Rend. Istit. Mat. Univ. Trieste
Vol. XXXVIII, 1–15 (2006)
A General View to Classification of
Almost Hermitian Manifolds
Mehmet Tekkoyun
(∗)
Summary. - In this paper, firstly, we obtain the higher order vertical and complete lifts of type (1,1) and type (0,2) by means of
lifted geometric structures on extended complex manifolds. Then
using higher order lifting theory, we generalize the classification
of almost Hermitian manifolds M to the extended complex manifolds k M .
1. Introduction and notations
In studies before, the classes of almost Hermitian manifolds M had
been given as follows:
Kähler manifolds
Nearly Kähler manifolds
Almost Kähler manifolds
Quasi Kähler manifolds
Hermitian manifolds
Semi Kähler manifolds
:
:
:
:
:
:
[∇X J0 ] Y = 0
[∇X , J0 ] X = 0
dΦ = 0
[∇J0 X , J0 ] = −J0 [∇X , J0 ]
[∇
PJm0 X , J0 ] = J0 [∇X , J0 ]
{∇Ei (J0 )Ei + ∇J0 Ei (J0 )J0 Ei } = 0
i=1
for all X, Y ∈ χ(M ), where χ(M ) denotes the lie algebra of C ∞
vector fields on M, ∇ the Riemannian connection, J0 the almost
complex structure, Φ the fundamental 2-form, and {Ei , J0 Ei } a local orthonormal frame field [1, 2]. Lifts of complex structures on
Hermitian and Kähler manifolds and the first order lift classes of
the classes given in [4] were studied.
(∗)
Author’s address:
Mehmet Tekkoyun, Department of Mathematics, Faculty of Arts and Sciences ,
Pamukkale University, 20070 Denizli, Turkey; E-mail: [email protected]
Keywords: Complex Structure, Complex Manifold,Extensions,Lifting Theory
AMS Subject Classification: 32Q60, 32Q35, 20K35, 28A51
2
MEHMET TEKKOYUN
The paper is structured as follows. In section 2, firstly we recall extended complex manifold and give higher order vertical and
complete lifts of differential objects a complex manifold to extended
complex manifold [3, 5]. Then higher order lifts of an almost complex structure will be obtained. In sections 3 and 4, we obtain the
vertical and complete lifts of complex tensor fields of type (1,1) and
type (0,2) defined on any complex manifold M to extended complex
manifolds. In section 5, we give definitions about higher order vertical and complete lifts of a Hermitian metric and Kähler form defined
on a complex manifold M to its extensions.
In section 6 it is generalized the classification of almost Hermitian
manifolds M to the extended complex manifolds k M.
Along this study, all mappings and manifolds will be assumed to
be of class C ∞ and the sum is taken over repeated indices. Also,
v (resp. c) denotes the vertical (resp. complete) lift of any differentiable geometric complex structure defined on extended complex
manifolds k−1 M to k M .
2. Preliminaries
2.1. Extended complex manifolds
Let k M be extension of order k of manifold M. A tensor field Jk
on k M is called an extended almost complex structure on k M if
at every point p of k M, Jk is endomorphism of the tangent space
Tp (k M ) such that (Jk )2 = −I. An extended manifold k M with
fixed extended almost complex structure Jk is called extended almost
complex manifold. If k = 0, J0 is called almost complex structure
and the manifold M with fixed almost complex structure J0 is called
almost complex manifold.
Let (xri , y ri ) be a real coordinate system on a neighborhood k U
of any point p of k M . In this case, it is respectively defined by
{ ∂x∂ri p , ∂y∂ri } and { dxri p , dy ri p } natural bases over R of tangent
p
space Tp (k M ) and cotangent space Tp∗ (k M ) of k M.
Let k M be extended almost complex manifold with fixed extended almost complex structure Jk . Then k M n
is called
extended
o
k
k
complex manifold if there exists an open covering U of M sat-
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 3
isfying the following condition: There is a local coordinate system
(xri , y ri ) on each k U, such that
Jk (
∂
∂
∂
∂
) = ri , Jk ( ri ) = − ri ,
ri
∂x
∂y
∂y
∂x
(1)
for each point of k U.
If k = 0, then a manifold M with fixed canonical almost complex
structure J0 is called complex manifold.
Let (z ri , z ri ) be an extended complex local coordinate system on
a√ neighborhood k U of any point p of k M where z ri = xri +i y ri , i=
−1. The real dimension of k M is equal to 2(k+1)m. Therefore, these coordinates are locally defined by xri , y ri :k M →
R(k+1)m ,∀A, A ∈k M, A = ari +i bri , xri (A) = ari , y ri (A) = bri
and z ri , z ri :k M → C(k+1)m , z ri (A) = A, z ri (A) = A.
We define the vector fields
∂
∂
1 ∂
∂
1 ∂
∂
= {
− i ri }, ri = { ri + i ri }
∂z ri p 2 ∂xri p
∂y p ∂z p 2 ∂x p
∂y p
(2)
and the dual covector fields
dz ri = dxri + i dy ri , dz ri = dxri − i dy ri p
p
p
p
p
p
(3)
which represent bases of the tangent space Tp (k M ) and cotangent
space Tp∗ (k M ) of k M respectively. Then the endomorphism Jk is
given as:
∂
∂
∂
∂
Jk ( ri ) = i ri , Jk ( ri ) = −i ri
(4)
∂z
∂z
∂z
∂z
In this study, we recall the higher order vertical and complete
lifts of functions, vector fields, 1-forms on M to k M given in [3].
Then we obtain higher order vertical and complete lifts of complex
tensor fields of type (1,1) and of type (0,2) defined on M to k M.
Throughout the paper, all mappings and manifolds are assumed
to be differentiable of class C ∞ and the sum is taken over repeated
indices.
4
MEHMET TEKKOYUN
2.2. Higher order lifts of complex structures
In this section, we recall extensions of definitions and properties
about vertical and complete lifts of complex geometrical structures
defined on a complex manifold M to extended manifolds k M .
The vertical lift of function f ∈ F( M ) to k M is the function
k
f v on k M such that
k
f v = f ◦ τ0 M ◦ τ1 M ◦ ... ◦ τk−1 M .
(5)
where, for any h ∈ {1, 2, ..., k − 1}, τh : T (h M ) →h M denotes
k
the canonical projection. The function f c ∈ F(k M ) such that
k−1
f
k−1
. ri ∂f c
∂f c
v
)v .
)
+
=z (
z
(
∂z ri
∂z ri
. ri
ck
(6)
is called complete lift of function f to k M. The vertical and
k
k
complete lifts of Z to k M are the complex vector fields Z v and Z c
on k M such that
k
k
k
k
k
Z v (f c ) = (Zf )v and Z c (f c ) = (Zf )c
k
(7)
Proposition 2.1. Let M be a complex manifold and k M its k order
extended complex manifold. If the complex vector field Z defined on
M is taken by
∂
0i ∂
,
Z = Z 0i 0i + Z
∂z
∂z 0i
the vertical and complete lifts of Z to
Zv
k
=
(Z 0i )v
and
Zc
k
=
k
k
M are
∂
0i k ∂
+ (Z )v
ki
∂z
∂z ki
!
(8)
!
k−r r ∂
k
k
0i k−r r ∂
(Z 0i )v c
+
(Z )v c
.
ri
r
r
∂z
∂z ri
The vertical and complete lifts of ω ∈ χ∗ ( M ) to
k
k
complex 1-forms ω v and ω c on k M determined by
k
k
k
k
k
k
k
ω v (Z c ) = (ωZ)v and ω c (Z c ) = (ωZ)c .
M are the
(9)
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 5
Proposition 2.2. Let k M be extension of order k of complex manifold M . If the complex 1-form ω defined on M is given
ω = ω0i dz 0i + ω 0i dz 0i ,
the vertical and complete lifts of ω to
ωv
k
k
M are
k
k
(ω0i )v dz 0i + (ω 0i )v dz 0i
=
and
ω
ck
(10)
ck−r vr
=
(ω0i )
ck−r vr
dz ri + (ω 0i )
dz ri .
3. Higher order lifts of complex tensor fields of type
(1,1)
In this section, we obtain the definitions and properties about vertical
and complete lifts of a complex tensor field of type (1,1) defined on
any complex manifold M to extended complex manifolds k M .
Let M be any complex manifold and k−1 M its (k − 1) order
extended complex manifold. Denote by Fe a complex tensor field of
type (1,1) and by Ze be a complex vector field defined on k−1 M. Then
the vertical lift of a complex tensor field of type (1,1) Fe ∈ ℑ11 (k−1 M )
to k M is the structure Fe v ∈ ℑ11 (k M ) such that
e v.
Fe v (Ze c ) = (Fe Z)
(11)
k−1
k−1
and F v be respectively complete and vertical
Now, let Z c
lifts of a complex vector field Z ∈ χ(M ) and a complex tensor field of
k−1
k−1
and Fe = F v ,
type (1,1) F ∈ ℑ11 (M ) to k−1 M. In (11), if Ze = Z c
k
then the vertical lift of F ∈ ℑ11 (M ) to k M is the tensor field F v on
k M such that
k
k
k
(12)
F v (Z c ) = (F Z)v .
Local components of vertical lift of a complex tensor field
0i
F = F0j
∂
0i ∂
⊗ dz 0j + F 0j 0i ⊗ dz 0j
0i
∂z
∂z
(13)
∂
0i k ∂
⊗ dz 0j ,
⊗ dz 0j + (F 0j )v
ki
∂z
∂z ki
(14)
are
k
0i v
)
F v = (F0j
k
6
MEHMET TEKKOYUN
i.e.,
F
where
vk
:



K:


K 0
0 K
!

0
..
.
0
0i )v
i(F0j
k
0 ··· 0
..
.. 
.
. 

0 ··· 0 

0 ... 0
and K is the conjugate of K.
If F is taken as complex structure J0 =
then it is
k
J0v
where
:

0
 .
 ..

H :

0
iIm
(15)
H 0
0 H
!
∂
∂z 0i
⊗ dz 0j + ∂z∂0i ⊗ dz 0j ,
(16)
,

0 ... 0
..
.. 
.
. 


0 ... 0 
0 ... 0
and H is the conjugate of H.
Vertical lift of order k of complex structure J0 is tangent structure
k
k
on M such that (J0v )2 = 0.
Let M be a complex manifold and k−1 M its (k−1) order extended
complex manifold. Denote by Fe a complex tensor field of type (1,1)
and by Ze a complex vector field defined on k−1 M. Then the complete
lift of Fe ∈ ℑ11 (k−1 M ) to k M is the complex tensor field Fe c ∈ ℑ11 (k M )
such that
k−1
e c.
Fe c (Ze c ) = (Fe Z)
k−1
(17)
and F c be respectively complete lifts of a comNow, let Z c
plex vector field Z ∈ χ(M ) and a complex tensor field of type (1,1)
k−1
k−1
and Fe = F c , then
F ∈ ℑ11 (M ) to k−1 M. In (17), if Ze = Z c
k
the complete lift of F ∈ ℑ11 (M ) to k M is the tensor field F c on k M
such that
k
k
k
(18)
F c (Z c ) = (F Z)c .
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 7
The local components of the complete lift of a complex tensor field
expressed as in (13) are
Fc
k
∂
0i vk ∂
⊗ dz 0j + (F0j
)
⊗ dz 0j +
ki
∂z
∂z 0i
0i k ∂
0i vk ∂
⊗ dz 0j +
+(F0j
)
⊗ dz kj + (F 0j )c
ki
∂z ki
∂z
0i vk ∂
0i vk ∂
0j
+(F 0j )
⊗ dz + (F 0j )
⊗ dz kj ,
∂z 0i
∂z ki
0i c
= (F0j
)
k
(19)
i.e.,
F
ck
L 0
0 L
:
!
(20)
,
where

!
k
0i vk c0
 i 0 (F0j )



.

.
L:



! .


k
0i )v0 ck
i
(F0j
k

0
.
.
.
. .
.
!
k
0i )vk c0
i
(F0j
k











and L is the conjugate of L.
If F is taken as complex structure J0 = ∂z∂0i ⊗ dz 0j + ∂z∂0i ⊗ dz 0j ,
then complete lift of order k of complex structure J is
J0 :
where




T :


0
T
T
0
ck
!

iIm
and T is the conjugate of T .
0
.
.
0
(21)
,
.
iIm






8
MEHMET TEKKOYUN
Complete lift of order k of complex structure J0 is complex struck
ture on k M such that (J0c )2 = −I.
The higher order vertical and complete lifts of a complex tensor
field of type (1,1) on any complex manifold M obey the following
generic properties
k
k
F v (Z v ) = 0
k
k
k
k
k
F v (Z c ) = F c (Z v ) = (F Z)v
k
k
k
F c (Z c ) = (F Z)c ,
i)
ii)
iii)
for all Z ∈ χ(M ) and F ∈ ℑ11 (M ).
4. Higher order lifts of complex tensor fields of type
(0,2)
In this section, we obtain the definitions and properties about vertical
and complete lifts of a complex tensor field of type (0,2) defined on
any complex manifold M to extended complex manifolds k M .
Let M be any complex manifold and k−1 M its (k − 1) order
e be a tensor field of type (0,2) and
extended complex manifold. Let G
e
f
Z, W be complex vector fields defined on k−1 M. Then the vertical lift
e v ∈ ℑ0 (k M )
e ∈ ℑ0 (k−1 M ) to k M is the tensor field of type (0,2) G
of G
2
2
such that
e v (Z
ec, W
f c ) = G(
e Z,
e W
f )v .
G
(22)
k−1
k−1
k−1
and Gv be respectively complete and
Now, let Z c , W c
vertical lifts of complex vector fields Z, W ∈ χ(M ) and a tensor field
k−1
f = W ck−1
of type (0,2) G ∈ ℑ02 (M ) to k−1 M. In (22), if Ze = Z c , W
e = Gvk−1 , then the vertical lift of G ∈ ℑ0 (M ) to k M is the
and G
2
k
tensor field of type (0,2) Gv ∈ ℑ02 (k M ) such that
k
k
k
k
Gv (Z c , W c ) = (G(Z, W ))v .
(23)
Local components of vertical lift of a complex tensor field
g = g0i0j dz 0i ⊗ dz 0j + g0i0j dz 0i ⊗ dz 0j
are
(24)
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 9
k
k
k
gv = (g0i0j )v dz 0i ⊗ dz 0j + (g0i0j )v dz 0i ⊗ dz 0j ,
(25)
i.e.,
g
where
vk

0
M
:
(g0i0j )v
0
..
.


M :


k
0
M
0
!
(26)
,

0 ... 0

0 ··· 0 

..
.. 
.
. 
0 ··· 0
and M is the conjugate of M.
e ∈ ℑ0 (k−1 M ) to k M is the tensor
We say the complete lift of G
2
k
c
0
e
field of type (0,2) G ∈ ℑ2 ( M ) such that
k−1
k−1
e c (Z
ec, W
f c ) = G(
e Z,
e W
f )c .
G
(27)
k−1
and Gc be respectively complete lifts of comLet Z c , W c
plex vector fields Z, W and a tensor field of type (0,2) G ∈ ℑ02 (M )
k−1
f = W ck−1 and
defined on M to k−1 M. In (27), if Ze = Z c , W
e = Gck−1 , then the complete lift of G ∈ ℑ0 (M ) to k M is the tensor
G
2
k
field of type (0,2) Gc on k M such that
k
k
k
k
Gc (Z c , W c ) = (G(Z, W ))c .
(28)
Local components of vertical lift of a complex tensor field g given
in (24) are
gc
k
k
k
= (g0i0j )c dz 0i ⊗ dz 0j + (g0i0j )v dz ki ⊗ dz 0j
k
k
k
k
+(g0i0j )v dz 0i ⊗ dz kj + (g0i0j )c dz 0i ⊗ dz 0j
(29)
+(g0i0j )v dz ki ⊗ dz 0j + (g0i0j )v dz 0i ⊗ dz kj
i.e.,
g
ck
:
0
N
N
0
!
,
(30)
10
MEHMET TEKKOYUN
where

!
k
k
(g0i0j )c
0
..
.
!
···






 k
k−r r

(g0i0j )c v
N :
 r


..

.

!

k

vk
k
!
k−r r
k
(g0i0j )c v · · ·
r
!

k
k
(g0i0j )v 
k















0
(g0i0j )
and N is the conjugate of N.
The general properties of higher order vertical and complete lifts
of complex tensor fields of type (0,2) on complex manifold M are
i)
ii)
iii)
k
k
k
Gv (Z v , W v )
k
k
k
Gc (Z c , W v )
k
k
k
Gc (Z c , W c )
=
=
=
0
k
k
k
k
k
k
k
Gc (Z v , W c ) = Gv (Z c , W c ) = (G(Z, W ))v
k
(G(Z, W ))c ,
for all Z, W ∈ χ(M ) and G ∈ ℑ02 (M ).
5. Higher order lifts of hermitian metric and Kähler
form
In this section, we give definitions about higher order vertical and
complete lifts of an Hermitian metric and Kähler form defined on
any complex manifold M to extended complex manifolds k M .
In order to define higher order vertical and complete lifts of an
Hermitian metric defined on M , we first define higher order vertical
and complete lifts of complex tensor fields of type (0,2).
Let M be any complex manifold and k−1 M its (k − 1) order
e be a tensor field of type (0,2) and
extended complex manifold. Let G
e
f
Z, W be complex vector fields defined on k−1 M. Then the vertical
e to k M is the tensor field of type (0,2) G
e v on k M such that
lift of G
k−1
e v (Z
ec, W
f c ) = G(
e Z,
e W
f )v .
G
k−1
k−1
(31)
and Gv be respectively complete and
Now, let Z c , W c
vertical lifts of complex vector fields Z, W and a tensor field of type
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 11
k−1
k−1
f = Wc
(0,2) G defined on M to k−1 M. In (31), if Ze = Z c , W
k−1
e = Gv , then the vertical lift of G to k M is the tensor field
and G
k
of type (0,2) Gv on k M such that
k
k
k
k
Gv (Z c , W c ) = (G(Z, W ))v .
(32)
Given a Hermitian metric g and an almost complex structure J0
defined on any complex manifold M. Since g is a tensor field of type
(0,2), we have equality
k
k
k
k
k
k
k
k
gv (Z c , W c ) = gv (J0c Z c , J0c W c ).
(33)
for any complex vector fields Z, W on M. Hence we call the vertical lift of g to extended complex manifold k M as Hermitian metric
k
gv on k M. extended complex manifold k M with fixed Hermitian metk
ric gv is called vertical lift of order k of Hermitian manifold which
is complex manifold M with fixed Hermitian metric g.
e to k M is the tensor field of
We say that the complete lift of G
c
k
e on M such that
type (0,2) G
k−1
k−1
e c (Z
ec, W
f c ) = G(
e Z,
e W
f )c .
G
(34)
k−1
and Gc be respectively complete lifts of comLet Z c , W c
plex vector fields Z, W and a tensor field of type (0,2) G defined on
k−1
e = Gck−1 , then
f = W ck−1 and G
M to k−1 M. In (34), if Ze = Z c , W
k
the complete lift of G to k M is the tensor field of type (0,2) Gc on
k M given by equality
k
k
k
k
Gc (Z c , W c ) = (G(Z, W ))c .
(35)
Given a Hermitian metric g and an almost complex structure J0
defined on any complex manifold M. Since g is a tensor field of type
(0,2), we have equality
k
k
k
k
k
k
k
k
gc (Z c , W c ) = gc (J0c Z c , J0c W c ).
(36)
for any complex vector fields Z, W on M. Hence we call the
k
complete lift of g to extended complex manifold k M as gc on k M.
k
extended complex manifold k M with fixed Hermitian metric gc is
12
MEHMET TEKKOYUN
called complete lift of order k of Hermitian manifold which is complex
manifold M with fixed Hermitian metric g.
Given a Kähler form Φ, Hermitian metric g and an almost complex structure J0 defined on any Hermitian manifold M. Since Φ is
a complex tensor field of type (0,2), we have equality
k
k
k
k
k
k
k
Φv (Z c , W c ) = gv (Z c , J0c W c )
(37)
for any complex vector fields Z, W on M. Hence we call the vertik
cal lift of Φ to extended complex manifold k M as Kähler form Φv on
k M.
In this way we shall definition the following.
Given a Kähler form Φ,Hermitian metric g and an almost complex structure J0 defined on any Hermitian manifold M. Since Φ is
a complex tensor field of type (0,2), we have equality
k
k
k
k
k
k
k
Φc (Z c , W c ) = gc (Z c , J0c W c )
(38)
for any complex vector fields Z, W on M. Hence we call the complete lift of Φ to extended complex manifold k M as Kähler form
k
Φc on k M.
Now, let define the complete lift to k M of an Affine connection ▽
on M for using in Section 6. So, we extend using induction method
the properties about vertical and complete lifts of complex tensor
fields given the following such that all X, Y, Z ∈ ℑ10 (M ), f ∈ ℑ00 (M ),
ω ∈ ℑ01 (M ), K ∈ ℑrs (M ) :
i)
ii)
k
k
fv
X vk
k
k
▽c v k f c
X
k
k
▽c ck f c
X
k
k
▽c v k Y v
X
k
k
▽c v k Y c
X
k
k
▽c ck Y c
X
▽c
= 0,
= ▽c
k
X ck
k
k
f v = (▽X f )v ,
= (▽X f )c
k
= 0,
= ▽c
k
X ck
k
k
Y v = (▽X Y )v ,
= (▽X Y )c
k
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 13
k
k
ωv
X vk
k
k
▽c v k ω c
X
k
k
▽c ck ω c
X
k
k
▽c v k K v
X
k
k
▽c v k K c
X
k
k
▽c ck K c
X
▽c
iii)
iv)
= 0,
= ▽c
k
X ck
k
k
ω v = (▽X ω)v ,
= (▽X ω)c
k
= 0,
= ▽c
k
X ck
k
k
K v = (▽X K)v ,
= (▽X K)c
k
6. Classification of extended almost Hermitean
manifold
In this section we may generalize the classification made for almost
Hermitian manifold to extended almost Hermitian manifolds using
lift theory as follows.
Proposition 6.1. Let M be a Kähler manifold with almost comk
plex structure J0 . Then the extended manifold (k M, J0c ) is a Kähler
manifold.
Proof. Given by complex vector fields Z, W and by J0 almost comk
k
plex structure on any Kähler manifold M . Also, let Z c , W c , and
k
J0c be respectively complete lifts of this tensor fields to extended
complex manifold k M. Then we have
h
k
▽cZ ck , J0c
k
i
Wc
k
k
k
k
k
k
= ▽cZ ck (J0c W c ) − J0c ▽cZ ck W c
= ([▽Z , J0 ] W )c
k
k
= 0.
Finally, the proof is completed.
Proposition 6.2. Let M be a nearly Kähler manifold with almost
k
complex structure J0 . Then the extended manifold (k M, J0c ) is a
nearly Kähler manifold.
Proof. We easily make as similar the proof of proposition 6.1.
14
MEHMET TEKKOYUN
Proposition 6.3. Let M be an almost Kähler manifold with almost
k
complex structure J0 . Then the extended manifold (k M, J0c ) is an
almost Kähler manifold.
Proof. For any 2-form ω, one has:
3dω(X, Y, Z) = (∇X ω)(Y, Z) + (∇Y ω)(Z, X) + (∇Z ω)(X, Y ). (39)
So, we consider an almost Kähler manifold (M, g, J0 ) with fundamenk
k
k
k
k
tal form Φ. Since (∇c ck Φc )(Y c , Z c ) = ((∇X Φ)(Y, Z))c (which
X
follows from iv) in Section 5, using eq. (39), we get:
k
k
k
k
k
dΦc (X c , Y c , Z c ) = (dΦ(X, Y, Z))c .
Then one proves the statement, since Φ is closed, M being almost
Kähler.
Proposition 6.4. Let M be a quasi Kähler manifold with almost
k
complex structure J0 . Then the extended manifold (k M, J0c ) is a
quasi Kähler manifold.
Proof. Given by Z, W complex vector fields and by J0 almost comk
k
plex structure on any quasi Kähler manifold M . Also, let Z c , W c ,
k
and J0c be respectively complete lifts of this tensor fields to extended
complex manifold k M. Then we have
k
k
▽c ck ck , J0c
J0 Z
Wc
k
k
k
k
Jc W c
J 0 Z ck 0
= ▽c ck
k
k
− J0c ▽c ck
J 0 Z ck
= (▽J0 Z J0 W − J0 ▽J0 Z W )c
= ([▽J0 Z , J0 ] W )c
= −J0c
k
▽cZ ck , J0c
k
k
k
= (−J0 [▽Z , J0 ] W )c
h
k
Wc
i
k
k
.
Hence, the proof is finished.
Proposition 6.5. Let M be a Hermitian manifold with almost comk
plex structure J0 . Then the extended manifold (k M, J0c ) is a Hermitian manifold.
CLASSIFICATION OF ALMOST HERMITIAN MANIFOLDS 15
Proof. We easily make as similar the proof of proposition 6.4.
Proposition 6.6. Let M be a Semi Kähler manifold with almost
k
complex structure J0 . Then the extended manifold (k M, J0c ) is a
Semi Kähler manifold.
Proof. Given a local orthonormal frame {Ei , J0 Ei }1≤i≤m , one has:
k
k
k
k
▽c ck (J0c )Eic +▽c ck
Ei
k
J0 Eic
k
k
k
(J0c )J0c Eic = (▽Ei (J0 )Ei +▽J0 Ei (J0 )J0 Ei ),
for any i ∈ {1, ..., m}.
Therefore, if M is Semi Kähler , k M is Semi Kähler.
References
[1] J.J. Etayo, Derivations in the tangent bundle. Differential geometry (Peñı́scola, 1982), 43–52, Lecture Notes in Math., 1045, Springer,
Berlin, 1984.
[2] A. Gray, Curvature identities for hermitian and almost hermitian
manifolds, Tôhoku Math. Journ. 28 (1976), 601-612.
[3] M. Tekkoyun, The higher order lifts of lagrangian and hamiltonian
equations to the extended Kähler manifolds, Ph.D. Thesis, Osmangazi
Univ., Eskişehir (2002).
[4] M. Tekkoyun and Ş. Civelek, On lifts of structures on complex
manifolds, Differential Geometry-Dynamics Systems (DGDS), 5 n. 1
(2003), 59-64.
[5] M. Tekkoyun, Ş. Civelek and A. Görgülü, Higher order lifts
of complex structures, Rend. Istit. Mat. Univ. Trieste, vol. XXXVI
(2004), 85-95.
[6] K. Yano and S. Ishihara, Tangent and cotangent bundles, Marcel
Dekker Inc., New York (1973).
Received May 20, 2005.