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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 34, Num. 2 (2019), 255 – 268

doi:10.17398/2605-5686.34.2.255

Available online April 29, 2019

Ricci solitons on para-Kähler manifolds

Sunil Kumar Yadav

Department of Mathematics, Poornima College of Engineering

ISI-6, RIICO Institutional Area, Sitapura, Jaipur-302022, Rajasthan, India

prof sky16@yahoo.com

Received December 18, 2018 Presented by Anna Maria Fino
Accepted March 30, 2019

Abstract: The main purpose of the paper is to study the nature of Ricci soliton on para-Kähler
manifolds satisfying some certain curvature conditions. In particular, if we consider certain pseu-

dosymmetric and parallel symmetric tensor on para-Kähler manifolds we prove that V is solenoidal

if and only if it is shrinking or steady or expanding depending upon the sign of scalar curvature for
dimension M > 4, where (g,V,λ) be a Ricci soliton in a paraholomorphic projectively, pseudosym-

metric para-Kähler manifolds. Moreover, we obtain some results related to the quasi-conformal

curvature tensor on such manifolds.

Key words: Para-Kähler manifold, Ricci soliton, pseudosymmetry, paraholomorphic projective
curvature, quasi-conformal curvature.

AMS Subject Class. (2010): Primary 53C15; Secondary 53C50 and 53C56.

1. Introduction

Ricci flow is an excellent tool in simplifying the structure of the manifolds.
It is defined for Riemannian manifolds of any dimension. It is a process that
deforms the metric of a Riemannian manifold analogous to the diffusion of
heat there by smoothing out the irregularity in the metric which is given by

∂g(t)

∂t
= −2 Ric(g(t)),

where g is the Riemannian metric dependent on time t and Ric(g(t)) is the
Ricci tensor. We consider φt : M → M, t ∈ R, be a family of diffeomorphisms
and (φt : t ∈ R) is a one parameter family of abelian group called flow. It
generates a vector field Xq given by

Xqf =
df(φt(q))

dt
, f ∈ C∞(M).

If Y is a vector field then LXY = limt→0
φ•tY−Y

t
is known as Lie-derivative

of Y with respect to X. Ricci solitons move under the Ricci flow under

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.34.2.255
mailto:prof_sky16@yahoo.com
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


256 sunil k. yadav

φt : M → M of the initial metric, that is, they are stationary points of the
Ricci flow in the space of metric. If g0 is a metric on the co-domain, then
g(t) = φ•tg0 is the pullback of g0, is a metric on the domain. Thus if g0
is a soliton of the Ricci flow on the co-domain, subjected to the condition
LV g0 + 2 Ric g0 + 2λg0 = 0 on the co-domain then g(t) is the soliton of Ricci
flow on the domain subjected to the condition LV g + 2 Ric g + 2λg = 0 on the
domain by [13] under suitable conditions. So g0 and g(t) are metrics which
satisfy Ricci flow. Thus the following equation

LV g + 2S + 2λg = 0, (1.1)

is called Ricci soliton. It is said to be shrinking, steady or expanding according
as λ < 0, λ = 0 and λ > 0 respectively. Therefore, Ricci solitons are gen-
eralization of Einstein manifolds and they are also known as quasi-Einstein
manifolds by theoretical physicists.

Para-Kähler manifolds are examples of symplectic, locally product and
semi-Riemannian manifolds. Some authors studied on paracomplex geometry
[3]. Besides, many authors considered the notion of “hyperbolic” instead of
“para”. It was first used by Prvanovic [8]. In this paper, the references [4, 5]
have been our motivation in studying the para-Kähler manifolds, where the
structure tensor P is an almost complex and metric g is positive definite.

This paper is organized as follows: In Section 2, we give the basic concept
of para-Kähler manifolds and some certain curvature tensor. In Section 3, we
introduce certain pseudosymmetric conditions on such manifolds. In Section
4, we discuss the parallel symmetric second order tensor field. In Section 5,
we consider quasi-conformally flat para-Kähler manifolds. Finally in Section
6, we study the parallel quasi-conformal para-Kähler manifolds.

2. Para-Kähler manifolds

By a para-Kählerian manifold we mean a triple (M,P,g), where M is a
connected differentiable manifold of dimension n = 2m, P is a (1, 1)-tensor
field and g is a pseudo-Riemannian metric on M satisfying the conditions

P 2 = I, g(PX,PY ) = −g(X,Y ), ∇P = 0, (2.1)

for any X,Y ∈ℵ(M), where ℵ(M) is the Lie algebra of vector fields on M, ∇
is the Levi-Civita connection of g and I is the identity tensor field.

Let (M,P,g) be a para-Kählerian manifold. The Riemann-Christoffel cur-
vature tensor R, the Ricci curvature S and the scalar curvature r are defined



ricci solitons on para-kähler manifolds 257

by

R(X,Y,Z,W) = g(R(X,Y )Z,W),

S(X,Y ) = Tr{Z → R(Z,X)Y},
r = Trg S.

Let Q be the Ricci operator given by S(X,Y ) = g(QX,Y ), for these tensor
fields, the following identities are satisfied

R(PX,PY ) = −R(X,Y ), R(PX,Y ) = −R(X,PY ),
S(PX,Y ) = −S(PY,X), S(PX,PY ) = −S(X,Y ).

(2.2)

Tr{Z → R(X,Y )PZ} = −2S(X,PY ),
Tr{Z → R(PZ,X)Y} = S(X,PY ),

QY = −
∑
i

εiR(ei,Y )ei.

For any (0, 2)-type tensor field Φ on M and X,Y ∈ ℵ(M), we define the
endomorphism X ∧Φ Y of ℵ(M) by

(X ∧Φ Y )Z = Φ(Y,Z)X − Φ(X,Z)Y, Z ∈ℵ(M).

The paraholomorphic projective curvature tensor P̃ of (M,P,g) is defined as
follows ([7, 8, 9]):

P̃(X,Y ) = R(X,Y )−
1

n + 2
{X∧SY−(PX)∧S(PY )+2g(QPX,Y )P}. (2.3)

We recall that

P̃(X,Y ) = −P̃(Y,X), Tr{Z → P̃(Z,X)Y} = 0,∑
i

εi P̃ (X,ei,ei,W) =
1

n + 2
{nS(X,W) −rg(X,W)},

(2.4)

where {e1,e2, . . . ,en} is an orthonormal frame and εi is the indicator of ei,
εi = g(ei,ei) = ±1. In [8], Prvanovic defined the following (0, 4)-type tensor
field: given X,Y,Z,V ∈ℵ(M),

R0(X,Y,Z,V ) =
1

4

{
g(X,Z)g(Y,V ) −g(X,V )g(Y,Z)

−g(X,PZ)g(Y,PV ) + g(X,PV )g(Y,PZ) (2.5)

− 2g(X,PY )g(Z,PV )
}
.



258 sunil k. yadav

For any q ∈ M, a subspace U ⊂ TqM is called non-degenerate if g restricted
to U is non-degenerate. If {u,v} is a basis of a plane σ ⊂ TqM, then σ is
non-degenerate if and only if g(u,u)g(v,v)−[g(u,v)]2 6= 0. Thus the sectional
curvature of σ = span{u,v} is

k(σ) =
R(u,v,u,v)

g(u,u)g(v,v) − [g(u,v)]2
.

From (2.1) it follows that X and PX are orthogonal for any X ∈ℵ(TM). By
a P-plane we mean a plane which is invariant by P . For any q ∈ M, a vector
u ∈ TqM is isotropic if g(u,u) = 0. If u ∈ TqM is not isotropic, then the
sectional curvature K(u) of the P-plane span{u,Pu} is called the P-sectional
curvature defined by u. When K(u) is constant, then (M,P,g) is called of
constant P-sectional curvature, or a para-Kähler space form.

The notion of a quasi-conformal curvature tensor C is given by Yano and
Sawaki [14] and is defined by

C(X,Y )Z = αR(X,Y )Z + β
[
S(Y,Z)X −S(X,Z)Y

+ g(Y,Z)QX −g(X,Z)QY
]

(2.6)

−
r

n

(
α

n− 1
+ 2β

)
{g(Y,Z)X −g(X,Z)Y} ,

where α and β are constants. If α = 1 and β = − 1
n−2 , then (2.6) reduces to

conformal curvature tensor [6]. A manifold (Mn,g) (n > 3) is said to be quasi-
conformally flat manifold if C = 0. In [1], it is known that a quasi-conformally
flat manifold is either conformally flat under α 6= 0 or Einstein manifold under
the conditions α = 0 and β 6= 0. The authors give no restrictions for α = 0
and β = 0. However, we consider the condition α 6= 0, or β 6= 0 in this study.

In view of the equations (2.2), (2.4) and (2.6), we have∑
i

εi g(C (Pei,PY )ei,W) = −
α

2
g(PY,W) + β[2S(PY,PW)−τg(PY,W)]

−
r

n

{
α

n− 1
+ 2β

}
g(PY,PW), (2.7)

which implies∑
i

εi C (Pei,PY )ei = −
α

2
PY + β[2QPY − τ PY ]

+
r

n

{
α

n− 1
+2β

}
Y,

(2.8)



ricci solitons on para-kähler manifolds 259

where τ is the special scalar curvature, which is defined as the trace of PQ.
It is remarked that

∑
i εig(Pei,ei) = 0.

For any (0,k)-type tensor (k ≥ 1) field T on a pseudo-Riemannian manifold
(M,g), we define a (0,k + 2)-tensor field R ·T by the following condition

(R ·T)(U,V,X1, . . . ,Xk) = −
k∑
s=1

T(X1, . . . ,R(U,V )Xs, . . . ,Xk). (2.9)

A pseudo-Riemannian manifold (M,g) is called semisymmetric if R · R = 0;
Ricci-symmetric if R ·S = 0 ([2, 4, 11, 12]).

We also define a (0,k + 2)-tensor (k ≥ 1) field Q(g,T) as follows

Q(g,T)(U,V,X1, . . . ,Xk) = −
k∑
s=1

T(X1, . . . , (U ∧V )Xs, . . . ,Xk). (2.10)

A pseudo-Riemannian manifold (M,g) is Ricci-pseudosemisymmetric [4] if
there exists a function LS : M → R such that

R ·S = LSQ(g,S). (2.11)

It is clear that every Ricci-semisymmetric manifold is Ricci-pseudosymmetric.
In general, the converse is not true [4]. The Riemannian curvature (1, 3)-tensor
field associated to the Levi-Civita connection ∇ of g is given by R = [∇,∇]−∇.
Then

R(X,Y,Z,V ) = −R(Y,X,Z,V )
= −R(X,Y,V,Z) = R(PX,PY,Z,V ),∑

σ

R(X,Y,Z,V ) = 0,
(2.12)

where σ represents the sum over all cyclic permutations.

3. Ricci-pseudosymmetric and pseudosymmetric

In this section, we study the Ricci-pseudosymmetric and pseudodymmetric
condition on para-Kähler manifolds and deduce some results.

Theorem 3.1. Every Ricci-pseudosymmetric para-Kählerian manifold is
Ricci-semisymmetric.



260 sunil k. yadav

Proof. Let the manifold (M,P,g) be para-Kähler satisfying the condition

(R ·S)(X,Y,U,V ) = LSQ(g,S)(X,Y,U,V ). (3.1)

In view of (2.2) and (2.9), we have

(R ·S)(PX,PY,U,V ) = −(R,S)(X,Y,U,V ). (3.2)

Using (3.2) in (3.1), it follows that

LSQ(g,S)(X,Y,U,V ) = −LSQ(g,S)(PX,PY,U,V ). (3.3)

Since LS is non-zero at a certain point q ∈ M, from (3.3), we get

Q(g,S)(X,Y,U,V ) = −Q(g,S)(PX,PY,U,V ).

In view of (2.10), we have

S(X,V )g(Y,U) −S(Y,V )g(X,U) + S(X,U)g(Y,V ) −S(Y,U)g(X,V )
= −S(V,PX)g(U,PY ) + S(V,PY )g(U,PX) (3.4)
−S(U,PX)g(V,PY ) + S(U,PY )g(V,PX).

Then taking contraction of (3.4) with respect to Y and U, and using (2.2),
we obtain

S(X,V ) =
r

n
g(X,V ). (3.5)

Which implies (M,P,g) is an Einstein manifold with R · S = 0. Thus it
completes the proof.

Corollary 3.2. Let (g,V,λ) be a Ricci soliton in a Ricci-pseudosymmetric
para-Kählerian manifold (M,P,g). Then V is solenoidal iff it is shrinking or
steady or expanding depending on the sign of the scalar curvature.

Proof. In view of (1.1) and (3.5), we get

(LV g)(X,V ) + 2
r

n
g(X,V ) + 2λg(X,V ) = 0. (3.6)

Taking X = V = ei where {ei} is an orthonormal basis of the tangent space
at each point of the manifold and taking summation over i (1 ≤ i ≤ n), we
have

(LV g)(ei,ei) + 2
r

n
g(ei,ei) + 2λg(ei,ei) = 0, (3.7)



ricci solitons on para-kähler manifolds 261

which implies
div V + r + λn = 0. (3.8)

If V is solenoidal, then div V = 0. Thus (3.8) reduces to λ = −( r
n

). Therefore,
we obtain the desired result.

A pseudo-Riemannian manifold (M,g) is said to be pseudosymmetric [4]
if there exists a function LR : M → R such that

R ·R = LRQ(g,R). (3.9)

It is well-known that every semisymmetric manifold is also pseudosymmetrc
but converse is not true, in general [4].

Theorem 3.3. Let (M,P,g) be a pseudosymmetric para-Kähler mani-
fold. Then

(i) (M,P,g) is Ricci flat, for dim M = 4.

(ii) (M,P,g) is semisymmetric, for dim M > 4.

Proof. Let (M,P,g) be para-Kähler manifold satisfying the condition (3.9).
Then using the analogy with Theorem 3.1, we get

(n− 4)R(X,U,V,W) − 2 S(V,PW)g(U,PX) + S(W,PU)g(V,PX) (3.10)
−S(V,PU)g(W,PX) + S(U,W)g(U,V ) −S(U,V )g(X,W) = 0.

Putting PU instead of U in (3.10) and contacting with respect to U and X
we have S = 0, for n = 4.

On the other hand, for n > 4, taking contraction (3.10) with respect to U
and V we have

S(X,W) =
r

n
g(X,W), (3.11)

which implies R ·S = 0. In view of (3.10), we get R ·R = 0. So Theorem 3.3
is proved.

Corollary 3.4. Let (g,V,λ) be a Ricci soliton in a pseudosymmetric
para-Kähler manifold (M,P,g) (n = 4). Then V is solenoidal iff it is always
steady.

Corollary 3.5. Let (g,V,λ) be a Ricci soliton in a pseudosymmetric
para-Kählerian manifold (M,P,g) (n > 4). Then V is solenoidal iff it is
shrinking or steady or expanding depending on the sign of the scalar curvature.



262 sunil k. yadav

A para-Kähler manifold with paraholomorphic projective curvature tensor
satisfies the condition

R · P̃ = LPQ(g,P̃),

where LP is a function on M. Such type of manifold is called paraholomorphic
projective-pseudosymmetric. If R·P̃ = 0, then it is said to be paraholomorphic
projective-semisymmetric manifold (see [7]).

Theorem 3.6. Let (M,P,g) be a paraholomorphic projective-
pseudosymmetric para-Kähler manifold. Then we have

(i) (M,P,g) is Ricci flat, for dim M = 4.

(ii) (M,P,g) is semisymmetric, for dim M > 4.

Proof. It is well know that, if R·P̃ = 0 at a certain point q of the manifold
M, then R ·R = 0 at this point (see [7]). Now, we suppose that R ·P̃ 6= 0 and
let the contraction of tensor P̃ is W̃ , defined by

W̃(X,W) =
∑
i

εiP̃(X,ei,ei,W). (3.12)

From (2.4), we have

W̃(X,W) =
1

n + 2
{nS(X,W ) −rg(X,W)}. (3.13)

Thus (M,A,g) is paraholomorphic projective-pseudosymmetric. So we have

(R · P̃)(X,Y,Z,U,V,W) = LPQ(g,P̃)(X,Y,Z,U,V,W). (3.14)

Taking contraction of (3.14) with respect to U and V we get

(R ·W̃)(X,Y,Z,W) = LPQ(g,P̃)(X,Y,Z,W).

Also, from (2.9) and (3.13), we obtain

(R ·S)(X,Y,Z,W) = LPQ(g,S)(X,Y,Z,W).

According to Theorem 3.1, we get R · S = 0. Since LP does not vanish at
point q, we have Q(g,S) = 0 by the help of above equation. From (2.4) and
(3.14), we conclude that R·R = LPQ(g,R), i.e., (M,P,g) is pseudosymmetric.
Therefore, the proof is completed.



ricci solitons on para-kähler manifolds 263

Corollary 3.7. Let (g,V,λ) be a Ricci soliton in a paraholomorphic
projectively-pseudo symmetric para-Kähler manifold (M,P,g) (n = 4). Then
V is solenoidal iff it is always steady.

Corollary 3.8. Let (g,V,λ) be a Ricci soliton in a paraholomorphic
projectively pseudo symmetric para-Kähler manifold (M,P,g) (n > 4). Then
V is solenoidal iff it is shrinking or steady or expanding depending on the sign
of the scalar curvature.

4. Parallel symmetric second order covariant tensor

In this section, we study the second order parallel tensor on a para-Kähler
manifold. Thus we give the following results.

Theorem 4.1. A second order parallel tensor in a para-Kähler space form
is a linear combination of para-Kähler metric and para-Kähler 2-form.

Proof. Let ~ be a (0, 2)-tensor which is parallel in view of ∇, that is,
∇~ = 0. Then from Ricci identity [10], we have

~(R(X,Y )Z,V ) + ~(Z,R(X,Y )V ) = 0. (4.1)

Using (2.5) in (4.1) and replacing X = V = ei, 1 ≤ i ≤ n, on simplification,
we get{

~(Y,Z) −g(Y,Z)(trH) + ~(PY,PZ) −g(Y,PZ)(tr.HP)
+2~(PY,PZ) + (n− 1)~(Y,Z) + 3~(Z,P 2Y )

}
= 0, (4.2)

where H is a (1, 1) tensor and tr.H =
∑n

i=1 ~(ei,ei). Using the notion of
symmetrization and anti-symmetrization. Then from (4.2), we obtain

(n + 3)~(Y,Z) + 3~(PY,PZ) = g(Y,Z)(tr.H) (4.3)

and
(n + 3)~(Y,Z) + 3~(PY,PZ) = g(Y,PZ)(tr.HP). (4.4)

Replacing Y and Z by PY and PZ in (4.3) and (4.4) respectively, using (2.1),
we have

~s(Y,Z) = −
1

(n + 6)
(tr.H)g(Y,Z) (4.5)

and

~a(Y,Z) = −
1

n
(tr.HP)g(PY,Z). (4.6)



264 sunil k. yadav

In view of (4.5) and (4.6), we get

~(Y,Z) = {ϑ (tr.H) g(Y,Z) + ω (tr.HP) ψ}, (4.7)

where ϑ = − 1
(n+6)

, ω = −1
n

and ψ = g(PY,Z). Thus it completes the
proof.

Corollary 4.2. A locally Ricci symmetric para-Kähler space form is an
Einstein manifold.

Proof. According to our hypothesis, if we put H = S, in (4.7) then we
have tr.H = r and tr.HP = 0. Then follows from (4.7), we obtain

S(Y,Z) = ϑrg(Y,Z). (4.8)

Which completes the proof.

Corollary 4.3. Let (g,V,λ) be a Ricci soliton in a para-Kähler space
form. Then V is solenoidal iff it is shrinking or steady or expanding depending
on the sign of the scalar curvature.

Proof. In view of (1.1) and (4.8), we get

(LV g)(Y,Z) + 2ϑrg(Y,Z) + 2λg(Y,Z) = 0. (4.9)

Taking Y = Z = ei in (4.9), where {ei} is an orthonormal basis of the tangent
space at each point of the manifold and taking summation over i (1 ≤ i ≤ n),
we have

(LV g)(ei,ei) + 2ϑrg(ei,ei) + 2λg(ei,ei) = 0, (4.10)

which implies
div V + ϑrn + λn = 0. (4.11)

Here, if V is solenoidal, then we get div V = 0. Thus (4.11) reduces to
λ = −ϑr. So this completes the proof.

Theorem 4.4. A Ricci semi-symmetric para-Kähler space form is an
Einstein manifold.

Proof. Let the para-Kähler space form holds the condition R·S = 0. Then
we have

(R(X,Y ) ·S)(V,U) = 0, (4.12)



ricci solitons on para-kähler manifolds 265

which reduces to

S(R(X,Y )V,U) + S(V,R(X,Y )U) = 0. (4.13)

Using (2.5) in (4.13) and putting Y = V = ei, where {ei} is an orthonormal
basis of the tangent space at each point of the manifold and again taking
summation over i, 1 ≤ i ≤ n, we get

S(X,U) =
r

(n + 2)
g(X,U). (4.14)

Thus this proves the theorem.

Corollary 4.5. Let (g,V,λ) be a Ricci soliton in a Ricci semi-symmetric
para-Kähler space form. Then V is solenoidal iff it is shrinking or steady or
expanding depending on the sign of the scalar curvature.

Proof. From (1.1) and (4.14), we get

(LV g)(X,U) + 2
r

(n + 2)
g(X,U) + λg(X,U) = 0. (4.15)

Then putting X = U = ei in (4.15) and taking summation over i (1 ≤ i ≤ n),
we have

(LV g)(ei,ei) + 2
r

(n + 2)
g(ei,ei) + 2λg(ei,ei) = 0, (4.16)

which implies

div V +
r

(n + 2)
n + λn = 0. (4.17)

In view of (4.17), if V is solenoidal then we have div V = 0. Thus (4.17)
reduces to λ = − r

(n+2)
. So the proof is clear.

Corollary 4.6. If the (0, 2)-type tensor field LV + 2S is parallel, where
V is a vector field on a para-Kähler space form. Then (g,V ) admits a Ricci
soliton if PV is solenoidal. Moreover, it is shrinking or steady or expanding
depending on the sign of the scalar curvature.

5. Quasi-conformally flat para-Kähler manifolds

In this section, we consider the notion of quasi-conformally flat para-Kähler
space form, i.e., C(X,Y )Z = 0. Thus we can state the following result.



266 sunil k. yadav

Theorem 5.1. Let (g,V,λ) be a Ricci soliton in a quasi-conformally flat
para-Kähler space form. Then V is solenoidal iff it is always steady.

Proof. In view of (2.1), (2.2) and (2.7), we have

α

2
g(PY,W) + β

[
2S( Y,W) + τg(PY,W)

]
−
r

n

{
α

n− 1
+ 2β

}
g(Y,W) = 0.

(5.1)

Contracting (5.1) over the pair of argument Y and W , we obtain

r

{
α

(n− 1)

}
= 0, (5.2)

which implies
r = 0, α 6= 0. (5.3)

In view of (5.1) and (5.3), we get

S(Y,W) = −
{
τ

2
+

α

4β

}
g(PY,W). (5.4)

Then using (5.3) and (5.4) in (2.6), we obtain

R(X,Y )Z =
β

α

[(
τ

2
+
α

β

)(
g(PY,Z)X −g(PX,P)Y

+ g(Y,Z)PX −g(X,Z)PY
)]
.

(5.5)

Again from (1.1) and (5.4), it yields

(LV g)(Y,W) − 2
{
τ

2
+

α

4β

}
g(PY,W) + 2λg(Y,W) = 0. (5.6)

Taking contraction with respect to Y and W for i (1 ≤ i ≤ n), we have

(LV g)(ei,ei) − 2
{
τ

2
+

α

4β

}
g(Pei,ei) + 2λg(ei,ei) = 0, (5.7)

which reduces to
div V + λn = 0. (5.8)

Suppose that V is solenoidal. Then (5.8), takes the form λ = 0. Thus the
proof is obvious.

Corollary 5.2. In a quasi-conformally flat para-Kähler space, the Ricci
and the curvature tensors have the shapes (5.4) and (5.5), respectively.



ricci solitons on para-kähler manifolds 267

6. Parallel quasiconformal curvature tensor

This section deals with the parallelity condition (∇C = 0) of quasi-conformal
curvature tensor on para-Kähler space form. It is essential to state the fol-
lowing result.

Theorem 6.1. A para-Kähler space form is quasi-conformally symmetric
iff it is locally symmetric.

Proof. In view of the condition ∇C = 0, (2.7) reduces to

−
α

2
g(PY,W) + β

[
2S(Y,W) − τg(PY,W)

]
+
r

n

{
α

n− 1
+ 2β

}
g(Y,W) = 0.

(6.1)

Taking covariant derivation of (6.1) along the vector field Z, we get

β
[
2(∇ZS)(Y,W) −dτ(Z)g(PY,W)

]
+
dr(Z)

n

{
α

n− 1
+ 2β

}
g(Y,W) = 0.

(6.2)

Taking contraction in (6.2) with respect to Y and W , multiplying by εi, we
obtain

dr(Z)

{
α

n− 1
+ 4β

}
= 0. (6.3)

Thus (6.3) implies that dr(Z) = 0. Using this equation in (6.2), we have

(∇ZS)(Y,W) =
1

2
dτ(Z)g(PY,W). (6.4)

Replacing Y by AY in (6.4) and using (2.1), we get

(∇ZS)(PY,W) =
1

2
dτ(Z)g(Y,W). (6.5)

Again taking contraction in (6.2) with respect to Y and W , multiplying by
εi, we obtain

(∇ZS)(PY,W) = 0. (6.6)

Taking covariant derivation of (2.6) and using (6.6), we get

(∇ZC)(X,Y )W = α(∇ZR)(X,Y )W, α 6= 0. (6.7)

Thus it completes the proof.



268 sunil k. yadav

On the other-hand, if we consider the condition R ·C = 0, then we have
R ·Q = 0 by using (2.7). This implies that R ·S = 0. Taking into account of
R·C = 0 and R·S = 0, we get R·R = 0. Thus we can state the following result.

Corollary 6.2. A para-Kähler space form is quasi-conformally semisym-
metric iff it is semisymmetric.

Acknowledgements

The author would like to thank the referee for reading the manuscript
in great detail and for his/her valuable suggestions and useful comments.

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	Introduction
	Para-Kähler manifolds
	Ricci-pseudosymmetric and pseudosymmetric
	Parallel symmetric second order covariant tensor
	Quasi-conformally flat para-Kähler manifolds
	Parallel quasiconformal curvature tensor