On M-projectively flat LP-Sasakian manifolds
In the present paper, we study the nature of LP-Sasakian manifolds admitting the M-projective curvature tensor. It is examined whether this manifold satisfies the condition W(X, Y ).R = 0. Moreover, it is proved that, in the M-projectively flat LP-Sasakian manifolds, the conditions R(X, Y ).R = 0 an...
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irk-123456789-1657052020-02-17T01:25:58Z On M-projectively flat LP-Sasakian manifolds Zengin, F.Ö. Короткі повідомлення In the present paper, we study the nature of LP-Sasakian manifolds admitting the M-projective curvature tensor. It is examined whether this manifold satisfies the condition W(X, Y ).R = 0. Moreover, it is proved that, in the M-projectively flat LP-Sasakian manifolds, the conditions R(X, Y ).R = 0 and R(X, Y ).S = 0 are satisfied. In the last part of the paper, an M-projectively flat space-time is introduced, and some properties of this space are obtained. Вивчається природа многовидів Сасакяна, що допускають M-проективний тензор кривизни. Перевірено, чи задовольняє цей многовид умову W(X, Y ).R = 0. Більш того, доведено, що умови R(X, Y ).R = 0 та R(X, Y ).S = 0 виконуються для M-проективно плоских LP-многовидів Сасакяна. В останній частині роботи введено M-проективно плоский простір-час та встановлено деякі властивості цього простору. 2013 Article On M-projectively flat LP-Sasakian manifolds / F.Ö. Zengin // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1560–1566. — Бібліогр.: 17 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165705 517.91 en Український математичний журнал Інститут математики НАН України |
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In the present paper, we study the nature of LP-Sasakian manifolds admitting the M-projective curvature tensor. It is examined whether this manifold satisfies the condition W(X, Y ).R = 0. Moreover, it is proved that, in the M-projectively flat LP-Sasakian manifolds, the conditions R(X, Y ).R = 0 and R(X, Y ).S = 0 are satisfied. In the last part of the paper, an M-projectively flat space-time is introduced, and some properties of this space are obtained. |
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Zengin, F.Ö. |
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Zengin, F.Ö. |
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On M-projectively flat LP-Sasakian manifolds |
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On M-projectively flat LP-Sasakian manifolds |
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On M-projectively flat LP-Sasakian manifolds |
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On M-projectively flat LP-Sasakian manifolds |
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On M-projectively flat LP-Sasakian manifolds |
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on m-projectively flat lp-sasakian manifolds |
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Інститут математики НАН України |
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2013 |
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Короткі повідомлення |
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On M-projectively flat LP-Sasakian manifolds / F.Ö. Zengin // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1560–1566. — Бібліогр.: 17 назв. — англ. |
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UDC 517.91
F. Ö. Zengin (Istanbul Techn. Univ., Turkey)
ON M -PROJECTIVELY FLAT LP-SASAKIAN MANIFOLDS
ПРО M -ПРОЕКТИВНО ПЛОСКI LP-МНОГОВИДИ САСАКЯНА
The object of the present paper is to study the nature of LP-Sasakian manifolds admitting the M -projective curvature
tensor. It is examined whether this manifold satisfies the condition W (X,Y ).R = 0. Moreover, it is proved that, in the
M -projectively flat LP-Sasakian manifolds, the conditions R(X,Y ).R = 0 and R(X,Y ).S = 0 are satisfied. In the last
part of our paper, M -projectively flat space-time is introduced and some properties of this space are obtained.
Вивчається природа многовидiв Сасакяна, що допускають M -проективний тензор кривизни. Перевiрено, чи задо-
вольняє цей многовид умовуW (X,Y ).R = 0. Бiльш того, доведено, що умовиR(X,Y ).R = 0 таR(X,Y ).S = 0 ви-
конуються для M -проективно плоских LP-многовидiв Сасакяна. В останнiй частинi роботи введено M -проективно
плоский простiр-час та встановлено деякi властивостi цього простору.
1. Introduction. A Riemannian manifold (M, g) is called a Sasakian manifold if there exists a
Killing vector field ξ of unit length on M so that tensor field Φ of type (1,1), defined by Φ(X) =
= −∇Xξ, satisfies the condition (∇XΦ)(Y ) = g(X,Y )ξ − g(ξ, Y )X for any pair of vector fields
X and Y on M. This is a curvature condition which can be easily expressed in terms the Riemann
curvature tensor as R(X, ξ)Y = g(ξ, Y )X − g(X,Y )ξ. Equivalently, the Riemannian cone defined
by (C(M), ḡ,Ω) = (R+XM, dr2 + r2g, d(r2η)) is Kähler with the Kähler form Ω = d(r2η), where
η is the dual 1-form of ξ. The 4-tuple s = (ξ, η,Φ, g) is commonly called a Sasakian structure on M
and ξ is its characteristic or Reeb vector field.
Sasakian geometry is a special kind of contact metric geometry such that the structure transverse
to the Reeb vector field ξ is Kähler and invariant under the flow of ξ. On the analogy of Sasakian
manifolds, in 1989 Matsumoto [1, 2], introduced the notion of LP-Sasakian manifolds. Again the
same notion is introduced by Mihai and Rosca [3] and obtained many interesting results. LP-Sasakian
manifolds are also studied by De et al. [4], Shaikh et al. [5 – 8], Taleshian and Asghari [9], Venkatesha
and Bagewadi [10] and many others.
The M -projective curvature tensor of a Riemannian manifold M defined by Pokhariyal and
Mishra [11] is in the following form:
W (X,Y )Z = R(X,Y )Z − 1
2(n− 1)
(
S(Y,Z)X − S(X,Z)Y + g(Y, Z)QX − g(X,Z)QY
)
,
(1.1)
where R(X,Y )Z and S(X,Y ) are the curvature tensor and the Ricci tensor of M, respectively and
Q is the Ricci operator defined by S(X,Y ) = g(QX,Y ). Some properties of this tensor in Sasakian
and Kähler manifolds have been studied before [12, 13]. In 2010, Chaubey and Ojha [14] investigated
the M -projective curvature tensor of a Kenmotsu manifold.
The object of the present paper is to study LP-Sasakian manifolds admitting M -projective curva-
ture tensor. The paper is organized as follows. Section 2 is concerned with some preliminaries about
LP-Sasakian manifolds. Section 3 deals with LP-Sasakian manifolds withM -projective curvature ten-
sor. Section 4 is devoted to M -projectively flat LP-Sasakian manifolds. In Section 5, M -projectively
flat LP-Sasakian spacetimes are introduced.
c© F. Ö. ZENGİN, 2013
1560 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
ON M -PROJECTIVELY FLAT LP-SASAKIAN MANIFOLDS 1561
2. Preliminaries. An n-dimensional differentiable manifold M is called an LP-Sasakian man-
ifold [1, 2] if it admits a (1, 1) tensor field ϕ, a contravariant vector field ξ, a 1-form η and a
Lorentzian metric g which satisfy:
ϕ2 = I + η ⊗ ξ, (2.1)
η(ξ) = −1, (2.2)
g(ϕX,ϕY ) = g(X,Y ) + η(X)η(Y ), (2.3)
∇Xξ = ϕX, g(X, ξ) = η(X), (2.4)
(∇Xϕ)Y = g(X,Y )ξ + 2η(X)η(Y )ξ, (2.5)
where ∇ denotes the operator of the covariant differentiation with respect to the Lorentzian metric g.
It can be easily seen that in an LP-Sasakian manifold, the following relations hold:
ϕξ = 0, η(ϕX) = 0,
rankϕ = n− 1.
Again if we put
Ω(X,Y ) = g(X,ϕY )
for any vector fields X and Y, then Ω(X,Y ) is symmetric (0, 2) tensor field [1]. Also since the
1-form η is closed in an LP-Sasakian manifold, we have [1, 4]
(∇Xη)(Y ) = Ω(X,Y ), Ω(X, ξ) = 0
for any vector fields X and Y.
Also, in an LP-Sasakian manifold, the following conditions hold [2, 4]:
g(R(X,Y )Z, ξ) = η(R(X,Y )Z) = g(Y,Z)η(X)− g(X,Z)η(Y ), (2.6)
R(ξ,X)Y = g(X,Y )ξ − η(Y )X, (2.7)
R(X,Y )ξ = η(Y )X − η(X)Y, (2.8)
R(ξ,X)ξ = X + η(X)ξ, (2.9)
S(X, ξ) = (n− 1)η(X), (2.10)
S(ϕX,ϕY ) = S(X,Y ) + (n− 1)η(X)η(Y ) (2.11)
for any vector fields X, Y, Z where R(X,Y )Z is the curvature tensor and S(X,Y ) is the Ricci
tensor.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1562 F. Ö. ZENGİN
3. LP-Sasakian manifold satisfying W (X,Y ).S = 0. Let us consider an LP-Sasakian mani-
fold (M, g) satisfying the condition
W (X,Y ).S = 0. (3.1)
Now, we have
S(W (ξ,X)Y,Z) + S(Y,W (ξ,X)Z) = 0. (3.2)
From (1.1), (2.7) and (2.10), we get
W (ξ,X)Y =
1
2
g(X,Y )ξ − 1
2
η(Y )X − 1
2(n− 1)
S(X,Y )ξ +
1
2(n− 1)
η(Y )QX. (3.3)
By using (2.10) and (3.3), (3.2) takes the form
1
2
(n− 1)g(X,Y )η(Z) +
1
2
(n− 1)g(X,Z)η(Y )− S(X,Z)η(Y )−
−S(X,Y )η(Z) +
1
2(n− 1)
S(QX,Z)η(Y ) +
1
2(n− 1)
S(QX,Y )η(Z) = 0. (3.4)
Let λ be the eigenvalue of the endomorphism Q corresponding to an eigenvector X. Then
QX = λX. (3.5)
By using (3.5) in (3.4), we obtain
1
2
(n− 1)g(X,Y )η(Z) +
1
2
(n− 1)g(X,Z)η(Y )− S(X,Z)η(Y )−
−S(X,Y )η(Z) +
λ
2(n− 1)
S(X,Z)η(Y ) +
λ
2(n− 1)
S(X,Y )η(Z) = 0. (3.6)
Remembering that g(QX,Y ) = S(X,Y ) and using (3.6), we have
g(QX,Y ) = g(λX, Y ) = λg(X,Y ) = S(X,Y ). (3.7)
Thus, from (3.6) and (3.7), taking Z = ξ in (3.6) and using (2.2), it can be easily seen that(
λ2
2(n− 1)
− λ+
n− 1
2
)
(g(X,Y )− η(X)η(Y )) = 0. (3.8)
Finally, taking Y = ξ in (3.8) and using the properties (2.2) and (2.4)2, we obtain(
λ2
2(n− 1)
− λ+
n− 1
2
)
η(X) = 0. (3.9)
In this case, as η(X) 6= 0, we have from (3.9)
λ2 − 2(n− 1)λ+ (n− 1)2 = 0. (3.10)
From (3.10), it follows that the non-zero eigenvalues of the endomorphism Q are congruent such as
(n− 1). Thus we can state the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
ON M -PROJECTIVELY FLAT LP-SASAKIAN MANIFOLDS 1563
Theorem 3.1. If an n-dimensional (n ≥ 3) LP-Sasakian manifold admitting M -projective cur-
vature tensor and with non-zero Ricci tensor S satisfies
W (X,Y ).S = 0,
then the non-zero eigenvalues of the symmetric endomorphism Q of the tangent space corresponding
to S are congruent such as (n− 1).
4. M -projectively flat LP-sasakian manifolds. Let us consider that M be an M -projectively
flat LP-Sasakian manifold. Thus, we have W (X,Y )Z = 0 for all vector fields X, Y, Z. Then, we
get from (1.1)
R(X,Y )Z =
1
2(n− 1)
(
S(Y, Z)X − S(X,Z)Y + g(Y,Z)QX − g(X,Z)QY
)
. (4.1)
Taking Z = ξ in (4.1) and using the relations (2.4), (2.8) and (2.10), we find
η(Y )X − η(X)Y =
1
n− 1
[
η(Y )QX − η(X)QY
]
. (4.2)
Again taking Y = ξ in (4.2) and applying (2.2), (4.2) reduces to
QX = (n− 1)X. (4.3)
Hence in view of (2.7), (4.1) and (4.3), we get
S(X,Y )ξ = (n− 1)g(X,Y )ξ. (4.4)
Taking the inner product of both sides (4.4) with ξ and using (2.2), we have
S(X,Y ) = (n− 1)g(X,Y ). (4.5)
Next, we have the following theorem.
Theorem 4.1. Let M be an n-dimensional M -projectively flat LP-Sasakian manifold. Then M
is an Einstein manifold and the Ricci tensor of M is in the form S(X,Y ) = (n− 1)g(X,Y ).
In this case, by the use of (4.3) and (4.5) in (4.1), we obtain
R(X,Y )Z = g(Y, Z)X − g(X,Z)Y. (4.6)
According to Karcher [15], a Lorentzian manifold is called infinitesimally spatially isotropic relative
to a unit timelike vector field U if its Riemann curvature tensor R satisfies the relation
R(X,Y )Z = δ
[
g(Y, Z)X − g(X,Z)Y
]
for all X,Y, Z ∈ U⊥ and R(X,U)U = γX for X ∈ U⊥ where δ, γ are real valued functions on the
manifold. Hence, we can obtain the following theorem.
Theorem 4.2. An n-dimensional M -projectively flat LP-Sasakian manifold is infinitesimally
spatially isotropic relative to the unit timelike vector field ξ.
Theorem 4.3. Let M be an n-dimensional M -projectively flat LP-Sasakian manifold. Then M
is semisymmetric, i.e., the condition R(X,Y ).R = 0 holds.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1564 F. Ö. ZENGİN
Proof. Let M be an n-dimensional M -projectively flat LP-Sasakian manifold. Thus, we can
write
R(X,Y ).R = R(X,Y )R(Z,U)V −R(R(X,Y )Z,U)V−
−R(Z,R(X,Y )U)V −R(Z,U)R(X,Y )V (4.7)
for all vector fields X, Y, Z, U, V on M. So from (4.6), we get
R(R(X,Y )Z,U)V = g(U, V )g(Y, Z)X − g(Y,Z)g(X,V )U−
−g(X,Z)g(U, V )Y + g(X,Z)g(Y, V )U. (4.8)
Again, we obtain
R(Z,R(X,Y )U)V = g(U, Y )g(X,V )Z − g(U, Y )g(Z, V )X−
−g(U,X)g(Y, V )Z + g(X,U)g(Z, V )Y (4.9)
and finally
R(Z,U)R(X,Y )V = g(U,X)g(Y, V )Z − g(X,Z)g(Y, V )U−
−g(X,V )g(U, Y )Z + g(X,V )g(Z, Y )U. (4.10)
So from (4.7) – (4.10), one can easily get
R(X,Y ).R = 0.
Theorem4.3 is proof is proved.
Corollary 4.1. Let M be an n-dimensional M -projectively flat LP-Sasakian manifold. Then M
is Ricci semisymmetric, i.e., the condition R(X,Y ).S = 0 holds.
Proof. Let M be an n-dimensional M -projectively flat LP-Sasakian manifold. Since a semisym-
metric manifold is also Ricci semisymmetric, [16], from Theorem 4.2, the proof is clear.
5. M -projectively flat LP-Sasakian spacetimes. In this section, we consider that M is an
M -projectively flat LP-Sasakian spacetime (M4, g) satisfying the Einstein’s equations with a cosmo-
logical constant. Further let ξ be the unit time-like velocity vector of the fluid. It is known that the
Einstein’s equations with a cosmological constant can be written as [17]
S(X,Y )− r
2
g(X,Y ) + λg(X,Y ) = kT (X,Y ) (5.1)
for all vector fields X and Y. Here, S(X,Y ) and T (X,Y ) denote the Ricci tensor and the energy-
momentum tensor, respectively. In addition, λ is the cosmological constant and k is the non-zero
gravitational constant.
Hence by use of (4.5), (5.1) forms into
T (X,Y ) =
(
λ− 3
k
)
g(X,Y ). (5.2)
Thus, we have the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
ON M -PROJECTIVELY FLAT LP-SASAKIAN MANIFOLDS 1565
Theorem 5.1. Let M4 be an M -projectively flat LP-Sasakian spacetime satisfying the Einstein’s
equations with a cosmological constant. Then the energy momentum tensor of this space is found as
in (5.2).
In a perfect fluid spacetime, the energy momentum tensor is in the form
T (X,Y ) = (σ + p)u(X)u(Y ) + pg(X,Y ), (5.3)
where σ is the energy density, p is the isotropic pressure and u(X) is a non-zero 1-form such that
g(X,V ) = u(X) for all X, V being the velocity vector field of the flow, that is, g(V, V ) = −1.
Also, σ + p 6= 0.
With the help of (5.2) and (5.3), we obtain
(λ− 3− kp)g(X,Y ) = k(σ + p)u(X)u(Y ). (5.4)
Contraction of (5.4) over X and Y leads to
λ = 3− k
4
(σ − 3p). (5.5)
If we put X = Y = V in (5.4) then we find
λ = 3− kσ. (5.6)
Combining the equations (5.5) and (5.6), we get
σ + p = 0. (5.7)
Hence we have the following theorem.
Theorem 5.2. In an M -projectively flat LP-Sasakian spacetime M4 satisfying the Einstein’s field
equations with a cosmological term then the matter contents of M4 satisfy the vacuum-like equation
of state.
If we assume a dust in a perfect fluid, we have
σ = 3p. (5.8)
By putting (5.8) in (5.7), we get
p = 0.
Thus, we can state the following theorem.
Theorem 5.3. The M -projectively flat LP-Sasakian spacetime admitting a dust for a perfect
fluid is filled with radiation.
In a relativistic spacetime, the energy-momentum tensor is in the form
T (X,Y ) = µu(X)u(Y ). (5.9)
From (5.2), (5.9) takes the form
(λ− 3)g(X,Y ) = kµu(X)u(Y ). (5.10)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1566 F. Ö. ZENGİN
Contraction of (5.10) over X and Y leads to
λ = 3− 1
4
kµ. (5.11)
And, if we put X = Y = V in (5.10), we get
λ = 3− kµ. (5.12)
Thus, combining the equations (5.11) and (5.12), we finally get that µ = 0. From this relation and
(5.9), we find T (X,Y ) = 0. This means that the spacetime is devoid of the matter. In this case, we
can give the following theorem.
Theorem 5.4. A relativistic M -projectively flat LP-Sasakian manifold satisfying the Einstein’s
field equations with a cosmological term is vacuum.
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Press, 1983. – 103.
Received 22.08.11,
after revision — 19.08.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
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