Bulletin T.CXLI de l’Académie serbe des sciences et des arts − 2010
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 35
SOME PROPERTIES OF THE LOCALLY CONFORMAL KÄHLER
MANIFOLD
MILEVA PRVANOVIĆ
(Presented at the 1st Meeting, held on February 29, 2010)
A b s t r a c t. We determine the holomorphic curvature tensor of locally conformal Kähler manifold and find the expression of the Riemannian
curvature tensor of such manifold if it is of pointwise constant holomorphic
sectional curvature. We examine the geodesic mapping using the holomorphic curvature tensor and apply the obtained results to the locally conformal
Kähler manifolds.
AMS Mathematics Subject Classification (2000): 53C56
Key Words: locally conformal Kähler manifold, holomorphic curvature
tensor, geodesic mapping
1. The locally conformal Kähler manifolds
Let (M, g, J) be a real 2n-dimensional almost Hermitian manifold, where
J is almost complex structure and g is Hermitian metric. Then
J 2 = −Id. ,
g(JX, JY ) = g(X, Y ) ,
for any vector fields X, Y tangent to M. The fundamental 2-form F (X, Y )
is
F (X, Y ) = g(JX, Y ) = −F (Y, X) .
10
Mileva Prvanović
(M, g, J) is a Kähler manifold if ∇J = 0, where ∇ denotes the covariant
differentiation with respect to Levi-Civita connection.
The manifold (M, g, J) is called locally conformal Kähler (bf. l.c.Kähler)
manifold if each point p ∈ M has an open neighborhood U with a differentiable function
σ : U → R2n
such that
ġ = e−2 g|U
σ
(1.1)
is a Kähler metric on U.
The necessary and the sufficient condition for (M, g, J) to be l.c. Kähler
manifold is that it admits a global 1-form α satisfying the condition [2]:
(∇Z F )(X, Y ) = α(JX)g(Z, Y )−α(JY )g(Z, X)−α(X)F (Z, Y )+α(Y )F (Z, X).
(1.2)
With respect to the local coordinates, (1.2) is
∇s Fij = −αi Fsj + αj Fsi − αa Fai gsj + αa Faj gsi ,
∇i αj = ∇j αi ,
αi = αp g pi ,
(1.3)
and the indices run over the range {1, 2, . . . , 2n}.
From (1.3) we obtain
∇r ∇s Fij − ∇s ∇r Fij =
= Prs Jja gsi − Pra Jia gsj − Psa Jja gri + Psa Jia grj
−Prj Fsi + Pri Fsj + Psj Fri − Psi Frj
where
,
1
Pri = −∇r αi − αr αi + αp αp gri .
2
We note that Pri = Pir .
Using the Ricci identity, we get
−Rsria Jja + Rsrja Jia =
= Pri Fsj + Psj Fri − Prj Fsi − Psi Frj
−Pra Jia gsj − Psa Jja gri + Pra Jja gsi + Psa Jia grj ,
(1.4)
11
Some properties of the locally conformal Kähler manifold
or
Rsrab Jja Jhb = Rsrjh
+Prh gsj − Prj gsh + Psj grh − Psh grj
(1.5)
+Pra Jha Fsj − Pra Jja Fsh + Psa Jja Frh − Psa Jha Frj .
Transecting (1.4) with g ir and denoting by ρ the Ricci tensor, we get
−ρsa Jja + Rsrja Jia g ir =
= −(2n − 3)Psa Jja − Pja Jsa + (Pab g ab )Fsj ,
from which, taking the symmetric part, we obtain [3]:
ρia Jja + ρja Jia = 2(n − 1)(Pja Jia + Pia Jja ).
(1.6)
2. Holomorphic curvature tensor
A 2-plane π in Tp (M ), p ∈ M, is said to be holomorphic if Jπ = π.
The manifold M has pointwise constant holomorphic sectional curvature if
the sectional curvature relative to π does not depend on the holomorphic
2-plane π in Tp (M ).
The curvature tensor of Kähler manifold satisfies the condition
R(X, Y, JZ, JW ) = R(X, Y, Z, W ).
(2.1)
Using it, in [4] (Proposition 7.3, p.167) is determined the curvature tensor of
Kähler manifold of constant holomorphic sectional curvature. But, if ∇J ̸=
0, (2.1) does not hold. Nevertheless, there exist for any almost Hermitian
manifold, the algebraic curvature tensor, satisfying the condition of type
(2.1). It is ([4],[7]):
(HR)(X, Y, Z, W ) =
1
{3 [R(X, Y, Z, W ) + R(JX, JY, Z, W ) + R(X, Y, JZ, JW )
16
+R(JX, JY, JZ, JW )] − R(X, Z, JW, JY ) − R(JX, JZ, W, Y )
=
−R(X, W, JY, JZ) − R(JX, JW, Y, Z) + R(JX, Z, JW, Y )
+R(X, JZ, W, JY ) + R(JX, W, Y, JZ) + R(X, JW, JY, Z)} .
(2.2)
12
Mileva Prvanović
Namely, it is easy to see that
(HR)(X, Y, Z, W ) = −(HR)(Y, X, Z, W ) =
= −(HR)(X, Y, W, Z) = (HR)(Z, W, X, Y ) ,
(2.3)
(HR)(X, Y, Z, W ) + (HR)(X, Z, W, Y ) + (HR)(X, W, Y, Z) = 0 ,
as well as
(HR)(X, Y, JZ, JW ) = (HR)(X, Y, Z, W ),
(2.4)
(HR)(X, JX, JX, X) = R(X, JX, JX, X).
(2.5)
The tensor (2.2) is said to be the holomorphic curvature tensor.
The relation (2.5) shows that the holomorphic sectional curvatures relative to R and HR are the same. Thus, applying the way of [4] to the tensor
HR, we can state
Proposition 2.1 If an almost Hermitian manifold (M, g, J) is of pointwise constant holomorphic sctional curvature H(p), p ∈ M, then the holomorphic curvature tensor is
H(p)
[g(X, W )g(Y, Z) − g(X, Z)g(Y, W )
4
+F (X, W )F (Y, Z) − F (X, Z)F (Y, W ) − 2F (X, Y )F (Z, W )] .
(HR)(X, Y, Z, W ) =
(2.6)
Now, we shall determine the holomorphic curvature tensor for a l.c.
Kähler manifold.
With respect to the local coordinates, (2.2) reads
(HR)ijhk =
=
]
1 { [
3 Rijhk + Rabhk Jia Jjb + Rijab Jha Jkb + Rabcd Jia Jjb Jhc Jkd
16
−Rihab Jka Jjb − Rabkj Jia Jhb − Rikab Jja Jhb − Rabjh Jia Jkb
}
+Rahbj Jia Jkb + Riakb Jha Jjb + Rakjb Jia Jhb + Riabh Jka Jjb .
(2.7)
13
Some properties of the locally conformal Kähler manifold
In view of (1.5), we find
[
3 Rijhk + Rabhk Jia Jjb + Rijab Jha Jkb + Rabcd Jia Jjb Jhc Jkd
]
= 3 [4Rijhk
+gih (3Pjk − Pab Jja Jkb ) − gik (3Pjh − Pab Jja Jhb )
+gjk (3Pih − Pab Jia Jhb ) − gjh (3Pik − Pab Jia Jkb )
(2.8)
+Fih (Pja Jka − Pka Jia ) − Fik (Pja Jha − Pha Jja )
+Fjk (Pia Jha − Pha Jia ) − Fjh (Pia Jka − Pka Jia )].
Using (1.5), we also obtain
−(Rihab Jka Jjb + Rabkj Jia Jhb + Rikab Jja Jhb + Rabjh Jia Jkb )
= 2Rijhk
−2(gik Pjh + gjh Pik − gih Pjk − gjk Pih )
−Fik (Pha Jja − Pja Jha ) − Fjh (Pka Jia − Pia Jka )
−Fih (Pja Jka − Pka Jja ) − Fjk (Pia Jha − Pha Jia )
−2Fij (Pka Jha − Pha Jka ) − 2Fhk (Pja Jia − Pia Jja ) .
Using (1.4), as well as the relation
Rakht − Rahkt = −Rkhat ,
we find
Rahbj Jia Jkb + Rakjb Jia Jhb =
= −Rkhab Jia Jjb
−Phj gik + Pka Jia Fhj + Phk gij − Pja Jia Fhk
−Pha Jja Fik − Pab Jia Jkb gjh + Pha Jka Fij + Pab Jia Jjb ghk
+Pkj gih − Pha Jia Fkj − Phk gij + Pja Jia Fkh
+Pka Jja Fih + Pab Jia Jhb gjk − Pka Jha Fij − Pab Jia Jjb ghk ,
(2.9)
14
Mileva Prvanović
such that, applying (1.5), we have
Rahbj Jia Jkb + Rakjb Jia Jhb =
= Rijhk + 2gih Pjk − 2gik Pjh
+gjk (Pih + Pab Jia Jhb ) − gjh (Pik + Pab Jia Jkb )
(2.10)
−2Pja Jia Fhk + (Pha Jka − Pka Jha )Fij .
In a similar way, we obtain
Riakb Jha Jjb + Riabh Jka Jjb =
= Rijhk − 2Pik gjh + gih (Pjk + Pab Jja Jkb )
(2.11)
−gik (Pjh + Pab Jja Jhb ) + 2Pih gjk
+2Pia Jja Fhk + (Pha Jka − Pka Jha )Fij .
Substituting (2.8)-(2.11) into (2.7), we get
(HR)ijhk = Rijhk
+
1{
gih (7Pjk − Pab Jja Jkb ) − gik (7Pjh − Pab Jja Jhb )
8
+gjk (7Pih − Pab Jia Jhb ) − gjh (7Pik − Pab Jia Jkb )
(2.12)
+Fih (Pja Jka − Pka Jja ) − Fik (Pja Jha − Pha Jja )
+Fjk (Pia Jha − Pha Jia ) − Fjh (Pia Jka − Pka Jia )
}
+2Fij (Pha Jka − Pka Jha ) + 2Fhk (Pia Jja − Pja Jia ) .
Thus, we can state
Theorem 2.1 The holomorphics curvature tensor of l.c.Kähler manifold
has the form (2.12).
In view of Proposition (2.1), l.c.Kähler manifold has pointwise constant
15
Some properties of the locally conformal Kähler manifold
holomorphic sectional curvature if its curvature tensor has the form
H(p)
[gik gjh − gih gjk + Fik Fjh − Fih Fjk − 2Fij Fhk ]
4
{
Rijhk =
1
gik (7Pjh − Pab Jja Jhb ) − gih (7Pjk − Pab Jja Jkb )
8
+gjh (7Pik − Pab Jia Jkb ) − gjk (7Pih − Pab Jia Jhb )
+
(2.13)
+Fik (Pja Jha − Pha Jja ) − Fih (Pja Jka − Pka Jja )
+Fjh (Pia Jka − Pka Jia ) − Fjk (Pia Jha − Pha Jia )
}
−2Fij (Pha Jka − Pka Jha ) − 2Fhk (Pia Jja − Pja Jia ) .
Conversely, if (2.13) holds, then the holomorphic curvature tensor has
the form (2.6). Thus, we can state
Theorem 2.2 L.c. Kähler manifold has pointwise constant holomorphic
sectional curvature if and only if its curvature tensor can be expressed in the
form (2.13).
If the tensor Pij is hybrid, i.e. if
Pab Jia Jjb = Pij ,
Pia Jja = −Pja Jia ,
the relations (2.12) and (2.13) reduce, respectively to
(HR)ijhk = Rijhk
3
+ (Pjk gih + Pih gjk − Pjh gik − Pik gjh )
4
1
+ (Fih Pja Jka + Fjk Pia Jha − Fik Pja Jha − Fjh Pia Jka
4
+2Fhk Pia Jja + 2Fij Pha Jka )
(2.14)
and
Rijhk =
H(p)
[gik gjh − gih gjk + Fik Fjh − Fih Fjk − 2Fij Fhk ]
4
3
+ (gik Pjh + gjh Pik − gih Pjk − gjk Pih )
4
1
+ (Fik Pja Jha + Fjh Pia Jka − Fih Pja Jka − Fjk Pia Jha
4
−2Fij Pha Jka − 2Fhk Pia Jja ).
(2.15)
16
Mileva Prvanović
The relation (2.15) is just the form of the curvature tensor of l.c.Kähler
manifold obtained in [3] and [5].
According (1.6), the tensor Pij is hybrid if and only if the Ricci tensor
ρij is hybrid. Thus, we can state ([3], [5]):
Corrollary 2.1. L.c.Kähler manifold whose Ricci tensor is hybrid is of
pointwise constant holomorphic sectional curvature if and only if its curvature tensor has the form (2.15).
3. The expression of the holomorphic curvature tensor which does not
include the tensor Pij
We shall show that, if n > 2, the tensors HR and R in the relations
(2.12)-(2.15) can be written in the form which include the second Ricci
tensor instead of the tensor Pij .
The second Ricci tensor is defined as follows
*jh = Rijab J a J b g ik .
ρ
h k
It has, for any almost Hermitian manifold, the property
*ab J a J b = ρ
*ji .
ρ
i j
In general, it is not symmetric, but for l.c.Kähler manifold, it is. Namely,
the relation (1.5) yields
*jh − P gjh ,
(2n − 3)Pjh − Pab Jja Jhb = ρjh − ρ
(3.1)
where
P = Pab g ab =
*
κ− κ
,
4(n − 1)
and
κ = ρab g ab
and
*= ρ
*ab g ab
κ
are the first and the second scalar curvatures.
*jh = ρ
*hj . Thus, for any l.c.Kähler
From (3.1) immediately follows ρ
manifold, we have
*jh = ρ
*hj ,
ρ
*ab J a J b = ρ
*jh .
ρ
j h
17
Some properties of the locally conformal Kähler manifold
The relation (3.1), together with
*jh − P gjh ,
−Pjh + (2n − 3)Pab Jja Jhb = ρab Jja Jhb − ρ
implies
*jh −2(n−1)P gjh , (3.2)
4(n−1)(n−2)Pjh = (2n−3)ρjh +ρab Jja Jhb −2(n−1) ρ
such that, if n > 2, we have
Pjh =
{
}
1
*jh−2(n−1)P gjh .
(2n−3)ρjh+ρab Jja Jhb−2(n − 1) ρ
4(n−1)(n−2)
(3.3)
Substituting (3.3) into (2.12) and (2.13), we get, respectively,
(HR)ijhk = Rijhk
P
{3(gik gjh − gih gjk ) − (Fik Fjh − Fih Fjk − 2Fij Fhk )}
4(n − 2)
7n − 11
+
(gih ρjk + gjk ρih − gik ρjh − gjh ρik )
16(n − 1)(n − 2)
n−5
(gih ρab Jja Jkb + gjk ρab Jia Jhb − gik ρab Jja Jhb − gjh ρab Jia Jkb )
−
16(n − 1)(n − 2)
3
*jk + gjk ρ
*ih − gik ρ
*jh − gjh ρ
*ik )
−
(gik ρ
8(n − 2)
{
1
*ja J a )
+
Fih (ρja Jka − ρka Jja − 2 ρ
k
16(n − 2)
a
a
a
*
+Fjk (ρia Jh − ρha Ji − 2 ρia Jh )
+
*ja J a )
−Fik (ρja Jha − ρha Jja − 2 ρ
h
*ia J a )
−Fjh (ρia Jka − ρka Jia − 2 ρ
k
*ha J a )
+2Fij (ρha Jka − ρka Jha − 2 ρ
k
}
*ia J a )
+2Fhk (ρia Jja − ρja Jia − 2 ρ
j
and
(3.4)
18
Mileva Prvanović
1
3P
Rijhk = (H −
)(gik gjh − gih gjk )
4
n−2
1
P
+ (H +
)(Fik Fjh − Fih Fjk − 2Fij Fhk )
4
n−2
7n − 11
+
(gik ρjh + gjh ρik − gih ρjk − gjk ρih )
16(n − 1)(n − 2)
n−5
−
(gik ρab Jja Jhb + gjh ρab Jia Jkb − gih ρab Jja Jkb − gjk ρab Jia Jhb )
16(n − 1)(n − 2)
3
*jh + gjh ρ
*ik − gih ρ
*jk − gjk ρ
*ih )
−
(gik ρ
8(n − 2)
{
1
*ja J a )
Fik (ρja Jha − ρha Jja − 2 ρ
+
h
16(n − 2)
*ia J a )
+Fjh (ρia Jka − ρka Jia − 2 ρ
k
*ja J a )
−Fih (ρja Jka − ρka Jja − 2 ρ
k
(3.5)
*ia J a )
−Fjk (ρia Jha − ρha Jia − 2 ρ
h
*ha J a )
−2Fij (ρha Jka − ρka Jha − 2 ρ
k
}
*ia J a )
−2Fhk (ρia Jja − ρja Jia − 2 ρ
j
.
Thus, we can state
Theorem 3.1. If n > 2, the holomorphic curvature tensor of a l.c.Kähler
manifold can be expressed in the form (3.4). If such a manifold has pointwise
constant holomorphic sectional curvature, its Riemannian curvature tensor
can be expressed in the form (3.5).
Remark. If n = 2, we can not eliminate the tensor Pij from the equations (2.12)-(2.15). On the other hand, according (3.2), we have
*jh + 2P gjh ,
ρjh + ρab Jja Jhb = 2 ρ
and, if ρjh is hybrid,
*jh = P gjh .
ρjh − ρ
19
Some properties of the locally conformal Kähler manifold
4. Geodesic mapping
The diffeomorphism transforming all geodesics of the Riemannian manifold (M̄ , ḡ) onto the geodesics of the Riemannian manifold (M, g) is said
to be the geodesic mapping. If we consider the manifolds M̄ and M with
respect to the coordinate system which is common for M̄ and M, then the
necessary and the sufficient condition for geodesic mapping in [6]
Γ̄tij = Γtij + δit ψj + δjt ψi .
(4.1)
where Γ̄ and Γ are the Levi-Civita connections of the metrics ḡ and g respectively, and ψi is the gradient vector field. As for the curvature tensors
R̄ and R, they are related as follows
R̄tjhk = Rtjhk + δkt ψjh − δht ψjk
(4.2)
where
ψjh = ∇h ψj − ψj ψh .
Because ψj is a gradient, we have ψjh = ψhj .
Now, let us consider the geodesic mapping
f : (M̄ , ḡ, J) → (M, g, J)
of an almost Hermitian manifold (M̄ , ḡ, J) onto an almost Hermitian manifold (M, g, J).
For (M̄ , ḡ, J), the holomorphic curvature tensor is
(H R̄)tjhk =
]
1 { [ t
3 R̄ jhk + R̄tjab Jha Jkb − R̄abhk Jat Jjb − R̄abcd Jat Jjb Jhc Jkd
16
−R̄thab Jka Jjb + R̄abkj Jat Jhb − R̄tkab Jja Jhb + R̄abjh Jat Jkb
−R̄ahbj Jat Jkb + R̄takb Jha Jjb − R̄akjb Jat Jhb + R̄tabh Jka Jjb
}
.
Substituting (4.2), we find
8(H R̄)tjhk = 8(HR)tjhk
δkt Qjh − δht Qjk + Jkt Qja Jha − Jht Qja Jka − 2Jjt Qha Jka ,
where
Qjh = ψjh + ψab Jja Jhb .
(4.3)
20
Mileva Prvanović
Contracting (4.3) with respect to t and k, we get
Qjh =
]
4 [
ρ(H R̄)jh − ρ(HR)jh ,
n+1
(4.4)
where ρ(H R̄) and ρ(HR) are the Ricci tensors of H R̄ and HR respectively.
Substituting (4.4) into (4.3), we find
(H W̄ )tjhk = (HW )tjhk ,
where
(HW )tjhk = (HR)tjhk
−
{
1
δ t ρ(HR)jh − δht ρ(HR)jk
2(n + 1) k
(4.5)
}
+Jkt ρ(HR)ja Jha − Jht ρ(HR)ja Jka − 2Jjt ρ(HR)ha Jka ,
and (H W̄ )tjhk is constructed in the same way, but with respect to the tensor
H R̄.
Thus we can state
Theorem 4.1. For an almost Hermitian manifold, the tensor (4.5) is
invariant with respect to the geodesic mapping (4.1), preserving the complex
structure.
Now, let us suppose that HW = 0. Then
(HR)ijhk =
1
[gik ρ(HR)jh − gih ρ(HR)jk
2(n + 1)
−Fik ρ(HR)ja Jha
+
Fih ρ(HR)ja Jka
+
(4.6)
2Fij ρ(HR)ha Jka ] ,
from which, transvecting with g jh and taking into account that, as a consequence of (2.4), the Ricci tensor ρ(HR) is hybrid, we get
ρ(HR)ik =
κ(HR)
gik ,
2n
where κ(HR) = ρ(HR)jh g jh .
Substituting this into (4.6), we obtain
(HR)ijhk =
κ(HR)
(gik gjh − gih gjk + Fik Fjh − Fih Fjk − 2Fij Fhk ).
4n(n + 1)
21
Some properties of the locally conformal Kähler manifold
But, this is just the equation (2.6). Conversely, if (2.6) holds, the tensor
(4.5) vanishes. Thus, we can state
Theorem 4.2. For an almost Hermitian manifold (M, g, J) the tensor
(4.5) vanishes if and only if (M, g, J) is of pointwise constant holomorphic
sectional curvature.
The l.c.Kähler manifold can serve as an example. Indeed, for such a
manifold the holomorphic curvature tensor has the form (2.12). Therefore
3
7n − 10
n+2
ρ(HR)ik = ρik − P gik −
Pik +
Pab yia ykb .
4
4
4
Substituting this and (2.12) into (4.5), we find that for l.c.Kähler manifold, the tensor (HW )ijhk is
(HW )ijhk = Rijhk
+
1{
gih (7Pjk − Pab Jja Jkb ) − gik (7Pjh − Pab Jja Jhb )
8
gjk (7Pih − Pab Jia Jhb ) − gjh (7Pik − Pab Jia Jkb )
+Fih (Pja Jka − Pka Jja ) − Fik (Pja Jha − Pha Jja )
+Fjk (Pia Jha − Pha Jia ) − Fjh (Pia Jka − Pka Jia )
}
+2Fij (Pha Jka − Pka Jha ) + 2Fhk (Pia Jja − Pja Jia )
{
7n − 10
n+2
1
3
Pjh +
Pab Jja Jhb )
−
gik (ρjh − P gjh −
2(n + 1)
4
4
4
7n − 10
n+2
3
Pjk +
Pab Jja Jkb )
−gih (ρjk − P gjk −
4
4
4
3
7n − 10
n+2
−Fik (ρja − P gja −
Pja +
Pst Jjs Jat )Jha
4
4
4
7n − 10
n+2
3
+Fih (ρja − P gja −
Pja +
Pst Jjs Jat )Jka
4
4
4
}
3
7n − 10
n+2
s t a
+2Fij (ρha − P gha −
Pha +
Pst yh ya )yk .
4
4
4
(4.7)
If this tensor vanishes, then
n2 + 14n − 12
7n2 + 2n − 12
Pik −
Pab Jia Jkb
2
2
[
]
3(n − 2)
κ−
P gik .
2
(2n + 3)ρik − 3ρab Jia Jkb =
22
Mileva Prvanović
This relation, together with
−3ρik + (2n + 3)ρab Jia Jkb = −
[
n2 + 14n − 12
Pik
2
]
7n2 + 2n − 12
3(n − 2)
+
Pab Jia Jkb + κ −
P gik ,
2
2
yields
3
7n − 10
n+2
ρik − P gik −
Pik +
Pab Jia Jkb
4
4
4
1
=
[κ − 3(n − 1)P ] gik ,
2n
and the relation (4.7), if HW = 0, gives
Rijhk =
κ − 3(n − 1)P
(gik gjh − gih gjk + Fik Fjh − Fih Fjk − 2Fij Fhk )
4n(n + 1)
1{
gik (7Pjh − Pab Jja Jhb ) + gjh (7Pik − Pab Jia Jkb )
8
−gih (7Pjk − Pab Jja Jkb ) − gjk (7Pih − Pab Jia Jhb )
+Fik (Pja Jha − Pha Jja ) + Fjh (Pia Jka − Pka Jia )
−Fih (Pja Jka − Pka Jja ) − Fjk (Pia Jha − Pha Jia )
}
−2Fij (Pha Jka − Pka Jha ) − 2Fhk (Pia Jja − Pja Jia ) .
But, this is just the relation (2.13), where
H(p) =
*
κ − 3(n − 1)P
κ+3 κ
=
.
n(n + 1)
4n(n + 1)
Thus, and also as a consequence of theorems 4.1 and 4.2, we have
Theorem 4.3 Let (M, g, J) be l.c.Kähler manifold. Then the tensor
(4.7) is invariant with respect to the geodesic mapping. The tensor (4.7)
vanishes if and only if (2.13) holds.
5. The remark on Kähler manifolds
A curve xi = xi (t) of an almost Hermitian manifold (M, g, J) having
the property that the holomorphic planes determined by its tangent vectors
Some properties of the locally conformal Kähler manifold
23
are parallel along the curve itself is called a holomorphically planer curve.
The manifold (M, ḡ, J) and (M, g, J) are said to be holomorphically projectively related if they have all holomorphically planer curves in common, and
the corresponding invariant tensor is holomorphically projective curvature
tensor. In the case of Kähler manifolds, this tensor is [8]:
(
)
1
Rijhk −
δkt ρjh − δht ρjk + Jkt ρja Jha − Jht ρja Jka − 2Jjt ρha Jka
2(n + 1)
(5.1)
On the other hand, if (M, g, J) is a Kähler manifold, then
(HR)tjhk = Rtjhk ,
ρ(HR)jh = ρjh ,
such that (4.5) reduces just to the tensor (5.1).
Thus, in the case of Kähler manifolds, the tensor (5.1) can be obtained
considering holomorphically planer curves or applying geodesic mapping to
the holomorphic curvature tensor (2.2).
This does not hold for l.c.Kähler manifolds.
REFERENCES
[1] G. G a n c h e v, On Bochner curvature tensors in almost Hermitian manifolds, Pliska,
Studia math. Bulgarica., 9 (1987), 33–43.
[2] A. G r a y, L. H e r v e l l a, The sixteen classes of almost Hermitian manifolds and
their linear invariants, Annali di Math. pura ed applicata, (IV), CXXIII, 1980, 35–58.
[3] T. K a s h i w a d a, Some properties of locally conformal Kähler manifold, Hokkaido
Math. J., 8 (1979), 191–198.
[4] S. K o b a y a s h i, K. N o m i z u, Fondation of differential gormetry,vol. II,
Interscience Publishers, New York - London, 1969
[5] K. M a t s u m o t o, Locally conformal Kähler manifolds and their submanifolds,
Mem. Secţ. Stiinţ. Acad. Românâ, Ser. IV, 14, 1991, No. 2, 7–49.
[6] J. M i k e š, V. K i o s a k, A. V a n ž u r o v a, Geodesic mappings of manifolds with
affine connection, Palasky Univ. Olomouc, Faculty of Science, 2008.
[7] M. P r v a n o v i ć, On the curvature tensors of Kähler type in an almost Hermitian
and almost para-Hermitian manifold, Matematički vesnik, 50 (1998), 57–64.
[8] K. Y a n o, Differential geometry on complex and almost complex spaces, Pergamon
Press, Oxford, 1965.
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