∇π π) π = πΌ {π (π, π) + π (π) π (π)} ,
for all vector fields π, π, π on π, where π
and π denote the
Riemannian curvature tensor and Ricci curvature, respectively, π is also the Ricci operator given by π(π, π) =
π(ππ, π).
Now let (π, π) be an π-dimensional Riemannian manΜ the Weyl
ifold; then the concircular curvature tensor πΆ,
conformal curvature tensor πΆ, and the pseudo projective
Μ are, respectively, defined by
curvature tensor π
πΌ =ΜΈ 0
(6)
for all π, π ∈ Γ(ππ), where πΌ is a nonzero scalar function
satisfying
∇π πΌ = π (πΌ) = ππΌ (π) = ππ (π) ,
(7)
π being a certain scalar function given by π = −π(πΌ).
Let us put
∇ππ = πΌππ,
−
πΆ (π, π) π = π
(π, π) π −
(8)
(9)
π (π) = π (π, π) = −1,
(10)
π2 π = π + π (π) π,
(11)
π (π, π) = π (π) ,
(12)
πβπ=0
(13)
for all π ∈ Γ(ππ). Particularly, if we take πΌ = 1, then we can
obtain the πΏπ-Sasakian structure of Matsumoto [2].
Also, in an (LCS)π -manifold π, the following relations
are satisfied (see [3–6]):
π (π
(π, π) π) = (πΌ2 − π) [π (π, π) π (π) − π (π, π) π (π)] ,
(14)
π
(π, π) π = (πΌ2 − π) [π (π, π) π − π (π) π] ,
(15)
π
(π, π) π = (πΌ2 − π) [π (π) π − π (π) π] ,
(16)
(∇π π) π = πΌ [π (π, π) π + 2π (π) π (π) π + π (π) π] , (17)
π (π, π) = (π − 1) (πΌ2 − π) π (π) ,
π (ππ, ππ) = π (π, π) + (π − 1) (πΌ2 − π) π (π) π (π)
1
π−2
+π (π, π) ππ − π (π, π) ππ]
which tell us that π is a symmetric (1, 1)-tensor. Thus the
Lorentzian manifold π together with the unit timelike
concircular vector field π, its associated 1-form π, and (1, 1)type tensor field π is said to be a Lorentzian concircular
structure manifold.
A differentiable manifold π of dimension π is called
(LCS)-manifold if it admits a (1, 1)-type tensor field π, a
covariant vector field π, and a Lorentzian metric π which
satisfy
ππ = 0,
(20)
× [π (π, π) π − π (π, π) π
+
then from (6) and (8), we can derive
ππ = π + π (π) π,
π
[π (π, π) π − π (π, π) π] ,
π (π − 1)
(18)
(19)
π
[π (π, π) π − π (π, π) π] ,
(π − 1) (π − 2)
(21)
Μ (π, π) π = ππ
(π, π) π
π
+ π [π (π, π) π − π (π, π) π]
−
π
π
[
+ π] [π (π, π) π − π (π, π) π] ,
π π−1
(22)
where π and π are constants such that π, π =ΜΈ 0, and π is also
the scalar curvature of π [7].
For an π-dimensional (LCS)π -manifold the quasiΜ is given by
conformal curvature tensor C
Μ (π, π) π = ππ
(π, π) π
C
+ π [π (π, π) π − π (π, π) π
+π (π, π) ππ − π (π, π) ππ]
−
π
π
[
+ 2π] [π (π, π) π − π (π, π) π]
π π−1
(23)
for all π, π, π ∈ Γ(ππ) [14].
The notion of quasi-conformal curvature tensor was
defined by Yano and Swaki [12]. If π = 1 and π = −1/(π − 1),
then quasi-conformal curvature tensor reduces to conformal
curvature tensor.
3. Quasi-Conformally Flat (LCS)π -Manifolds
and Some of Their Properties
For an π-dimensional quasi-conformally flat (LCS)π -manifold, we know for π = π from (23),
Abstract and Applied Analysis
3
ππ
(π, π) π + π [π (π, π) π − π (π, π) π
In view of (28) and (29), we obtain
+π (π, π) ππ − π (π, π) ππ]
−
π
π
[
+ 2π] [π (π, π) π − π (π, π) π] = 0.
π π−1
ππ = (π − 1) (πΌ2 − π) π
(25)
π
π
(
+ 2π)
π π−1
(26)
Now let us consider an (LCS)π -manifold π which is
conformally flat. Thus we have from (21) that
π
(π, π) π =
+ π [−ππ − π (π) (π − 1) (πΌ2 − π) π] = 0.
−
[π (π, π) + π (π) π (π)] [π (πΌ2 − π) + π (π − 1)
π
π
(
+ 2π)]
π π−1
π
{π (π, π) π − π (π, π) π} ,
(π − 1) (π − 2)
(33)
for all vector fields π, π, π tangent to π. Setting π = π in (33)
and using (16), (18) we have
[
+ π [π (π, π) + π (π) π (π) (πΌ2 − π) (π − 1)] = 0,
(27)
that is,
1
{π (π, π) π − π (π, π) π
π−2
+π (π, π) ππ − π (π, π) ππ}
Taking the inner product on both sides of the last equation by
π, we obtain
× (πΌ2 − π) −
(32)
Theorem 3. A quasi-conformally flat (LCS)π -manifold M (π >
1) is a manifold of constant curvature (πΌ2 − π) provided that
π + π(π − 2) =ΜΈ 0.
Let π = π be in (25); then also by using (18) we obtain
+ π (π − 1) (πΌ − π) ]
π
(π, π) π = (πΌ2 − π) {π (π, π) π − π (π, π) π} ,
for all π, π, π ∈ Γ(ππ). If we consider Schur’s Theorem, we
can give the following the theorem.
+ π [π (π] ππ − π (π) ππ] = 0.
2
(31)
for any π ∈ Γ(ππ).
By using (29) and (31) in (23) for a quasi-conformally flat
(LCS)π -manifold π, we get
[π (π) π − π (π) π] [π (πΌ2 − π) + π (π − 1) (πΌ2 − π)
[−π − π (π) π] [π (πΌ2 − π) −
(30)
which is equivalent to
Here, taking into account of (16), we have
π
π
− (
+ 2π)]
π π−1
π (π, π) = (π − 1) (πΌ2 − π) π (π, π) ,
(24)
π
− (πΌ2 − π)] [π (π) π − π (π) π]
π−1
(34)
= [π (π) ππ − π (π) ππ] .
If we put π = π in (34) and also using (18), we obtain
π (π, π) = π (π, π)
×[
ππ = [
π
π
π
(
+ 2π) − (πΌ2 − π) ( + (π − 1))]
ππ π − 1
π
π
π
+ π (π) π (π) [ (
+ 2π)
ππ π − 1
π
− (πΌ2 − π) ( + 2 (π − 1))] .
π
(28)
Now we are in a proposition to state the following.
Theorem 2. If an π-dimensional (LCS)π -manifold π is quasiconformally flat, then π is an π-Einstein manifold.
Now, let {π1 , π2 , . . . , ππ−1 , π} be an orthonormal basis of the
tangent space at any point of the manifold. Then putting π =
π = ππ , π in (28), and taking summation for 1 ≤ π ≤ π − 1, we
have
π = π (π − 1) (πΌ2 − π)
if π + (π − 2) π =ΜΈ 0.
(29)
π
π
− (πΌ2 − π)] π + [
− π (πΌ2 − π)] π (π) π.
π−1
π−1
(35)
Corollary 4. A conformally flat (LCS)π -manifold is an πEinstein manifold.
Generalizing the notion of a manifold of constant curvature, Chen and Yano [15] introduced the notion of a manifold
of quasi-constant curvature which can be defined as follows:
Definition 5. A Riemannian manifold is said to be a manifold
of quasi-constant curvature if it is conformally flat and its
Μ of type (0, 4) is of the form
curvature tensor π
Μ (π, π, π, π)
π
= π {π (π, π) π (π, π) − π (π, π) π (π, π)}
+ π {π (π, π) π΄ (π) π΄ (π) − π (π, π) π΄ (π) π΄ (π)
+π (π, π) π΄ (π) π΄ (π) − π (π, π) π΄ (π) π΄ (π)} ,
(36)
4
Abstract and Applied Analysis
for all π, π, π, π ∈ Γ(ππ), where π, π are scalars of which
π =ΜΈ 0 and π΄ is a nonzero 1-form (for more details, we refer to
[13, 16]).
Thus we have the following theorem for (LCS)π -conformally flat manifolds.
Theorem 6. A conformally flat (LCS)π -manifold is a manifold
of quasi-constant curvature.
Μ (π, π, π, π)
π
π − 2 (π − 1) (πΌ2 − π)
(π − 1) (π − 2)
+ π (π, ∇ππ) + (π − 1) π (πΌ2 − π) π (π) π (π)
Here taking account of (17), we arrive at
(π − 1) (π − 2)
(∇ππ) (ππ, ππ) − (∇ππ) (π, π)
)
= −π (πΌ {π (π, π) π + 2π (π) π (W) π + π (π) π} , ππ)
− π (ππ, πΌ {π (π, π) π + 2π (π) π (π) π + π (π) π})
× {π (π, π) π (π) π (π) − π (π, π) π (π) π (π)
+π (π, π) π (π) π (π) − π (π, π) π (π) π (π)} .
(37)
− π (ππ, π∇ππ) + π (∇ππ, π)
+ π (π, ∇ππ) + (π − 1) π (πΌ2 − π) π (π) π (π)
− π (π∇ππ, ππ) + (π − 1) (πΌ2 − π) π (π)
This implies (36) for
π − 2 (π − 1) (πΌ2 − π)
(π − 1) (π − 2)
× {π (∇ππ) + πΌπ (π, ππ)} + (π − 1) (πΌ2 − π) π (π)
,
(38)
2
π=
− π (ππ, (∇ππ) π + π∇ππ) + π (∇ππ, π)
)
π − π (π − 1) (πΌ2 − π)
π=
= −π ((∇ππ) π + π∇ππ, ππ)
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)} .
(41)
× {π (π, π) π (π, π) − π (π, π) π (π, π)}
+(
(∇ππ) (ππ, ππ) − (∇ππ) (π, π)
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)}
Proof. From (33) and (35), we obtain
=(
Thus we have
π − π (π − 1) (πΌ − π)
(π − 1) (π − 2)
,
× {π (∇ππ) + πΌπ (π, ππ)}
= −πΌ {π (π, π) π (π, ππ) + 2π (π) π (π) π (π, ππ)
π΄ = π.
+π (π) π (π, ππ)}
This proves our assertion.
− πΌ {π (π, π) π (ππ, π) + 2π (π) π (π) π (ππ, π)
Next, differentiating the (19) covariantly with respect to
π, we get
2
∇ππ (ππ, ππ) = ∇ππ (π, π) + (π − 1) π (πΌ − π)
+ (π − 1) (πΌ2 − π) π [π (π) π (π)] ,
(39)
for any π, π ∈ Γ(ππ). Making use of the definition of ∇π and
(8), we have
(∇ππ) (ππ, ππ) + π (∇πππ, ππ) + π (ππ, ∇πππ)
= (∇ππ) (π, π) + π (∇ππ, π) + π (π, ∇ππ)
+ (π − 1) π (πΌ2 − π) π (π) π (π)
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)}
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)} .
(40)
+π (π) π (ππ, π)}
− π (ππ, π∇ππ) + π (∇ππ, π) + π (π, ∇ππ)
− π (π∇ππ, ππ) + (π − 1) π (πΌ2 − π) π (π) π (π)
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)}
+ (π − 1) (πΌ2 − π) π (π) {π (∇ππ) + πΌπ (π, ππ)} .
(42)
Again, by using (13), (18), and (19), we reach
(∇ππ) (ππ, ππ) − (∇ππ) (π, π)
= −πΌπ (π) π (π, ππ) − πΌπ (π) π (ππ, π)
− (π − 1) (πΌ2 − π) π (π) π (∇ππ)
− (π − 1) (πΌ2 − π) π (π) π (∇ππ)
Abstract and Applied Analysis
5
+ (π − 1) π (πΌ2 − π) π (π) π (π)
Corollary 8. If an π-dimensional (πΏπΆπ)π -manifold π is
locally symmetric, then πΌ2 − π is constant.
+ (π − 1) (πΌ2 − π)
Now, taking the covariant derivation of the both sides of
(18) with respect to π, we have
× {π (∇ππ) π (π) + πΌπ (π) π (π, ππ)
+π (∇ππ) π (π) + πΌπ (π) π (π, ππ)}
ππ (π, π) = (π − 1) π [(πΌ2 − π) π (π)] .
= −πΌπ (π) π (π, ππ) − πΌπ (π) π (ππ, π)
From the definition of the covariant derivation of Riccitensor, we have
2
+ πΌ (π − 1) (πΌ − π)
(∇π π) (π, π) = ∇ππ (π, π) − π (∇π π, π) − π (π, ∇ππ)
× {π (π) π (π, ππ) + π (π) π (π, ππ)}
= (π − 1) {π (πΌ2 − π) π (π) + (πΌ2 − π)
+ (π − 1) π (πΌ2 − π) π (π) π (π) .
× [π (∇ππ) + πΌπ (π, ππ)] }
(43)
− (π − 1) (πΌ2 − π) π (∇π π) − πΌπ (π, ππ)
Thus we have the following theorem.
Theorem 7. If an (LCS)π -manifold π is Ricci-symmetric; then
πΌ2 − π is constant.
Proof. If π > 1-dimensional (LCS)π -manifold π is Riccisymmetric, then from (43) we conclude that
πΌ (π − 1) (πΌ2 − π) {π (π) π (π, ππ) + π (π) π (π, ππ)}
+ (π − 1) π (πΌ2 − π) π (π) π (π)
+ πΌ (π − 1) (πΌ2 − π) π (π, ππ) − πΌπ (π, ππ) .
(50)
If an (πΏπΆπ)π -manifold π Ricci symmetric, then Theorem 7
and (43) imply that
(51)
This leads us to state the following.
(44)
It follows that
πΌ (π − 1) (πΌ2 − π) {π (π, ππ) π − π (π) ππ}
(45)
Theorem 9. If an (LCS)π -manifold π is Ricci symmetric, then
it is an Einstein manifold.
Corollary 10. If an (LCS)π -manifold π is locally symmetric,
then it is an Einstein manifold.
In this section, an example is used to demonstrate that the
method presented in this paper is effective. But this example
is a special case of Example 6.1 of [6].
− πΌπ (π) πππ − πΌπ (ππ, π) π = 0,
from which
Example 11. Now, we consider the 3-dimensional manifold
− πΌ (π − 1) (πΌ2 − π) π (π, ππ)
− (π − 1) π (πΌ2 − π) π (π) + π (ππ, π) = 0,
= (π − 1) π (πΌ2 − π) π (π)
π (π, ππ) = (π − 1) (πΌ2 − π) π (ππ, π) .
− πΌπ (π) π (π, ππ) − πΌπ (π) π (ππ, π) = 0.
+ (π − 1) π (πΌ2 − π) π (π) π
(49)
(46)
π = {(π₯, π¦, π§) ∈ R3 , π§ =ΜΈ 0} ,
(52)
where (π₯, π¦, π§) denote the standard coordinates in R3 . The
vector fields
which is equivalent to
− πΌ (π − 1) (πΌ2 − π) ππ − (π − 1) π (πΌ2 − π) π
+ πΌπππ = 0,
(47)
π1 = ππ§ (π₯
π
π
+ π¦ ),
ππ₯
ππ¦
π2 = ππ§
π
,
ππ¦
π
π3 =
ππ§
that is,
π (πΌ2 − π) = 0,
(48)
which proves our assertion.
Since ∇π
= 0 implies that ∇π = 0, we can give the following corollary.
(53)
are linearly independent of each point of π. Let π be the
Lorentzian metric tensor defined by
π (π1 , π1 ) = π (π2 , π2 ) = −π (π3 , π3 ) = 1,
π (ππ , ππ ) = 0,
π =ΜΈ π,
(54)
6
Abstract and Applied Analysis
for π, π = 1, 2, 3. Let π be the 1-form defined by π(π) = π(π, π3 )
for any π ∈ Γ(ππ). Let π be the (1,1)-tensor field defined by
ππ1 = π1 ,
ππ2 = π2 ,
ππ3 = 0.
(55)
Then using the linearity of π and π, we have π(π3 ) = −1,
Since π(π) = −π3 and π(π) = −π3 and π(π, π) = π1 π1 +
π2 π2 − π3 π3 , we have
π (π, π) = −π2π§ π (π, π) − (π2π§ + 2) π (π) π (π) .
(63)
This tell us that π is an π-Einstein manifold.
2
π π = π + π (π) π3 ,
π (ππ, ππ) = π (π, π) + π (π) π (π) ,
(56)
for all π, π ∈ Γ(ππ). Thus for π = π3 , (π, π, π, π) defines a
Lorentzian paracontact structure on π.
Now, let ∇ be the Levi-Civita connection with respect
to the Lorentzian metric π, and let π
be the Riemannian
curvature tensor of π. Then we have
[π1 , π2 ] = −ππ§ π2 ,
[π1 , π3 ] = −π1 ,
[π2 , π3 ] = −π2 .
(57)
Making use of the Koszul formulae for the Lorentzian metric
tensor π, we can easily calculate the covariant derivations as
follows:
∇π1 π1 = −π3 ,
∇π2 π1 = ππ§ π2 ,
∇π2 π3 = −π2 ,
∇π1 π3 = −π1 ,
∇π2 π2 = −ππ§ π1 − π3 ,
(58)
∇π1 π2 = ∇π3 π1 = ∇π3 π2 = ∇π3 π3 = 0.
From the previously mentioned, it can be easily seen that
(π, π, π, π) is an (LCS)3 -structure on π, that is, π is an
(LCS)3 -manifold with πΌ = −1 and π = 0. Using the
previous relations, we can easily calculate the components of
the Riemannian curvature tensor as follows:
π
(π1 , π2 ) π1 = (π2π§ − 1) π2 ,
π
(π1 , π2 ) π2 = (1 − π2π§ ) π1 ,
π
(π1 , π3 ) π1 = −π3 ,
π
(π1 , π3 ) π3 = −π1 ,
π
(π2 , π3 ) π2 = −π3 ,
π
(π2 , π3 ) π3 = −π2 ,
π
(π1 , π2 ) π3 = π
(π1 , π3 ) π2 = π
(π2 , π3 ) π1 = 0.
(59)
By using the properties of π
and definition of the Ricci tensor,
we obtain
π (π1 , π1 ) = π (π2 , π2 ) = −π2π§ ,
π (π3 , π3 ) = −2,
π (π1 , π2 ) = π (π1 , π3 ) = π (π2 , π3 ) = 0.
(60)
Thus the scalar curvature π of π is given by
3
π = ∑π (ππ , ππ ) π (ππ , ππ ) = 2 (1 − π2π§ ) .
(61)
π=1
On the other hand, for any π, π ∈ Γ(ππ), π and π can be
written as π = ∑3π=1 ππ ππ and π = ∑3π=1 ππ ππ , where ππ and ππ
are smooth functions on π. By direct calculations, we have
π (π, π) = − π2π§ (π1 π1 + π2 π2 ) − 2π3 π3
= −π2π§ (π1 π1 + π2 π2 − π3 π3 ) − π3 π3 (π2π§ + 2) .
(62)
Acknowledgment
The authors would like to thank the reviewers for the extremely carefully reading and for many important comments,
which improved the paper considerably.
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