On a class of Lorentzian para-Sasakian manifold

Abstract

We classify Lorentzian para-Sasakian manifolds which satisfy P ¢ C = 0, Z ¢ C = LCQ(g;C); P ¢ Z ¡ Z ¢ P = 0, and P ¢ Z + Z ¢ P = 0; where P is the v¡Weyl projective tensor, Z is the concircular tensor, and C is the Weyl conformal curvature tensor.

Full text

Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 4, 210–219
On a class of Lorentzian para-Sasakian manifolds
Cengizhan Murathana, Ahmet Yıldızb, Kadri Arslana, and Uday Chand Dec
aDepartment of Mathematics, Uluda˘
g University, 16059 Bursa, Turkey;
cengiz@uludag.edu.tr, arslan@uludag.edu.tr
bDepartment of Mathematics, Dumlupınar University, Kütahya, Turkey;
ahmetyildiz@dumlupinar.edu.tr
cDepartment of Mathematics, Kalyani University, Kalyani, India; uc_de@yahoo.com
Received 23 March 2006
Abstract. We classify Lorentzian para-Sasakian manifolds which satisfy P·C= 0,Z·C=
LCQ(g, C ), P ·Z−Z·P= 0, and P·Z+Z·P= 0,where Pis the v−Weyl projective
tensor, Zis the concircular tensor, and Cis the Weyl conformal curvature tensor.
Key words: contact metric manifold, Lorentzian para-Sasakian manifold, Sasakian manifold,
v−Weyl projective tensor, concircular tensor.
1. INTRODUCTION
Matsumoto [1] introduced the notion of Lorentzian para-Sasakian (LP-Sasakian
for short) manifold. Mihai and Rosca defined the same notion independently in [2].
This type of manifold is also discussed in [3,4].
Let Mbe an n-dimensional Riemannian manifold of class C∞.Av-projective
symmetry is a projectable vector field Xwith the property in which every
diffeomorphism ϕof its one-parametric group is a projective map between leaves.
In the theory of the projective transformations of connections the Weyl projective
tensor plays an important role.
Recently, the authors of [5] studied the contact metric manifold Mnsatisfying
the curvature conditions Z(ξ, X)·R= 0 and R(ξ, X )·Z= 0, where Zis the
concircular tensor of Mndefined by
Z(X, Y )W=R(X, Y )W−τ
n(n−1)R0(X, Y )W, (1)
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where
R0(X, Y )W=g(Y, W )X−g(X, W )Y,
Rand τare the Riemannian–Christoffel curvature tensor and the scalar curvature
of Mn,respectively. They observed immediately from the form of the concircular
curvature tensor that Riemannian manifolds with a vanishing concircular curvature
tensor are of constant curvature. Thus one can think of the concircular curvature
tensor as a measure of the failure of a Riemannian manifold to be of constant
curvature.
In the theory of the projective transformations of connections the Weyl
projective tensor plays an important role. The v−Weyl projective tensor Pin a
Riemannian manifold (Mn, g)is defined by [6]
P(X, Y )W=R(X, Y )W−1
n−1R1(X, Y )W, (2)
where
R1(X, Y )W=S(Y, W )X−S(X, W )Y,
with Sbeing the Ricci tensor of M.
In the present study we give a classification of the LP-Sasakian manifold Mn
satisfying the curvature conditions P·C= 0,Z·C=LCQ(g, C), P ·Z−Z·P= 0,
and P·Z+Z·P= 0,where Zis the concircular tensor defined by (1), Pis the
v−Weyl projective tensor, and Cis the Weyl conformal curvature tensor of Mn.
2. PRELIMINARIES
A differentiable manifold of dimension nis called an LP-Sasakian mani-
fold [1,2] if it admits a (1,1)-tensor field φ, a contravariant vector field ξ, a covariant
vector field η, and a Lorentzian metric gwhich satisfy
η(ξ) = −1,(3)
φ2=I+η⊗ξ, (4)
g(φX, φY ) = g(X, Y ) + η(X)η(Y),(5)
g(X, ξ) = η(X),∇Xξ=φX, (6)
Φ(X, Y ) = g(X, φY ) = g(φX, Y ) = Φ(Y, X ),(7)
(∇XΦ)(Y, W ) = g(Y, (∇XΦ)W) = (∇XΦ)(W, Y ),(8)
where ∇is the covariant differentiation with respect to g. The Lorentzian metric
gmakes a timelike unit vector field, that is, g(ξ, ξ) = −1.The manifold Mn
equipped with a Lorentzian almost paracontact structure (φ,ξ,η,g)is said to be a
Lorentzian almost paracontact manifold (see [1,3]).
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If we replace in (3) and (4) ξby −ξ, then the triple (φ,ξ,η)is an almost
paracontact structure on Mndefined by Sato [7]. The Lorentzian metric given by
(6) stands analogous to the almost paracontact Riemannian metric for any almost
paracontact manifold (see [7,8]).
A Lorentzian almost paracontact manifold Mnequipped with the structure
(φ,ξ,η,g)is called a Lorentzian paracontact manifold (see [1]) if
Φ(X, Y ) = 1
2((∇Xη)Y+ (∇Yη)X).
A Lorentzian almost paracontact manifold Mn, equipped with the structure
(φ,ξ,η,g), is called an LP-Sasakian manifold (see [1]) if
(∇Xφ)Y=g(φX, φY )ξ+η(Y)φ2X.
In an LP-Sasakian manifold the 1-form ηis closed. In [1] it is also proved that if
an n-dimensional Lorentzian manifold (Mn, g)admits a timelike unit vector field
ξsuch that the 1-form ηassociated to ξis closed and satisfies
(∇X∇Yη)W=g(X, Y )η(W) + g(X, W )η(Y)+2η(X)η(Y)η(W),
then Mnadmits an LP-Sasakian structure.
Further, on such an LP-Sasakian manifold Mnwith the structure (φ,ξ,η,g)the
following relations hold:
g(R(X, Y )W, ξ) = η(R(X, Y )W) = g(Y , W )η(X)−g(X, W )η(Y),(9)
R(ξ, X )Y=g(X, Y )ξ−η(Y)X, (10)
R(X, Y )ξ=η(Y)X−η(X)Y, (11)
R(ξ, X )ξ=X+η(X)ξ, (12)
S(X, ξ) = (n−1)η(X),(13)
S(φX, φY ) = S(X, Y )+(n−1)η(X)η(Y)(14)
for any vector fields X, Y (see [1,2]), where Sis the Ricci curvature and Qis the
Ricci operator given by S(X, Y ) = g(QX, Y ).
An LP-Sasakian manifold Mnis said to be η-Einstein if its Ricci tensor Sis of
the form
S(X, Y ) = ag(X, Y ) + bη(X)η(Y)(15)
for any vector fields X, Y , where a, b are functions on Mn(see [9,10 ]).
Next we define endomorphisms R(X, Y )and X∧AYof χ(M)by
R(X, Y )W=∇X∇YW− ∇Y∇XW− ∇[X,Y ]W, (16)
(X∧AY)W=A(Y, W )X−A(X, W )Y, (17)
respectively, where X, Y, W ∈χ(M)and Ais the symmetric (0,2)-tensor.
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For a (0, k)-tensor field T,k≥1,on (M, g)we define P·T, Z ·T, and
Q(g, T )by
(P(X, Y )·T)(X1, ..., Xk) = −T(P(X, Y )X1, X2, ..., Xk)
−T(X1, P (X, Y )X2, ..., Xk)
−... −T(X1, X2, ..., P (X, Y )Xk),(18)
(Z(X, Y )·T)(X1, ..., Xk) = −T(Z(X, Y )X1, X2, ..., Xk)
−T(X1, Z(X, Y )X2, ..., Xk)
−... −T(X1, X2, ..., Z(X, Y )Xk),(19)
Q(g, T )(X1, ..., Xk;X, Y ) = −T((XΛY)X1, X2, ..., Xk)
−T(X1,(XΛY)X2, ..., Xk)
−... −T(X1, X2, ..., (XΛY)Xk),(20)
respectively [11].
By definition the Weyl conformal curvature tensor Cis given by
C(X, Y )Z=R(X, Y )Z−1
n−2·g(Y, Z )QX −g(X, Z)QY
+S(Y, Z )X−S(X, Z)Y¸
+τ
(n−1)(n−2) [g(Y, Z )X−g(X, Z)Y],(21)
where Qdenotes the Ricci operator, i.e., S(X, Y ) = g(QX, Y )and τis scalar
curvature [9]. The Weyl conformal curvature tensor Cis defined by C(X, Y, Z, W )
=g(C(X, Y )Z, W ).If C= 0, n ≥4,then Mis called conformally flat.
3. MAIN RESULTS
In the present section we consider the LP-Sasakian manifold Mnsatisfying the
curvature conditions P·C= 0,Z·C=LCQ(g, C), P ·Z−Z·P= 0, and
P·Z+Z·P= 0.
First we give the following proposition.
Proposition 1. Let Mbe an n-dimensional (n > 3) LP-Sasakian manifold. If the
condition P·C= 0 holds on M,then
S2(X, U ) = hτ
n−1−(n−1)2−1iS(X, U )
+(n−1)[τ−(n−1)]g(X, U )
+n[τ−n(n−1)]η(X)η(U)
is satisfied on M,where S2(X, U) = S(QX, U ).
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Proof. Assume that Mis an n-dimensional, n > 3,LP-Sasakian manifold
satisfying the condition P·C= 0.From (18) we have
(P(V, X )·C)(Y, U )W=P(V, X )C(Y, U )W
−C(P(V, X )Y, U )W−C(Y, P (V , X)U)W
−C(Y, U )P(V, X )W= 0,(22)
where X, Y, U, V, W ∈χ(M).Taking V=ξin (22), we have
(P(ξ, X )·C)(Y, U )W=P(ξ, X)C(Y, U )W
−C(P(ξ, X )Y, U )W−C(Y, P (ξ, X)U)W
−C(Y, U )P(ξ, X )W= 0.(23)
Furthermore, substituting (2), (9), (13), (21) into (23) and multiplying with ξ, we
get
−g(X, C(Y, U )W)−nη(C(Y, U )W)η(X)−g(X, Y )η(C(ξ, U )W)
+nη(Y)η(C(X, U )W)−g(X, U )η(C(Y, ξ)W)
+nη(U)η(C(Y, X)W) + nη(W)η(C(Y , U)X)
+1
n−1{S(X, C(Y, U )W) + S(X, Y )η(C(ξ, U )W)
+S(X, U )η(C(Y, ξ)W)}= 0.(24)
Thus, replacing Wwith ξin (24), we have
−g(X, C(Y, U )ξ)−nη(C(Y, U )X) + 1
n−1S(X, C(Y, U )ξ) = 0.(25)
Again, taking Y=ξin (25) and after some calculations, since n > 3,we get
S2(X, U ) =hτ
n−1−(n−1)2−1iS(X, U )
+ (n−1)[τ−(n−1)]g(X, U )
+n[τ−n(n−1)]η(X)η(U).
Our theorem is thus proved.
Theorem 2. Let Mbe an n-dimensional (n > 3) LP-Sasakian manifold. If the
condition Z·C=LCQ(g, C )holds on M,then either Mis conformally flat or
LC=τ
n(n−1) −1.
Proof. Let Mnbe an LP-Sasakian manifold.So we have
(Z(V, X )·C)(Y, U )W=LCQ(g, C)(Y, U, W ;V, X ).
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Then from (19) and (20) we can write
Z(V, X )C(Y, U )W−C(Z(V, X)Y, U )W−C(Y, Z(V , X)U)W
−C(Y, U )Z(V, X )W
=LC[(V∧X)C(Y, U )W−C((V∧X)Y, U )W
−C(Y, (V∧X)U)W−C(Y , U)(V∧X)W].(26)
Therefore, replacing Vwith ξin (26), we have
Z(ξ, X )C(Y, U )W−C(Z(ξ, X)Y, U )W−C(Y, Z (ξ, X)U)W
−C(Y, U )Z(ξ, X )W
=LC[(ξ∧X)C(Y, U )W−C((ξ∧X)Y, U )W
−C(Y, (ξ∧X)U)W−C(Y , U)(ξ∧X)W].(27)
Using (20), (9) and taking the inner product of (27) with ξ, we get
h1−τ
n(n−1) −LCi[−g(X, C(Y, U )W)−η(C(Y, U )W)η(X)
−g(X, Y )η(C(ξ, U )W) + η(Y)η(C(X, U )W)
−g(X, U )η(C(Y, ξ)W) + η(U)η(C(Y , X)W) + η(W)η(C(Y, U )X)] = 0.
(28)
Putting X=Yin (28), we have
h1−τ
n(n−1) −LCi[−g(Y, C (Y, U )W) + η(W)η(C(Y, U )Y)
−g(Y, Y )η(C(ξ, U )W)−g(Y, U )η(C(Y, ξ)W)] = 0.(29)
A contraction of (29) with respect to Ygives us
h1−τ
n(n−1) −LCiη(C(ξ, U )W) = 0.(30)
If LC6= 1 −τ
n(n−1) ,then Eq. (30) is reduced to
η(C(ξ, U )W) = 0,(31)
which gives
S(U, W ) = µτ
(n−1) −1¶g(U, W ) + µτ
(n−1) −n¶η(U)η(W).(32)
Therefore, Mis a η-Einstein manifold. So, using (31) and (32), we have Eq. (28)
in the form
C(Y, U, W, X ) = 0,
which means that Mis conformally flat.
If LC6= 0 and η(C(ξ , U)W)6= 0, then 1−τ
n(n−1) −LC= 0, which gives
LC= 1 −τ
n(n−1) .This completes the proof of the theorem.
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Corollary 3. Every n-dimensional (n > 3) nonconformally flat LP-Sasakian
manifold satisfies Z·C= (1 −τ
n(n−1) )Q(g, C ).
Theorem 4. Let Mbe an n-dimensional (n > 3) LP-Sasakian manifold. M
satisfies the condition
P·Z−Z·P= 0
if and only if Mis a η-Einstein manifold.
Proof. Let Msatisfy the condition P·Z−Z·P= 0.Then we can write
P(V, X )·Z(Y, U )W−Z(V, X)·P(Y, U )W
=1
n−1[R(V, X )R1(Y, U )W−R1(V, X)R(Y, U )W]
+τ
n(n−1)2[R1(V, X )R0(Y, U )W−R0(V, X)R1(Y, U )W]
+τ
n(n−1)[R0(V, X)R(Y, U )W−R(V, X )R0(Y, U )W] = 0.(33)
Therefore, replacing Vwith ξin (33), we have
P(ξ, X )·Z(Y, U )W−Z(ξ, X)·P(Y, U )W
=1
n−1[R(ξ, X )R1(Y, U )W−R1(ξ, X)R(Y, U )W]
+τ
n(n−1)2[R1(ξ, X )R0(Y, U )W−R0(ξ, X)R1(Y, U )W]
+τ
n(n−1)[R0(ξ, X)R(Y, U )W−R(ξ, X )R0(Y, U )W] = 0.(34)
Using (10), (13), we get
1
n−1[S(U, W )g(X, Y )ξ−S(U, W )η(Y)X−g(X, U )S(Y, W )ξ
+S(Y, W )η(U)X−S(X, R(Y, U )W)ξ+ (n−1)g(U, W )η(Y)X
−(n−1)g(Y, W )η(U)X]
+τ
n(n−1)2[g(U, W )g(X, Y )ξ−g(U, W )η(Y)X−g(Y , W )g(X, U )ξ
+g(Y, W )η(U)X−S(U, W )g(X, Y )ξ+S(U, W )η(Y)X
+S(Y, W )g(X, U )ξ−S(Y, W )η(U)X]
+τ
n(n−1)[g(X, R(Y , U)W)ξ+g(Y , W )η(U)X−g(U, W )g(X, Y )ξ
+g(Y, W )g(X, U )ξ−g(Y, W )η(U)X] = 0.(35)
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Again, taking U=ξin (35), we get
1
n−1[(n−1)g(X, Y )η(W)ξ−S(Y, W )η(X)ξ−S(Y, W )X
+(n−1)g(Y, W )η(X)ξ−S(X, Y )η(W)ξ+ (n−1)g(Y , W )X]
+τ
n(n−1)2[g(X, Y )η(W)ξ−η(W)η(Y)X−g(Y, W )η(X)ξ−g(Y, W )X
−(n−1)g(X, Y )η(W)ξ+ (n−1)η(W)η(Y)X
−S(Y, W )η(X)ξ+S(Y, W )X] = 0.(36)
Taking the inner product of (36) with ξ, we find
1
n−1[S(X, Y )η(W)−(n−1)g(X, Y )η(W)]
+τ(n−2)
n(n−1)2[g(X, Y )η(W) + η(X)η(Y)η(W)] = 0.(37)
Again, taking W=ξand using (3) in (37), we get
S(X, Y ) = ·(n−1) −(n−2)
n(n−1)τ¸g(X, Y )
−·(n−2)
n(n−1)τ¸η(X)η(Y).(38)
So, Mis a η-Einstein manifold.
Conversely, if Mnis a η-Einstein manifold, then it is easy to show that
P·Z−Z·P= 0.Our theorem is thus proved.
Theorem 5. Let Mbe an n-dimensional (n > 3) LP-Sasakian manifold. M
satisfies the condition
P·Z+Z·P= 0
if and only if Mis an Einstein manifold.
Proof. Let Msatisfy the condition P·Z+Z·P= 0.Then, from (33) and (34),
we can write
2R(ξ, X )R(Y, U )W
−1
n−1[R(ξ, X )R1(Y, U )W+R1(ξ, X)R(Y, U )W]
+τ
n(n−1)2[R1(ξ, X )R0(Y, U )W+R0(ξ, X)R1(Y, U )W]
−τ
n(n−1)[R0(ξ, X)R(Y, U )W+R(ξ, X )R0(Y, U )W] = 0.(39)
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Using (6), (10), and (13) in (39), we have
2[g(X, R(Y, U )W)ξ−g(U, W )η(Y)X+g(Y, W )η(U)X]
−1
n−1[S(U, W )g(X, Y )ξ−S(U, W )η(Y)X−S(Y , W )g(X, U )ξ
+S(Y, W )η(U)X+S(X, R(Y, U )W)ξ−(n−1)g(U, W )η(Y)X
+(n−1)g(Y, W )η(U)X]
+τ
n(n−1)2[g(U, W )S(X, Y )ξ−(n−1)g(U, W )η(Y)X
−g(Y, W )S(X, U )ξ+ (n−1)g(Y, W )η(U)X+S(U, W )g(X, Y )ξ
−S(U, W )η(Y)X−S(Y , W )g(X, U )ξ+S(Y, W )η(U)X]
−τ
n(n−1)[g(X, R(Y , U)W)ξ−2g(U, W )η(Y)X+ 2g(Y, W )η(U)X
+g(U, W )g(X, Y )ξ−g(Y , W )g(X, U )ξ] = 0.(40)
Replacing Ywith ξand using (3) in (40), we have
2[g(X, R(ξ, U )W)ξ+g(U, W )X+η(W)η(U)X]
−1
n−1[S(U, W )η(X)ξ+S(U, W )X−(n−1)g(X, U )η(W)ξ
+2(n−1)η(W)η(U)X+S(X, R(ξ, U )W)ξ+ (n−1)g(U, W )X]
+τ
n(n−1)2[(n−1)g(U, W )η(X)ξ+ (n−1)g(U, W )X
−S(X, U )η(W)ξ+ (n−1)η(W)η(U)X+S(U, W )η(X)ξ
+S(U, W )X−(n−1)g(X, U )η(W)ξ+ (n−1)η(W)η(U)X]
−τ
n(n−1)[g(X, R(ξ , U)W)ξ+ 2g(U, W )X+ 2η(W)η(U)X
+g(U, W )η(X)ξ−g(X, U )η(W)ξ] = 0.(41)
Taking the inner product of (41) with ξand using (7), (10), we get
·2−2τ
n(n−1)¸[g(X, U )η(W) + η(X)η(U)η(W)]
+·τ
n(n−1)2−1
n−1¸[(n−1)g(X, U )η(W) + 2(n−1)η(X)η(U)η(W)
+S(X, U )η(W)] = 0.(42)
Again, taking W=ξand using (3) in (42), we get
·2τ
n(n−1) −2¸[g(X, U ) + η(X)η(U)]
−·τ
n(n−1)2−1
n−1¸[(n−1)g(X, U )
+2(n−1)η(X)η(U) + S(X, U )] = 0.(43)
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Thus, from (43), we have
S(X, U ) = (n−1)g(X, U).
So, Mnis an Einstein manifold.
Conversely, if Mnis an Einstein manifold, then it is easy to show that
P·Z+Z·P= 0.Our theorem is thus proved.
ACKNOWLEDGEMENT
This study was supported by the Dumlupınar University research foundation
(project No. 2004-9).
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Ühest Lorentzi para-Sasaki muutkondade klassist
Cengizhan Murathan, Ahmet Yıldız, Kadri Arslan ja Uday Chand De
On käsitletud Lorentzi para-Sasaki muutkondi, mille puhul P·C= 0,Z·C=
LCQ(g, C ),P·Z−Z·P= 0 või P·Z+Z·P= 0, kus Con Weyli konformse
kõveruse tensor, Pon v−Weyli projektiivne tensor ja Zon kontsirkulaartensor.
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