FACTA UNIVERSITATIS
Series: Mechanics, Automatic Control and Robotics Vol.3, No 14, 2003, pp. 791 - 804
Invited Paper
LOCALLY CONFORMALLY KÄHLER MANIFOLDS
OF CONSTANT TYPE AND J-INVARIAN CURVATURE TENSOR
UDC 514.763.4+514.74+514.743.4
Mileva Prvanović
Serbian Academy of Sciences and Arts
11000 Beograd, Knez Mihailova 35
Abstract. We find the metrics of all locally conformally Kähler manifolds of constant type
and J-invariant curvature tensor and discuss the corresponding conformal invariants.
Key words: locally conformally Kähler manifold, almost Hermitian manifold of
constant type, J-invarian curvature tensor, Bochner and generalized
Bochner tensor, complex concircular curvature tensor.
1. ALMOST HERMITIAN MANIFOLD OF CONSTANT TYPE
AND J-INVARIANT CURVATURE TENSOR
Let (M,g,J) be an almost Hermitian manifold, where g is a Reimian metric and J is a
complex structure. Than
dimM = 2n , J 2 = −ident. , g (JX, JY ) = g ( X, Y)
(1.1)
∞
∈X(M), where X(M) is the Lie algebra of C vector fields on M. If
for all X,Y∈
F(X,Y) = g(JX,Y), then F(X,Y) = − F(X,Y).
The two-dimensional plane π of the tangent bundle Tp(M) at "p ∈ M" is holomorphic
if Jπ = π and antyholomorphic if Jπ ⊥ π.
The Kähler manifold is an almost Hermitian manifold satisfying the condition
∇J = 0 where ∇ denotes the operator of the covariant derivative with respect to the LeviCivita connection Γ on (M,g). The condition ∇J = 0 implies
(1.2)
R ( X, Y, JZ, JW ) = R ( X, Y, Z, W ) .
where R(X,Y,Z,W) is the Reimian curvature tensor of the metric g. The relation (1.2) is
the Kähler identity.
Received May 20, 2003
AMS Mathematics Subject Classification (1991): 53 C 25, 53C55.
792
M. PRVANOVIĆ
An almost Hermitian manifold is said to be of J-invariant curvature tensor if
R (JX, JY, JZ, JW ) = R ( X, Y, Z, W ) .
(1.3)
Such manifolds, called also RK-manifolds, are investigated in [6].
To define the constant type of an almost Hermitian manifold, we first consider the
tensor
λ( X, Y, Z, W ) = R ( X, Y, Z, W ) − R (X, Y, JZ, JW) .
So λ measures the defect from the Kähler identity. Now, we put
λ ( X, Y ) = λ ( X, Y , Y , X )
and say that an almost Hermitian manifold (M,g,J) is of constant type at p ∈ M provided
that for all X ∈Tp(M), we have
λ ( X, Y ) = λ ( X, Z )
(1.4)
whenever the planes {X, Y} , {X, Z} are antyholomorphic and
g ( X, Y ) = g ( X, Z ) = 0 , g ( Z , Z ) = g ( Y , Y ) ,
Y,Z ∈ Tp(M). If (1.4) hold for all p ∈ M, (M,g,J) has pointwise constant type. (M,g,J)
has global constant type if (1.4) is the constant function.
Now, let us put
L( X, Y, Z, W ) = R ( X, Y, Z, W ) − α[g( X, W )g(Y, Z) − g( X, Z)g ( X, W )] .
(1.5)
The following theorem is proved in [7].
The tensor (1.5) satisfies the Kähler identity
L( X, Y, Z, W ) = L( X, Y, JZ, JW )
if and only if the tensor R satisfies the following conditions
a) R (JX, JY, JZ, JW ) = R ( X, Y, Z, W)
b) R has constant type.
Thus, if the curvature tensor is J-invariant curvature tensor and has constant type,
then
R ( X, Y, Z, W ) − R ( X, Y, JZ, JW) =
(1.6)
= α[g( X, W)g(Y, Z) − g( X, Z)g(Y, W ) − F( X, W)F(Y, Z) + F( X, Z)F (Y, W )].
With respect to the local coordinates, (1.6) can be expressed as follows
R kjih − R kjab J ia J bh = α [g kh g ji − g ki g jh − Fkh F ji + Fki F jh ] .
(1.7)
2. LOCALLY CONFORMALLY KÄHLER MANIFOLDS
An almost Hermitian manifold (M, g, J ) is a locally conformally Kähler manifold if
the metric g is conformally related to the metric g of a Kähler manifold (M, g, J ) , i.e.
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
793
g ij = e 2 f g ij
(2.1)
where f = f ( x i ) is a scalar function.
Denoting by ∇ the operator of the covariant derivative with respect to the LeviChivita connection of the metric g , we find
∇ j J is = ∇ j J is + δ sj f a J ia + g ji f a J as − J sj f i − Fij f s
∂f
, f s = g st f t , Fij = J it g tj .
∂x i
Thus, if (M, g, J ) is a Kähler manifold, i.e. if ∇ j J is = 0 , then
where f i =
∇ j J is = δ sj f a J ia + g ji f a J as − J sj f i − Fij f
s
(2.2)
The relation (2.2) characterizes the locally conformally Kähler manifolds [2].
From (2.2), using the Ricci identity, we get:
R kjih − R kjab J ia J hb = − g ji Pkh + g ki Pjh − g kh Pji + g jh Pki − F ji Pka J ha +
+ Fki Pja J ha − Fkh Pja J ia + F jh Pka J ia
where
Pkh = ∇k f h + f k f h −
1
g kh f t f t
2
(2.3)
and R is the Riemannian curvature tensor of the metric g .
Thus and in view of (1.7) (M, g, J ) is locally conformally Kähler manifold of
constant type and J-invariant curvature tensor if and only if
− g ji Pkh + g ki Pjh − g kh Pji + g jh Pki − F ji Pka J ha + Fki Pja J ha − Fkh Pja J ia + F jh Pka J ia =
= α[ g kh g ji − g ki g jh − Fkh F ji + Fki F jh ]
(2.4)
Transvecting (2.4) with g ji and taking into account that Pja is symmetric, while
J ia g ji is skew-symmetric, we find
− (2n − 3) Pkh + Pab J ka J hb − g kh Pji g ji = 2(n − 1)αg kh
(2.5)
Transvecting (2.5) with g kh , we get:
Pkh g kh = − nα ,
because of which (2.5) reduces to:
− (2n − 3) Pkh + Pab J ka J hb = (n − 2)αg kh
The relation (2.6) implies
Pkh − (2n − 3) Pab J ka J hb = (n − 2)αg kh ,
which, together with (2.6), gives
−2(n − 1)(n − 2) Pkh = (n − 1)( n − 2)αg kh
(2.6)
794
M. PRVANOVIĆ
Thus, if n > 2 , we have
α
g kh
2
Conversely, if (2.7), (2.4) is identically satisfied.
In view of (2.7), (2.3) becomes
Pkh = −
∇k f h + f k f h = Φg kh
(2.7)
(2.8)
where
Φ=
1 st
( g f s f t − α)
2
Thus, we can state
Theorem: The relation (2.8) is the necessary and sufficient condition for locally
coformally Kähler manifold (M, g, J ) , dim M > 4 to be of constant type and have J invariant curvature tensor.
The relation (2.8) is the condition with respect to the metric g . With respect to the
metric g, it is:
(2.9)
∇ k f h − f k f h = ψg kh
1
ψ = − (αe 2 f + g st f s f t ) .
2
Now, we shall give the simpler form of (2.9). To do this, we put
1
f = log .
G
Then
∂G
∂f
G
1
f h = h = − h , ∇ k f h − f k f h = − ∇ k Gh , Gh = h ,
G
G
∂x
∂x
and the condition (2.9) obtains the form:
∇ k Gh = ϕg kh , (ϕ = −Gψ )
(2.10)
Thus, to find the metric g of a locally conformally Kähler manifold of constant type
and J-invarian curvature tensor, we have to find the metric g of the Kähler manifold
admiting the scalar function G such that (2.10) holds.
It is known ([8], p.95) that the Riemannian manifold (M,g), dim M = m , admitting
the scalar function G satisfying (2.10) has, with respect to the suitable local coordinates,
the metric of the form
ds 2 = (dx1 ) 2 + Q( x1 ) g~αβ dx α dx β ,
(2.11)
∂g~αβ
= 0,
where Q(x1) is nonconstant function of x1 only and g~αβ dx α dx β ,
∂x1
~
g) .
α, β, γ, δ = 2,3,..., m is a metric of (m − 1) - dimensional Riemannian manifold (M, ~
Indeed, with respect to the metric (2.11), the components of the Levi-Chivita
connection are:
1
α
Γ11
= Γα11 = Γ11
=0,
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
1
Γαβ
=−
Q′ ~
g αβ ,
2
Q′ β
δα ,
2Q
Γ1αβ =
795
~α
α
Γβγ
= Γβγ
~α
are the components of the Levi-Chivita connection with respect to the metric g~ .
where Γβγ
Then for
G=A
∫
Q dx1 + B ,
∇ k Gh =
we have
A, B const
A Q′
g kh ,
2 Q
and this is just the condition (2.10).
The metric (2.11) is the metric of a Kähler manifold if and only if [3], dim M = 2n ,
~
~
Q = (x1 ) 2 and (M, ~
g ) is a Sasakian manifold. If (ϕ, ξ, η, g ) is the Sasakian structure of
~
the manifold M , then the complex structure of the Kähler manifold (2.11) is given by
1
J11 = 0 , J1α = 1 ξα , J α1 = − x1ηα , J αβ = ϕβα
(2.12)
x
Thus, the metric of any locally conformally Kähler manifold (M, g, J ) , dim M > 4, of
constant type and J-invariant curvature tensor has, with respect to the suitable local
coordinates, the form
ds 2 = e 2 f ds 2 =
1

A 1 2
 2 ( x ) + B


2
[(dx1 ) 2 + ( x1 ) 2 g~αβ dx α dx β ]
(2.13)
where g~αβ dx α dx β is the metric of a (2n − 1) - dimensional Sasakian manifold.
Conversely, ds 2 being the metric of a Kähler manifold, (2.13) is the metric of locally
conformally Kähler manifold. Next, the components of the Levy-Civita connection Γ of
(2.13) are the folowing:
~α
Ax1
1
, Γ11α = Γ11α = 0 , Γβγα = Γβγ
Γ11
=−
A 1 2
(x ) + B
2




1
1 3

 α


Ax
1
(
)
A
x
1
~
Γ1αβ =  1 −
Γαβ
= − x1 +
 δβ
 g αβ ,
A
A
x

( x1 ) 2 + B 
( x1 ) 2 + B 
2
2




Thus, for
1
f = log
,
A 1 2
(x ) + B
2
A
Ax1
we have
,
f α = 0, ∇1 f1 = −
,
f1 = −
A 1 2
A 1 2
(x ) + B
(x ) + B
2
2
1

1 2
∇1 f α = 0 , ∇β f α =  ( Ax ) − AB  g~αβ ,
2


796
M. PRVANOVIĆ
and therefore
1

∇ k f h + f k f h =  ( Ax1 ) 2 − AB  g~kh .
2


But this is just the condition (2.8).
Thus we can state
Theorem: The metric
d~
s2 =
1

A
2
 2 ( x1 ) + B 


2
[(dx1 ) 2 + ( x1 ) 2 g~αβ dx α dx β ]
(2.13)
∂g~αβ
= 0 , α, β = 2,3,...,2n is the metric of a
where A and B are constants, g~αβ dx α dx β ,
∂x1
(2n − 1)-dimensional Sasakian manifold, is the metric of locally conformally Kähler
manifold of constant type and J-invariant curvature tensor. Conversely, any locally
conformally Kähler manifold of dimension 2n > 4, of constant type and J-invariant
curvature tensor has, with respect to some local coordinates (xi), the form (2.13).
3. HOLOMORPHIC CURVATURE TENSOR
Let (M,g,J) be an almost Hermitian manifold. Because of JX⊥X , any vector X
determines the holomorphic section. The sectional curvature with respect to the
holomorphic section,
R ( X, JX, JX, X)
H (X ) =
g ( X, X) 2
is the holomorphic sectional curvature.
We define the holomorphic curvature tensor (HR )( X, Y, Z, W ) as follows (see for
ex. [1],[4])
(HR )( X, Y, Z, W ) =
=
1
{3[R ( X.Y, Z, W ) + R (JX, JY, Z, W ) + R ( X, Y, JZ, JW ) + R (JX, JY, JZ, JW )] −
16
− 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)}
(HR )( X, Y, Z, W )
It is easy to see that (HR )( X, Y, Z, W ) is an algebraik curvature tensor, i.e. that
(HR )( X, Y, Z, W ) = −(HR )(Y, X, Z, W ) = −(HR )( X, Y, W, Z ) ,
(HR )( X, Y, Z, W ) = (HR )( Z, W, X, Y) ,
(HR )( X, Y, Z, W ) + (HR )(Y, Z, X, W ) + (HR )(Z, X, Y, W ) = 0
But the tensor (3.1) has also the following remarkable properties
(HR )( X, Y, JZ, JW ) = (HR )( X, Y, Z, W ) ,
(3.2)
(HR )( X, JX, JX, X) = R ( X, JX, JX, X) ,
(3.3)
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
797
The relation (3.3) shows that the holomorphic sectional curvatures with respect to R
and HR are the same. This is the reason to name HR the holomorphic curvature tensor.
The relation (3.2) shows that although the Riemannian curvature tensor of an almost
Hermitian manifold in general does not satisfy the Kähler identity (1.2) , nevertheless
there exists the tensor satisfying it: this is the tensor (3.1).
We underline that if (1.2) holds, then
(HR )( X, Y, Z, W) = R ( X, Y, Z, W)
(3.4)
Thus, for the Kähler manifolds, the holomorphic curvature tensor reduces to the
Riemannian curvature tensor.
If (M, g, J ) is of J-invariant curvature tensor, i.e. if (1.3) holds, then
(HR )( X, Y, Z, W ) =
1
= {3[R ( X.Y, Z, W ) + R ( X, Y, JZ, JW )] − R ( X, Z, JW, JY ) −
8
− R ( X, W, JY, JZ ) + R (JX, Z, JW, Y ) + R (JX, W, Y, JZ )}
If besides this (M, g, J ) is of constant type,i.e. if (1.6) holds, then
α
{3[ 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 ) = R ( X, Y, Z, W ) −
(3.5)
We define the Ricci tensor and Ricci tensor of (M, g, J ) by
ρ( x, y ) =
∗
∑ R (ei , X, Y, ei )
ρ( x, y ) =
∑ R (ei , X, JY, Jei )
i
i
where {e i } is an orthogonal basis. They have the properties:
∗
∗
ρ( X, JY ) = − ρ(Y, JX) .
ρ( X, Y ) = ρ(Y, X)
If (M,g,J) is a Kähler manifold, then
∗
ρ(JX, JY ) = ρ( X, Y )
ρ( X, Y ) = ρ( X, Y) .
If (M,g,J) is J-invariant and has the constant type,i.s. if (1.6) holds, than
*
ρ( X, Y) = ρ( X, Y) + 2(n − 1)α g ( X, Y)
The scalar curvature and * - scalar curvature are defined by
τ=
∑ ρ(ei , ei ) ,
i
∗
τ=
∗
∑ ρ(ei , e i )
i
∗
For the Kähler manifolds τ = τ .
We obtain from (3.6)
α=
∗
1
( τ − τ)
4n(n − 1)
(3.6)
798
M. PRVANOVIĆ
Therefore, if (M,g,J) is J-invariant and has the constant type, the Ricci tensor and *Ricci tensor are related as follows
∗
ρ( X, Y ) = ρ( X, Y) +
∗
1
(τ − τ)g ( X, Y)
2n
(3.7)
while (3.5) has the form
∗
(τ − τ)
(HR )( X, Y, Z, W ) = R ( X, Y, Z, W) −
{3[ g ( X, W) g (Y, Z) − g ( X, Z) g (Y, W )] − (3.8)
16n(n − 1)
− [F ( X, W)F (Y, Z) − F( X, Z)F(Y, W ) − 2F( X, Y)F(Z, W)]}
In the same manner, we define Ricci tensor, *-Ricci tensor, scalar curvature and
*-scalar curvature associated with holomorphic curvature tensor, namely,
∑ (HR )(ei , X, Y, e i )
ρ(HR )( X, Y) =
i
∗
ρ(HR )( X, Y ) =
∑ (HR )(e i , X, JY, Je i )
(3.9)
i
τ(HR ) =
∑ ρ(HR )(ei , ei ) ,
i
∗
τ(HR ) =
∗
∑ ρ(HR )(ei , ei )
i
Because (HR) is an algebraic curvature tensor satisfying the Kähler identity, we have
∗
ρ(HR )( X, Y) = ρ(HR )(Y, X) = ρ(HR )(JX, JY ) = ρ(HR )( X, Y )
∗
τ(HR ) = τ(HR )
Explicitly,
∗
∗
1
ρ(HR )( X, Y ) = [ρ( X, Y ) + 3 ρ( X, Y ) + 3 ρ(Y, X) + ρ(JX, JY )] ,
8
∗
1
τ(HR ) = [τ + 3 τ]
4
For the Kähler manifolds we have
ρ(HR )( X, Y ) = ρ( X, Y )
(3.10)
τ(HR ) = τ
(3.11)
∗
1
[ρ( X, Y ) + 3 ρ( X, Y )]
4
(3.12)
For (M,g,J) satisfying (1.6), we have
ρ(HR )( X, Y) =
or, taking into account (3.7)
ρ(HR )( X, Y) = ρ( X, Y) +
3 ∗
( τ − τ ) g ( X, Y )
8n
(3.13)
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
799
4. THE GENERALIZED BOCHNER CURVATURE TENSOR
Let us consider an almost Hermitian manifold (M,g,J), and conformal transformation
(2.1) of its metric. Then
( ∇ − ∇)( X, Y) = ω( X)Y + ω(Y ) X − g ( X, Y)U
for any vector fields X, Y, where ω is 1-form defined by ω = df and U is a vector field
satisfying g(U,X) = ω(X).
It is well known that the Riemannian curvature tensors of the metrics g and g
respectively, are related as follows
e − 2 f R ( X, Y , Z , W ) = R ( X, Y , Z , W ) + g ( X, W ) σ ( Y , Z ) + g ( Y , Z ) σ ( X, W ) −
− g ( X, Z)σ(Y, W) − g (Y, W )σ( X, Z)
where σ is the tensor field of type (0,2) defined by
σ( X, Y ) = (∇ x ω)Y − ω( X)ω(Y ) +
1
ω(U)g( X, Y)
2
(4.1)
We note that σ(X,Y) = σ(Y,X). As for the corresponding holomorphic curvature
tensors, we have, in view of (3.1)
8e −2 f (HR )( X, Y, Z, W ) = 8( HR )( X, Y, Z, W ) + g ( X, W )S( Y, Z) + g (Y, Z)S( X, W ) −
− g( X, Z)S( Y, W ) − g( Y, W )S( X, Z) + F ( X, W )S(JY, Z) + F (Y, Z)S(JX, W ) −
− F( X, Z)S( JY, W ) − F (Y, W )S( JX, Z) − 2F ( X, Y )S( JZ, W ) − 2F ( Z, W )S(JX, Y )
where
(4.3)
S( X, Y ) = σ( X, Y ) + σ(JX, JY )
Therefore
S(JX, Y ) = σ(JX, Y) − σ( X, JY )
Putting into (4.2) X = W = ei , summing up and taking into account (3.9), we obtain
8ρ(HR )( Y, Z) = 8ρ(HR )(Y, Z) + 2(n + 2)S( X, Z) + g (Y, Z)
∑ S(ei , e i )
(4.4)
i
Putting into (4.4) Y = Z = ei and summing up, we get
2
∑ S(e i , ei ) = n + 2 [e 2 f τ(HR ) − τ(HR )] ,
(4.5)
i
because of which (4.4) reduces to
S( Y , Z ) =
e 2 f τ( H R ) − τ( H R )
4
g (Y, Z )
[ρ(HR )(Y, Z) − ρ( HR )(Y, Z)] −
n+2
(n + 1)(n + 2)
(4.6)
Thus, (4.2) becomes
e −2 f B(HR )(X, Y, Z, W) = B(HR)( X, Y, Z, W)
where
(4,7)
800
M. PRVANOVIĆ
B(HR )( X, Y, Z, W) =
1
{g ( X, W)ρ(HR )( Y, Z) + g( Y, Z)ρ( HR )( X, W) −
2(n + 2)
− g( X, Z )ρ( HR )(Y, W) − g(Y, W )ρ(HR )( X, Z) + F ( X, W )ρ(HR )(JY, Z) +
+ F (Y, Z)ρ( HR )(JX, W) − F( X, Z)ρ( HR )(JY, W) − F( Y, W)ρ( HR )(JX, Z) −
− 2F ( X, Y)ρ( HR )(JZ, W ) − 2F (Z, W)ρ(HR )( JX, Y)} +
(HR )( X, Y, Z, W) −
(4.8)
τ( HR )
{g ( X, W)g (Y, Z) − g( X, Z)g (Y, W) + F( X, W)F ( X, Z) −
4( n + 1)(n + 2)
− F ( X, Z)F (Y, W) − 2F( X, Y)F( Z, W)}
+
and B(HR ) is constructed in the same way but using the tensor HR .
Thus, we see that: For any almost Hermitian manifold, the tensor (4.8) satisfies (4.7).
The tensor (4.8) is the generalized Bochner curvature tensor. In [1], it is obtained
applying the decomposition on a 2n-dimensional Hermitian vector space into orthogonal
components under the action of the unitary group. Here, we obtain it using the classical
way.
For the Kähler manifolds, (3.4) and (3.11) hold, because of which (4.8) become the
well known Bochner curvature tensor.
If (M,g,J) is J-invariant and has the constant type, then (3.8), (3.10) and (3.12) hold,
and (4.8) reduces to
∗
1
B( HR )( X, Y, Z, W ) = R ( X, Y, Z, W ) −
{g ( X, W )[ρ( Y, Z) + 3 ρ( Y, Z)] +
8(n + 2)
∗
∗
+ g ( Y, Z)[ρ( X, W ) + 3 ρ( X, W )] − g ( X, Z)[ρ(Y, W ) + 3 ρ( Y, W )] −
∗
∗
− g ( Y, W )[ρ( X, Z) + 3 ρ( X, Z )] + F ( X, W )[ρ( JY, Z ) + 3 ρ( JY, Z )] +
∗
∗
∗
∗
+ F ( Y, Z )[ρ( JX, W ) + 3 ρ(JX, W )] − F ( X, Z)[ρ( JY, W ) + 3 ρ(JY, W )] −
− F (Y, W )[ρ(JX, Z ) + 3 ρ( JX, Z )] − 2F ( X, Y )[ρ( JZ, W ) + 3 ρ(JZ , W )] −
∗
∗
τ + 3τ
{g( X, W)g (Y, Z) −
16(n + 1)(n + 2)
− g( X, Z)g (Y, W) + F( X, W )F(Y, Z) − F( X, Z)F(Y, W ) − 2F( X, Y)F(Z, W) +
− 2F(Z, W )[ρ(JX, Y) + 3ρ(JX, Y )] +
∗
τ− τ
{3[g ( X, W )g(Y, Z) − g( X, Z)g (Y, W )] −
16n(n − 1)
− [F( X, W)F(Y, Z) − F( X, Z)F(Y, W ) − 2F( X, Y)F(Z, W )]}
+
This is just the tensor find in [5], but using the decomposition of the vector space.
In view of (3.13), (4.9) can be rewritten in the form
(4.9)
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
801
1
{g ( X, W)ρ(Y, Z) + g(Y, Z)ρ( X, Z)
2(n + 2)
− g( X, Z)ρ(Y, W) − g(Y, W )ρ( X, Z) + F( X, W )ρ( JY, Z) + F( Y, Z)ρ(JX, W) −
− F( X, Z)ρ(JY, W ) − F(Y, W )ρ( JX, Z) − 2F ( X, Y)ρ(JZ, W) − 2F (Z, W)ρ(JX, Y )} +
B(HR )( X, Y, Z, W) = R ( X, Y, Z, W) −
∗


1
 τ + 3τ 3 *

+
−
τ
−
τ
(
)
{g ( X, W )g (Y, Z) − g ( X, Z )g (Y, W) +
8(n + 2)  2(n + 1) 4


+ F( X, W )F(Y, Z ) − F( X, Z)F(Y, W ) − 2F( X, Y)F(Z, W )} +
∗
τ− τ
+
{3[g ( X, W )g (Y, Z) − g ( X, Z )g (Y, W )] −
16n(n − 1)
− [F( X, W )F(Y, Z ) − F( X, Z)F(Y, W ) − 2F( X, Y)F(Z, W )]}
The components of the curvature tensor of the metric (2.13) which do not vanish, are
the following
R α11β = LPg~αβ
~
R αβγδ = ( x1 ) 2 L{R αβγδ + ( P − 1)( g~αδ g~βγ − g~αγ g~βδ )}
(4.11)
Where
A2 ( x1 ) 4
2 A( x1 ) 2
−
.
2
2
A 1 2
 A 1 2
A 1 2


+
x
B
(
)
2
  ( x ) + B
 2 ( x ) + B

 2



The Ricci tensor ρ of the metric (2.13) has the components
(2n − 1) P
ρ11 =
ρ1α = 0
,
( x1 ) 2
ραβ = ~
ραβ + [( 2n − 1) P − 2(n − 1)]g~αβ
~
ραβ is the Ricci tensor of the Sasakian manifold M . For the scalar curvature τ
where ~
L=
1
P=
,
∗
and *-scalar curvature τ , we have
1
τ = 1 2 [~τ + 2n(2n − 1) P − 2(n − 1)(2n − 1)]
(x ) L
(4.12)
∗
1
[~τ + 2nP − 2(n − 1)(2n − 1)]
(4.13)
(x ) L
~
where ~τ is the scalar curvature of the Sasacian manifold M .
Substituting this into the formula corresponding to (4.10) for the metric g , we find
τ=
1 2
that the components of the generalized Bochner curvature tensor B(HR ) of the metric
(2.13), which do not vanish, are the following
802
M. PRVANOVIĆ
2
~τ
1
A 1 2

~
~
+
x
B
(
)
2
 B(HR ) α11β = − 2(n + 2) ραβ + 4(n + 1)(n + 2) ( g αβ + 3ηα ηβ ) +


(4.14)
3(n − 1)
~
+
[ g αβ − (2n − 1)ηα ηβ ]
2(n + 1)(n + 2)
2

1
A 1 2

σ~
σ~
σ~
1
 2 ( x ) + B  B(HR )1αβγ = x − 2(n + 2) (ηγ ϕ α ρσβ − ηβ ϕα ρσγ − 2ηα ϕβ ρσγ )  +




~
1 τ + 6(n − 1)
[η γ ϕαβ − ηβ ϕαγ − 2ηα ϕβγ ]
+x
4(n + 1)(n + 2)
2
A 1 2

1 2 ~
~ ~
~ ~
 2 ( x ) + B  B(HR ) αβγδ = ( x ) {R αβγδ − ( g αδ g βγ − g αγ g βδ )} −


−
( x1 ) 2 ~ ~
~ − g~ ~
~ ~
[ g αδ ρβγ + g~βγ ρ
αδ
αγ ρβδ − g βδ ραγ ] +
2(n + 2)
( x1 ) 2
ρλγ + ϕβγ ϕλα ~
ρλδ − ϕ αγ ϕβλ ~
ρλδ − ϕβδ ϕ λα ~
ρλγ − 2ϕ αβ ϕλγ ~
ρλδ − 2ϕ γδ ϕ λα ~
ρλβ ] +
[ϕ αδ ϕβλ ~
2(n + 2)
~τ + 2(n − 1)(2n + 5)
+ ( x1 ) 2
[ g~αδ g~βγ − g~αγ g~βδ + ϕ αδ ϕβγ − ϕ αγ ϕβδ − 2ϕ αβ ϕ γδ ]
2(n + 1)(n + 2)
where ϕαβ = ϕλα g~λβ .
−
But the expression of the right hand sides of (4.14) are the components Bα11β, B1αβγ,
Bαβγδ of the Bochner tensor of the Kähler metric (2.11) (Q = (x1)2), respectively, that is,
we have
e −2 f B(HR )( X, Y, Z, W) = B( X, Y, Z, W)
as was to be expected.
5. CONCIRCULAR TRANSFORMATIONS
A geodesic circle of a Riemannian manifold (M,g), dim M = m is defined as a curve
whose first curvature is constant and other are identically zero. If a conformal
transformation (2.1) transforms every geodesicx circle unto geodesic circle, then the
function f must satisfy the partial differential equation [9]
σ ( X, Y ) = θ g ( X, Y )
(5.1)
where σ( X, Y ) is defined by (4.1) and θ is a scalar function.
The conformal transformation satisfying (5.1) is called concircular one. The tensor
τ
(5.2)
C( X, Y, Z, W ) = R ( X, Y, Z, W ) −
[g ( X, W )g (Y, Z) − g ( X, Z)g (Y, W )]
m(m − 1)
satisfies the following relation
e −2 f C ( X, Y, Z, W) = C( X, Y.Z.W)
and is called concircular curvature tensor [9].
Locally Conformally Kähler Manifolds of Constant Type and J-Invarian Curvature Tensor
803
Now, we consider an almost Hermitian manifold and tray to find the complex
concircular curvature tensor, i.e. the complex analog of the tensor (5.2).
From (5.1) , it follows σ(JX, JY) = θ g ( X, Y) . Therefore
S( X, Y) = σ( X, Y) + σ( JX, JY) = 2θ g ( X, Y) ,
because of which (4.2) becomes
2e −2 f ( HR )( X, Y, Z.W ) = 2(HR )( X, Y, Z.W ) +
+ θ [g ( X, W)g( Y, Z) − g ( X, Z)g (Y, W ) + F( X, W )F ( Y, Z) − F ( X, Z)F (Y, W ) − 2F( X, Y )F( Z, W)]
From (5.3), we find
ρ(HR )(Y, Z) = ρ(HR )(Y, Z) + (n + 1)θ g(Y, Z)
and
e 2 f τ(HR ) − τ(HR )
θ=
2n(n + 1)
because of which (5.3) reduces to
e −2 f C (HR )(X, Y, Z.W) = C(HR)(X, Y, Z.W)
where
C(HR )(X, Y, Z.W) = (HR )( X, Y, Z.W) −
τ(HR )
(5.4)
−
[g ( X, W )g (Y, Z) − g ( X, Z)g (Y, W )] −
4n(n + 1)
τ(HR )
−
[F ( X, W)F(Y, Z) − F ( X, Z)F (Y, W) − 2F ( X, Y )F(Z, W)]
4n(n + 1)
Thus, (5.4) is complex concircular curvature tensor of an almost Hermitian manifold
(M, g, J ) . If (M, g, J ) is a Kähler manifold (5.4) becomes
(CK )( X, Y.Z.W) = R ( X, Y.Z.W) −
τ
−
[g ( X, W )g (Y, Z) − g( X, Z)g (Y, W )] −
4n(n + 1)
τ
−
[F ( X, W)F (Y, Z) − F ( X, Z)F (Y, W) − 2F ( X, Y)F (Z, W)]
4n(n + 1)
If (M,g,J) is of constant type and J-invariant curvature tensor, (5.4) becomes
(5.5)
C(HR )(X, Y.Z.W) = R ( X, Y.Z.W) −
∗
τ + 3τ
−
[g ( X, W)g(Y, Z) − g( X, Z)g(Y, W)] −
16n(n + 1)
∗
τ + 3τ
−
[F( X, W)F(Y, Z) − F( X, Z)F(Y, W) − 2F( X, Y)F(Z, W)] +
16n(n + 1)
∗
τ− τ
+
3[g( X, W)g(Y, Z) − g( X, Z)g (Y, W)] −
16n(n + 1)
∗
τ− τ
−
[F( X, W)F(Y, Z) − F( X, Z)F(Y, W) − 2F( X, Y)F(Z, W)]
16n(n + 1)
(5.6)
804
M. PRVANOVIĆ
We have seen in §2 that the Kähler manifold admitting the concircular transformation
has, with respect to the suitable local coordinates, the metric of the form (2.11). The
corresponding locally conformally Kähler manifold has the metric of the form (2.13).
This last metric is the metric of the manifold of constant type and J -invariant
curvature tensor. Thus, to find its complex concircular curvature tensor, we have to use
the formula (5.6). Indeed, according to (4.11), (4.12) and (4.13), the non-zero
components of the tensor (5.6) are the following
2
x1
A 1 2

C
H
R
+
=
−
(
)
(
)
[~τ − 2(n − 1)(2n − 1)](η γ ϕαβ − ηβ ϕαγ − 2ηα ϕβγ ) ,
x
B
αβγ
1
2

4n(n + 1)


2
~τ − 2( n − 1)(2n − 1)
A 1 2

+
x
B
(
)
( g~αβ + 3ηα ηβ ) ,
2
 C(HR ) α11β = −
4n( n + 1)


2
~τ + 2(5n − 1)
~
A 1 2

~ ~
~ ~ 
+
x
B
(
)
2
 C(HR ) αβγδ = R αβγδ − 4n(n + 1) ( g αδ g βγ − g αγ g βδ ) −




~τ − 2(n − 1)(2n − 1)
−
[ϕαδ ϕβγ − ϕαγ ϕβδ − 2ϕαβ ϕ γδ ].
4n(n + 1)
But, on the right hand side we have just the components (CK)1αβγ , (CK)α11β , and
(CK) αβγδ , of the tensor (5.5).
REFERENCES
1. Gancharev, G., On Bochner Curvature Tensor in Almost Hermitian Manifold, Pliska, Studia mathematica
Bulgarica 9(1987), 33-43.
2. Gray, A., Hervella, L.M., The Sixteen Classes of Almost Hermitian Manifolds and their Linear Invariants,
Annali di Mathematica pura ed applicata (IV), Vol. CXXIII(1980), 35-58.
3. Микеш , Й, О Сасакиевих и еквидистантних Келеровых простарнствах, Док. А.Н. СССР, 291, Nо.
1(1986),33-36.
4. Prvanović, M., On the Curvature Tensor of Kähler Type in an Almost Hermitian and Almost Para Hermitian
Manifold, Matematički vesnik, 50(1998), 57-64.
5. Vanhecke, L., The Bochner Curvature Tensor on an Almost Hermitian Manifold, Rend. Sem Math. Univ.
Politec. Torino, 34 (1975-1976), 21-83.
6. Vanhecke, L., Almost Hermitian Manifolds with J-invarian Riemannian Curvature Tensor, Rend. Sem
Math. Univ. Politec. Torino, 34 (1975-1976), 487-498.
7. Vanhecke, L., and Bouten F., Constant type for Almost Hermitian Manifolds, Bull. Math. de la Soc. Sci.
Math. de Roumanie, 20(68), No,3-4, (1976), 415-422.
8. Синюков, Н.С., Геодезические отображения Римановыџ пространств, Наука Москва 1979.
9. Yano, K., Concircular Geometry I, Concircular Transformations, Proc. Imp. Acad. Japan 16(1940), 195-200.
LOKALNO KONFORMNE KÄHLEROVE MNOGOSTRUKOSTI
TIPA KONSTANTE I J-INVARIJANTNOG TENZORA KRIVINE
Mileva Prvanović
U radu su odredjene metrike svih lokalno konformnih Kählerovih mnogostrukosti tipa
konstante i J-invarijantnog tenzora krivine i diskutovane odgovarajuće conformalne invarijante.
Ključne reči: Lokalno konformna Kählerova mnogostrukost, skoro Hermitian mnogostrukost tipa
konstante, J-invarijantni tenzor krivine, Bochnerov i generalisani Bochner-ov
tenzor, kompleksni kocirkularni tenzor krivine.