Some remarks concerning Riemannian extensions
We study some properties of Riemannian extensions in cotangent bundles with the help of adapted frames.
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| author | Aslanci, S. Kazimova, S. Salimov, A. A. Асланці, С. Казімова, С. Салімов, А. А. |
| author_facet | Aslanci, S. Kazimova, S. Salimov, A. A. Асланці, С. Казімова, С. Салімов, А. А. |
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| description | We study some properties of Riemannian extensions in cotangent bundles with the help of adapted frames. |
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UDC 514.763
S. Aslanci (Ataturk Univ., Erzurum, Turkey),
S. Kazimova (Baku State Univ., Azerbaijan),
A. A. Salimov (Ataturk Univ., Erzurum, Turkey)
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS*
DEQKI ZAUVAÛENNQ WODO RIMANOVYX ROZÍYREN|
In this paper we investigate some properties of Riemannian extensions in the cotangent bundle using the
adapted frames.
DoslidΩeno deqki vlastyvosti rimanovyx rozßyren\ u kodotyçnomu rozßaruvanni z vykorystan-
nqm adaptovanyx reperiv.
1. Introduction. Let M n be an n-dimensional differentiable manifold of class C∞ ,
C
nT M( ) its cotangent bundle, and π the natural projection C
nT M( ) → M n . A
system of local coordinates ( ; )U xi , i = 1, … , n, in M n induces on C
nT M( ) a
system of local coordinates π−( 1( )U ; xi , x i = pi) , i = 1, … , n, i = n + i = n +
+ 1, … , 2n , where x pi
i= is the cartesian coordinates of covectors p in each co-
tangent space C
x nT M( ) , x U∈ with respect to the natural coframe dxi{ } .
We denote by ℑs
r
nM( ) ℑ ( )( )s
r C
nMΤ ( ) the modul over F M n( ) F T MC
n( )( )( )
of C∞ tensor fields of type ( , )r s , where F M n( ) F T MC
n( )( )( ) is the ring of real-
valued C∞ functions on M T Mn
C
n( )( ) . The so-called Einsteins summation conven-
tion is used.
Let X = X
x
i
i
∂
∂
and ω = ω i idx be the local expressions in U M n⊂ of a vector
field X ∈ ℑ1
0 ( )M n , and 1-form ω ∈ ℑ1
0 ( )M n respectively. Then the horizontal lift
H X ∈ ℑ ( )0
1 C
nT M( ) of X and the vertical lift V ω ∈ ℑ ( )0
1 C
nT M( ) of ω are given,
respectively, by
H i
i h
i
ij
h j
i
X X
x
p X
x
=
∂
∂
+
∂
∂
∑ Γ (1)
and
V
i
i
ix
ω ω=
∂
∂
∑ (2)
with respect to the natural frame
∂
∂
∂
∂
x xi i
, , where Γ ij
h are components of a sym-
metric (torsion-free) affine connection ∇ on M n .
*This paper is supported by Scientific and Technological Research Council of Turkey (TBAG-108T590).
© S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5 579
580 S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV
We now consider a tensor field R∇ ∈ ℑ ( )2
0 C
nT M( ) , whose components in
π−1( )U are given by
R∇ = R
JI∇( ) =
−
2
0
ph ji
h
j
i
i
j
Γ δ
δ
(3)
with respect to the natural frame, where δ j
i denotes the Kronecker delta. The indices
I, J, K, … = 1, … , 2n indicate the indices with respect to the natural frame
∂
∂
xi
,
∂
∂
x i
. This tensor field defines a pseudo-Riemannian metric in C
nT M( ) and the line
element of pseudo-Riemannian metric R∇ is given by
ds dx pi
i
2 2= δ ,
where
δp dp p dxi i h ji
h i= − Γ .
This metric is called the Riemannian extension of the symmetric affine connection
∇ [1, 2]. A number of results referring to the applications of the Riemannian exten-
sion are contained in [3, 4].
The complete lift of vector field X ∈ ℑ0
1 ( )M n to cotangent bundle C
nT M( ) is
defined by
C X = X
x
p X
x
i
i h
i
i
h
i
∂
∂
− ∂
∂
∂
∑ . (4)
Using (3) and (4), we easily see that
R C CX Y∇( ), = − ∇ + ∇( )γ X YY X , (5)
where
γ ∇ + ∇( )X YY X = p X Y Y Xh
i
i
h i
i
h∇ + ∇( ) .
Since the tensor field R∇ ∈ ℑ ( )2
0 C
nT M( ) is completely determined by its action on
vector fields of type C X and CY (see Proposition 4.2 of [2, p. 237]), we have an al-
ternative definition of R∇ : The tensor field R∇ is completely determined by the
condition (5).
On the other hand, the vector fields H X and V ω span the module ℑ ( )0
1 C
nT M( ) .
Hence tensor field R∇ is also determined by its action of H X and V ω .
From (1), (2) and (3) we have
R V V∇( ) =ω θ, 0 , (6)
R V H X∇( )ω, = V Xω( )( ) = ω π( )X( ) � , (7)
R H HX Y∇( ) =, 0 (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS 581
for any X, Y ∈ ℑ0
1 ( )M n and ω, θ ∈ ℑ1
0( )M n . Thus R∇ is completely determined
by the conditions (6), (7), (8) because of the above stated reasons.
In this paper we shall develop the Riemannian extension R∇ using the conditions
(6) – (8). Moreover, we find it more convenient to refer equations (6) – (8) to the
adapted frame.
2. Adapted frames. Let ∇ be a torsion-free affine connection on M n . In
U M n⊂ , we put
X
x
i i( ) =
∂
∂
, θ i idx( ) = , i = 1, … , n.
Then from (1) and (2) we see that H
iX( ) and V iθ( ) have respectively local ex-
pressions of the form
H
i i a
h
hi
a
h
X
x
p
x
( ) =
∂
∂
+
∂
∂
∑ Γ , (9)
V i
ix
θ( ) =
∂
∂
. (10)
We call the set H
i
V iX( )
( ), θ{ } = � �e ei i( ), ( ){ } = �e( )α{ } the frame adapted to the affi-
ne connection ∇ . The indices α, β, γ, … = 1, … , 2n indicate the indices with respect
to the adapted frame.
We now from equations (1), (2) and (9), (10) see that the lifts H X and V ω have
respectively components
H i
iX X e= �( ) , H
i
X
X=
0
, (11)
V
i i
i
eω ω= ( )∑ � , V
iω
ω
=
0
(12)
with respect to the adapted frame �e( )α{ } , where X ∈ ℑ0
1 ( )M n , ω ∈ ℑ1
0( )M n , Xi
and ω i being local components of X and ω, respectively. Also from (6) – (8) we
see that
R V i V j∇( )ω θ( ) ( ), = R
i je e∇ ( )( ) ( )� �, = R
ij
�∇ = 0,
R H
i
H
jX Y∇( )( ) ( ), = R
i je e∇ ( )� �( ) ( ), = R
ij
�∇ = 0,
R V i H jX∇( )ω( ) ( ), = R
i je e∇ ( )( )� �, ( ) = R
ij
�∇ = R
ji
�∇ = dx
x
i
j( ) ∂
∂
= δ j
i ,
R H
i
V jX∇( )( )
( ), ω = R
i je e∇ ( )( )� �( ), = R
ij
�∇ = R
ji
�∇ = dx
x
j
i( ) ∂
∂
= δi
j ,
i.e., R∇ has components
R∇ = R �∇( )βα =
R
ji
R
ji
R
ji
R
ji
� �
� �
∇ ∇
∇ ∇
=
0
0
δ
δ
j
i
i
j
(13)
with respect to the adapted frame �e( )α{ } .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
582 S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV
Using (9), (10), we now consider local vector fields �eβ and 1-forms �ωα in
π−1( )U defined by
�e A A
Aβ β= ∂ , �ωα α= A dxB
B ,
where
A A A= ( )β =
A A
A A
j
i
j
i
j
i
j
i
=
δ
δ
j
i
a ij
a
i
jp
0
Γ
, (14)
A A B
− = ( )1 α =
A A
A A
i
j
i
j
i
j
i
j
=
δ
δ
j
i
a ij
a
i
jp
0
−
Γ
. (15)
We easily see that the set �ωα{ } is the coframe dual to the adapted frame �eβ{ } , i.e.,
� �ωα
βe = A AB
Bα
β = δβ
α .
Since the adapted frame �eβ{ } is nonholonomic, we put
� � �e e eγ β γβ
α
α, = Ω
from which we have
Ωγβ
α
γ β β γ
α= −( )� �e A e A AA A
A .
According to (9), (10), (14) and (15), the components of nonholonomic object
Ωγβ
α are given by
Ω Ω Γlj
i
jl
i
li
j= − = − ,
(16)
Ωlj
i
a lji
ap R=
all the others being zero, where Rljk
h being local components of the curvature tensor
R of ∇ .
Let C∇ be the Levi-Civita connection determined by the Riemannian extension
R∇ . We call C∇ the complete lift of the symmetric affine connection ∇ to
C
nT M( ) . We put
C
e
Ce e∇ =� � �
γ β γβ
α
αΓ .
From the equation C
X Y∇ – C
Y X∇ = X Y,[ ] ∀ X , Y ∈ ℑ ( )0
1 C
nT M( ) we have
C CΓ Γ Ωγβ
α
βγ
α
γβ
α− = . (17)
The equation C
X
R Y Z∇ ∇( ) ( , ) = 0 has form
�e R C R C R
δ γβ δγ
ε
εβ δβ
ε
γ ε∇ − ∇ − ∇ =Γ Γ 0 (18)
with respect to the adapted frame �eβ{ } . We have from (17) and (18)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS 583
CΓγβ
α =
1
2
R R R Re e e∇ ∇ + ∇ − ∇( )αε
γ εβ β γ ε ε γβ� � � +
1
2
Ω Ω Ωγβ
α α
γβ
α
βγ+ +( ) ,
where Ωα
γβ = R R∇ ∇αε
δβ εγ
δΩ and R∇( )αε =
0
0
δ
δ
m
i
i
m
.
Taking account of (9), (10), (13) and (16) we obtain
C
k j
iΓ = C
k j
iΓ = C
k j
iΓ = C
k j
iΓ = C
k j
iΓ = 0,
C
kj
i
kj
iΓ Γ= , C
k j
i
ki
jΓ Γ= − , (19)
C
kj
i
a kji
a
jik
a
ikj
ap R R RΓ = − +( )1
2
.
Let X ∈ ℑ ( )0
1 C
nT M( ) and X = � �X eα
α = � �X ei
i( ) + � �X ei
i( ) . Then the covariant de-
rivative C X∇ has components
C CX e X X∇ = +γ
α
γ
α
γβ
α β� � � �Γ .
If X = H X and X = Vω , then using (9), (10), (11), (12) and (19) we see that cova-
riant derivatives C H X∇ and C V∇ ω have ω respectively components
C H X∇( )γ
α� =
∇
− +( )
k
i
a kji
a
jik
a
ikj
a i
X
p R R R X
0
1
2
0
, (20)
C V∇( )γ
αω� =
0 0
0∇
k iω
(21)
with respect to the adapted frame �eα{ } .
Taking account (4), (9) and (10), we find
C i
i h i
h
i
i
X X e p X e= + − ∇( ) ( )∑� �( ) (22)
for any X ∈ ℑ0
1 ( )M n .
Using now (19) and (22), by similar devices we can prove
C C
k
i
h k i
h
a kji
a
jik
a
i
X
X
p X p R R R
∇( ) =
∇
− ∇ ∇ + − +
γ
α�
0
1
2
kkj
a j
i
kX X( ) −∇
. (23)
From (21) we have the following theorem.
Theorem 1. The vertical lift of covector field ω ∈ ℑ1
0( )M n to C
nT M( ) with
metric R∇ is parallel if and only if the given covector field ω is parallel with respect
to ∇ .
If M n has pseudo-Riemannian metric g, then by virtue of
p R Xa kji
a j = p X R ga
j
kjis
sa( ) =
= p X R ga
j
iskj
sa( ) = p X R ga
j
isjk
sa−( ) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
584 S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV
= p X R g ga
j
isj
t
tk
sa−( ) = − ∇ ∇p g g Xa tk
sa
i s
t
[ ] , (24)
we have from (20) and (23) the following theorem.
Theorem 2. When M n has pseudo-Riemannian metric g and the Levi-Civita
connection ∇ of g and C
nT M( ) has the Riemannian extension R∇ as its metric,
the horizontal and the complete lifts of a vector field X ∈ ℑ0
1 ( )M n to C
nT M( ) with
the metric R∇ are parallel if and only if the given vector field X is parallel with res-
pect to the Levi-Civita connection ∇ .
3. The metric connection of R∇∇ . In Introduction and Section 2, we have given
to the cotangent bundle C
nT M( ) the metric R∇ and considered the Levi-Civita con-
nection C∇ of R∇ . This is the unique connection which satisfies C R∇ ∇( ) = 0,
and has no torsion. But there exists another connection which satisfies �∇ ∇( )R = 0,
and has nontrivial torsion tensor. We call this connection the metric connection
of R∇ .
The horizontal lift H ∇ of the non-torsion connection ∇ to the cotangent bundle
C
nT M( ) defined by
H
V
V∇ =θ ω 0 , H H
V Y∇ =θ 0 ,
(25)
H
H X
V V
X∇ = ∇( )ω ω , H
X
H H
XH Y Y∇ = ∇( )
for any X, Y ∈ ℑ0
1 ( )M n and ω, θ ∈ ℑ1
0( )M n .
We now put H ∇α = H
e∇ �( )α , where �e( )α{ } = � �e ei i( ), ( ){ } -adapted frame. Then ta-
king account of C e∇α β�( ) = H eΓαβ
γ
γ�( ) and writing H �Γαβ
γ for the different indices,
from (25) we have
H
ij
k
ij
k�Γ Γ= , H
i j
k
ik
j�Γ Γ= − ,
(26)
H
ij
k H
i j
k H
i j
k H
i j
k H
ij
k H
i j
� � � � � �Γ Γ Γ Γ Γ Γ= = = = =
kk = 0 .
Let T be the torsion tensor of the horizontal lift H ∇ . Then T is the skew-sym-
metric tensor field of type (1, 2) in C
nT M( ) determined by [2, p. 287]
Τ V Vω θ,( ) = 0 , Τ H VX, θ( ) = 0 , Τ H HX Y R X Y, ,( ) = − ( )γ ,
where R is curvature tensor of ∇ and γR X Y( , ) = p R X Y
x
h kli
h k l
ii
∂
∂
∑ . Thus the
connection H ∇ has nontrivial torsion even for Levi-Civita connection ∇ determined
by g, unless g is locally flat.
Using (6) – (8) and (25), we have
H
V
R V V∇ ∇( ) ( ) =ω θ ε, 0 ,
H
H X
R V V∇ ∇( ) ( )θ ε, = − ∇( )( )R V
X
Vg θ ε, = 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS 585
H
V
R V H Z∇ ∇( ) ( )ω θ, = V V Zω θ( )( ) = 0,
H
H X
R V H Z∇ ∇( ) ( )θ, = H VX Zθ( )( ) – R V
X
Hg Z
H
∇( )( )θ , –
– R V H
Xg Z
H
θ, ∇( )( ) = V
X XX Z Z Zθ θ θ( ) − ∇( ) − ∇( ) = 0,
H
V
R H VY∇ ∇( ) ( )ω ε, = V V Yω ε( )( ) = 0,
H
H X
R H VY∇ ∇( ) ( ), ε = H VX Yε( )( ) – R V
X
Vg Y
H
∇( )( ), ε –
– R H V
Xg Y , ∇( )( )ε = V
X XX Y Y Yε ε ε( ) ( )− ∇( ) − ∇( ) = 0,
H
V
R H HY Z∇ ∇( ) ( )ω , = 0,
H
H X
R H HY Z∇ ∇( ) ( ), = 0
for any X, Y, Z ∈ ℑ0
1 ( )M n and ω, θ, ε ∈ ℑ1
0( )M n .
Let now H R be a curvature tensor field of
H
∇ . The curvature tensor H R of
the metric connection
H
∇ of
R
∇ has components
H R�δγβ
α
= � �e H
δ γβ
α
( ) Γ – � �e H
( )γ δβ
αΓ + H H� �Γ Γδε
α
γβ
ε – H H� �Γ Γγ ε
α
δβ
ε – Ω Γδγ
ε
εβ
αH �
(27)
with respect to the adapted frame.
Using (9), (10), (16), (26), (27) and computing components of the contracted curva-
ture tensor field (Ricci tensor field) H R�γβ = H R�αγβ
α , we obtain
H
kjR� = H
kjR�α
α = H
ikj
i H
i kj
iR R� �+ = Rikj
i = Rkj ,
(28)
H
k jR� = 0 , H
k jR� = 0 , H
k jR� = 0 ,
where Rkj is the Ricci tensor field of ∇ in M n .
For the scalar curvature of C
nT M( ) with the metric connection H ∇ , we have
� � �R RR H= ∇ =γβ
γβ 0
by means of (28) and
R j
k
k
j
�∇( ) =
γβ δ
δ
0
0
.
Thus we have the following theorem.
Theorem 3. The cotangent bundle C
nT M( ) with the metric connection H ∇
has vanishing scalar curvature with respect to the metric R∇ .
4. Killing vector fields in C
n
RT M( ), ∇∇(( )) . In a manifold with a pseudo-Rieman-
nian metric g, a vector field is called a Killing vector field (or, an infinitesimal iso-
metry) if L gX = 0, where LX is the Lie derivative.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
586 S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV
The condition L gX = 0 can be rewritten as
L g Y ZX( ) ( , ) = g X ZY∇( ), + g X YZ∇( ), = 0 (29)
for any Y, Z ∈ ℑ0
1 ( )M n , where ∇ is the Levi-Civita connection of g.
We now compute the Lie derivative of the metric R∇ . In view of the adapted fra-
me �e( )α{ } , from (29) we obtain
R C X e e∇ ∇( )( )β
σ
σ γ� � �( ) ( ), + R C X e e∇ ∇( )( )γ
σ
σ β� � �( ) ( ), = 0
or
C CX X∇ + ∇ =β γ γ β� � 0 , (30)
where �Xγ( ) is an associated covector field of a vector field �Xσ( ) is given by
� � �X XR
γ γ σ
σ( ) = ∇( ) .
The associated covector fields of the vertical, horizontal and complete lifts to C
nT M( )
with the metric R∇ , with respect to the adapted frame �e( )α{ } , are given respectively
by
V X� γ( ) = R V� �∇( )γσ
σω = ( , ),ω k 0
H X� γ( ) = R H X� �∇( )γσ
σ = ( , ),0 Xk
C X� γ( ) = R C X� �∇( )γσ
σ = − ∇( )p X Xh k
h k, ,
because of (11), (12), (13) and (22).
Using (21) and (30) we see that the Lie derivative of R∇ with respect to V ω has
components
LV
R
ω βγ
∇( ) = C V C V∇ + ∇β γ γ βω ω� � =
∇ + ∇
j k k jω ω 0
0 0
(31)
with respect to the adapted frame �e( )α{ } . We put ω i = g Xij
j for any X ∈ ℑ0
1 ( )M n .
Then from (31) we have the following theorem.
Theorem 4. A necessary and sufficient condition for a vector field V ω in cotan-
gent bundle with metric R∇ to be a Killing vector field is that an associated vector
field is Xi = gij
jω is Killing vector field.
Also, using (20), (23) and (30), we see that L H X
R∇ and LC X
R∇ have respecti-
vely components
L H X
R∇( )βγ =
p R R X X Xa ksj
a
jsk
a s
k
j
j
k+( ) ∇ + ∇
0 0
,
LC X
R∇( )βγ =
=
− ∇ ∇ + ∇ ∇( ) + +( ) ∇ + ∇2 p X X p R R X Xh k j
h
j k
h
a ksj
a
jsk
a s
k
j
jj
k
k
j
j
k
X
X X−∇ − ∇
0
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS 587
with respect to the adapted frame �e( )α{ } . From these equations and (24) we have the
following theorem.
Theorem 5. The horizontal and complete lifts of vector fields in M n to C
nT M( )
with metric R∇ is Killing if the given vector field X ∈ ℑ0
1 ( )M n is parallel with re-
spect to the Levi-Civita connection ∇ of the metric g in M n .
5. Norden structures in C
nT M( ) with metric R∇∇ . Let ( , )M n2 ϕ be an al-
most complex manifold with almost complex structure ϕ. A pseudo-Riemannian met-
ric g ∈ ℑ2
0
2( )M n is a Norden metric with respect to structure ϕ if
g X Y( , )ϕ = g X Y( , )ϕ
for any X, Y ∈ ℑ0
1
2( )M n . Metrics of this kind have been also studied under the na-
mes: pure, anti-Hermitian and B-metrics (see, for example, [5 – 10]). If ( , )M n2 ϕ is
an almost complex manifold with Norden metric g, we say that ( , , )M gn2 ϕ is an
almost Norden manifold. If ϕ is integrable, we say that ( , , )M gn2 ϕ is a Norden
manifold.
Let ( , )M n2 ϕ be an almost complex manifold with almost complex structure ϕ.
This structure is said to be integrable if the matrix ϕ ϕ= ( )j
i is reduced to the con-
stant form in a certain holonomic natural frame in a neighborhood Ux of every point
x M n∈ 2 . In order that the almost complex structure ϕ be integrable, it is necessary
and sufficient that it is possible to introduce a torsion-free affine connection ∇ with
respect to which the structure tensor ϕ is covariantly constant, i. e., ∇ =ϕ 0 . Also,
we know that the integrability of ϕ is equivalent to the vanishing of the Nijenhuis ten-
sor Nϕ ∈ ℑ2
1
2( )M n . If ϕ is integrable, then ϕ is a complex structure and moreover
M n2 is a C -holomorphic manifold Xn ( )� whose transition functions are holo-
morphic mappings.
Let t
∗
be a complex tensor field on Xn ( )� . The real model of such a tensor field
is a tensor field on M n2 of the same order that is independent of whether its vector or
covector arguments is subject to the action of the affinor structure ϕ. Such tensor
fields are said to be pure with respect to ϕ. They were studied by many authors (see,
e. g., [10 – 14]). In particular, being applied to a ( , )0 q -tensor field ω, the purity
means that for any X1 , … , Xq ∈ ℑ0
1
2( )M n the following conditions should hold:
ω ϕ( , , , )X X Xq1 2 … = ω ϕ( , , , )X X Xq1 2 … = … = ω ϕ( , , , )X X Xq1 2 … .
We define an operator
Φϕ : ( ) ( )ℑ → ℑ +q n q nM M0
2 1
0
2
applied to the pure tensor field ω by (see [15])
( ) ( , , , , )Φϕω X Y Y Yq1 2 … = ( ) ( , , , )ϕ ωX Y Y Yq1 2 …( ) –
– X Y Y Yqω ϕ( , , , )1 2 …( ) + ω ϕ( ) , , ,L X Y YY q1 2 …( ) + …
… + ω ϕY Y L XYq1 2, , , ( )…( ) ,
where LY denotes the Lie differentiation with respect to Y.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
588 S. ASLANCI, S. KAZIMOVA, A. A. SALIMOV
When ϕ is a complex structure on M n2 and the tensor field Φϕω vanishes, the
complex tensor field ω∗ on Xn ( )� is said to be holomorphic (see [11, 15]). Thus a
holomorphic tensor field ω∗ on Xn ( )� is realized on M n2 in the form of a pure
tensor field ω, such that
( ) ( , , , , )Φϕω X Y Y Yq1 2 0… =
for any X, Y1 , … , Yq ∈ ℑ0
1
2( )M n . Therefore such a tensor field ω on M n2 is also
called holomorphic tensor field. When ϕ is an almost complex structure on M n2 , a
tensor field ω satisfying Φϕω = 0 is said to be almost holomorphic.
In a Norden manifold a Norden metric g is called a holomorphic if
( ) ( , , )Φϕg X Y Z = 0
for any X, Y, Z ∈ ℑ0
1
2( )M n .
If ( , , )M gn2 ϕ is a Norden manifold with holomorphic Norden metric g, we say
that ( , , )M gn2 ϕ is a holomorphic Norden manifold.
In some aspects, holomorphic Norden manifolds are similar to Kähler manifolds.
The following theorem is analogue to the next known result: An almost Hermitian ma-
nifold is Kähler if and only if the almost complex structure is parallel with respect to
the Levi-Civita connection.
Theorem 6 [6] (For paracomplex version see [9]). For an almost complex mani-
fold with Norden metric g, the condition Φϕg = 0 is equivalent to ∇ =ϕ 0 , where
∇ is the Levi-Civita connection of g.
A Kähler – Norden manifold can be defined as a triple ( , , )M gn2 ϕ which con-
sists of a manifold M n2 endowed with an almost complex structure ϕ and a pseudo-
Riemannian metric g such that ∇ =ϕ 0 , where ∇ is the Levi-Civita connection of
g and the metric g is assumed to be Nordenian. Therefore, there exist a one-to-one
correspondence between Kähler – Norden manifolds and Norden manifolds with a
holomorphic metric. Recall that in such a manifold, the Riemannian curvature tensor
is pure and holomorphic, also the curvature scalar is locally holomorphic function
(see [6, 9]).
Remark 1. We know that the integrability of the almost complex structure ϕ is
equivalent to the existing a torsion-free affine connection with respect to which the
equation ∇ =ϕ 0 holds. Since the Levi-Civita connection ∇ of g is a torsion-free
affine connection, we have: If Φϕg = 0 , then ϕ is integrable. Thus, almost Norden
manifold with conditions Φϕg = 0 and Nϕ ≠ 0 , i. e., almost holomorphic Norden
manifolds does not exist.
Remark 2. The Levi-Civita connection of Kähler – Norden metric g coincides
with the Levi-Civita connection of twin metric G = g � ϕ (nonuniquences of the met-
ric for the Levi-Civita connection in Kähler – Norden manifolds).
We define the horizontal lift H ϕ ∈ ℑ ( )( )1
1
2
C
nMΤ by [2, p.281]
H V Vϕ ω ω ϕ= ( )� ,
(32)
H H HX Xϕ ϕ= ( )
for any X ∈ ℑ0
1
2( )M n and ω ∈ ℑ1
0
2( )M n . We see from (9), (10) and (32) that, the
horizontal lift H ϕ has components of the form
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
SOME NOTES CONCERNING RIEMANNIAN EXTENSIONS 589
H j
i
i
j
ϕ ϕ
ϕ
ϕ
β
α= ( ) =
�
0
0
(33)
with respect to the adapted frame �e( )α{ } , ϕ j
i being local components of ϕ.
It is well known that if ϕ an almost complex structure in M n2 with torsion free
connection ∇ , then H ϕ is an almost complex structure in C
nT M( ) [2, p. 283].
From (6), (7), (8) and (32), we easily verify that
R H R HX Y X Y∇( ) = ∇( )ϕ ϕ� � � �, ,
for any �X XH= or V ω and �Y YH= or V θ , that is, Τ M n( )( , R∇ , H ϕ) is an
almost Norden manifold.
We now consider covariant derivative of the almost complex structure H F with
respect to Levi-Civita connection C∇ of R∇ . Taking account of (19) and (33), we
find that
C
i
H
j
k
i j
k∇ = ∇�ϕ ϕ , C
i
H
j
k
i k
j∇ = ∇�ϕ ϕ ,
(34)
C
i
H
j
k∇ �ϕ =
=
1
2
p R R R R R Ra imk
a
mki
a
kim
a
j
m
i jm
a
jmi
a
mi j− +( ) − − +ϕ aa
k
m( ) ϕ
the other being all zero, with respect to the adapted frame �e( )α{ } .
If a torsion free affine connection ∇ preserving the structure ϕ (∇ =ϕ 0 ) satis-
fies the condition ∇ϕXY = ϕ ∇( )XY ∀ X , Y ∈ ℑ ( )0
1
2M n , then ∇ is called a holo-
morphic connection [14, p. 185]. The purity of the curvature tensor field of a connecti-
on ∇ ( Rmjk
s
i
mϕ = Rimk
s
j
mϕ = Rijm
s
k
mϕ = Rijk
m
m
sϕ ) is a necessary and sufficient
condition for its holomorphy [11, 14]. Therefore, from (34) we have the following
theorem.
Theorem 7. The cotangent bundle C
nT M( ) is a Kähler – Norden with respect to
R∇ and the almost complex structure H ϕ if the a torsion-free connection ∇ is a
holomorphic connection with respect to the structure ϕ.
On the other hand it is well known that in a Kähler – Norden manifold the curvature
tensor of Norden-metric is pure [6]. Therefore, when M n2 has Kähler – Norden met-
ric g and the Levi-Civita connection ∇ of g and C
nT M( )2 has the Riemannian
extension R∇ as its metric, we have the following theorem.
Theorem 8. The cotangent bundle C
nT M( )2 of a pseudo-Riemannian manifold
M n2 is a Kähler – Norden with respect to R∇ and H ϕ , i f ( , , )M gn2 ϕ is a
Kähler – Norden.
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Received 21.09.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #5
|
| id | umjimathkievua-article-2889 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:32:16Z |
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| spelling | umjimathkievua-article-28892020-03-18T19:39:51Z Some remarks concerning Riemannian extensions Деякі зауваження щодо ріманових розширень Aslanci, S. Kazimova, S. Salimov, A. A. Асланці, С. Казімова, С. Салімов, А. А. We study some properties of Riemannian extensions in cotangent bundles with the help of adapted frames. Досліджено деякі властивості ріманових розширень у кодотичному розшаруванні з використанням адаптованих реперів. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2889 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 579–590 Український математичний журнал; Том 62 № 5 (2010); 579–590 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2889/2529 https://umj.imath.kiev.ua/index.php/umj/article/view/2889/2530 Copyright (c) 2010 Aslanci S.; Kazimova S.; Salimov A. A. |
| spellingShingle | Aslanci, S. Kazimova, S. Salimov, A. A. Асланці, С. Казімова, С. Салімов, А. А. Some remarks concerning Riemannian extensions |
| title | Some remarks concerning Riemannian extensions |
| title_alt | Деякі зауваження щодо ріманових розширень |
| title_full | Some remarks concerning Riemannian extensions |
| title_fullStr | Some remarks concerning Riemannian extensions |
| title_full_unstemmed | Some remarks concerning Riemannian extensions |
| title_short | Some remarks concerning Riemannian extensions |
| title_sort | some remarks concerning riemannian extensions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2889 |
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