On Four-Dimensional Paracomplex Structures with Norden Metrics
We study almost paracomplex structures with Norden metric on Walker 4-manifolds and try to find general solutions for the integrability of these structures on suitable local coordinates. We also discuss para-Kähler (paraholomorphic) conditions for these structures.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507861049671680 |
|---|---|
| author | İşcan, M. Özkan, M. Іскан, М. Озкан, М. |
| author_facet | İşcan, M. Özkan, M. Іскан, М. Озкан, М. |
| author_sort | İşcan, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:47:39Z |
| description | We study almost paracomplex structures with Norden metric on Walker 4-manifolds and try to find general solutions for the integrability of these structures on suitable local coordinates. We also discuss para-Kähler (paraholomorphic) conditions for these structures. |
| first_indexed | 2026-03-24T02:16:02Z |
| format | Article |
| fulltext |
UDC 512.662
M. İşcan (Ataturk Univ., Erzurum, Turkey),
M. Özkan (Gazi Univ., Ankara, Turkey)
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES
WITH NORDEN METRICS
ПРО ЧОТИРИВИМIРНI ПАРАКОМПЛЕКСНI СТРУКТУРИ
З МЕТРИКАМИ НОРДЕНА
We study the almost paracomplex structures with Norden metric on Walker 4-manifolds and try to find general solutions for
the integrability of these structures on suitable local coordinates. We also discuss para-Kähler (paraholomorphic) conditions
for these structures.
Вивчаються майже паракомплекснi структури з метрикою Нордена на 4-многовидах Уолкера. Встановлено загальнi
розв’язки щодо iнтегровностi таких структур у вiдповiдних локальних координатах. Також обговорюються пара-
келеровi (параголоморфнi) умови для таких структур.
1. Introduction. Let M2n be a Riemannian manifold with a neutral metric, i.e., with a pseudo-
Riemannian metric g of signature (n, n). We denote by =p
q(M2n) the set of all tensor fields of type
(p, q) on M2n. Manifolds, tensor fields and connections are always assumed to be differentiable and
of class C∞.
An almost paracomplex manifold is an almost product manifold (M2n, ϕ), ϕ2 = id, such that
the two eigenbundles T+M2n and T−M2n associated to the two eigenvalues +1 and −1 of ϕ,
respectively, have the same rank. Note that the dimension of an almost paracomplex manifold is
necessarily even. Considering the paracomplex structure ϕ, we obtain the following set of affinors
on M2n : {id, ϕ} , ϕ2 = id, which form a bases of a representation of the algebra of order 2 over
the field of real numbers R, which is called the algebra of paracomplex (or double) numbers and is
denoted by R(j) =
{
a0 + a1j : j2 = 1; a0, a1 ∈ R
}
. Obviously, it is associative, commutative and
unitial, i.e., it admits principal unit 1. The canonical bases of this algebra has the form {1, j}.
Let (M2n, ϕ) be an almost paracomplex manifold with almost paracomplex structure ϕ. For
almost paracomplex structure the integrability is equivalent to the vanishing of the Nijenhuis tensor
Nϕ(X,Y ) = [ϕX,ϕY ]− ϕ[ϕX, Y ]− ϕ[X,ϕY ] + [X,Y ].
This structure is said to be integrable if the matrix ϕ = (ϕi
j) is reduced to the constant form in a certain
holonomic natural frame in a neighborhood Ux of every point x ∈M2n. On the other hand, in order
that an almost paracomplex structure be integrable, it is necessary and sufficient that we can introduce
a torsion free linear connection such that∇ϕ = 0. A paracomplex manifold is an almost paracomplex
manifold (M2n, ϕ) such that the G-structure defined by the affinor field ϕ is integrable. We can give
another-equivalent-definition of paracomplex manifold in terms of local homeomorphisms in the
space Rn(j) =
{
(X1, . . . , Xn) : Xi ∈ R(j), i = 1, . . . , n
}
and paraholomorphic changes of charts
in a way similar to [2] (see also [6]), i.e., a manifold M2n with an integrable paracomplex structure
ϕ is a real realization of the paraholomorphic manifold Mn(R(j)) over the algebra R(j).
c© M. İŞCAN, M. ÖZKAN, 2015
32 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES WITH NORDEN METRICS 33
1.1. Norden metrics. A metric g is a Norden metric [15] if
g(ϕX,ϕY ) = g(X,Y )
or equivalently
g(ϕX, Y ) = g(X,ϕY )
for any X,Y ∈ =1
0(M2n). Metrics of this kind have been also studied under the names: pure, anti-
Hermitian and B-metric (see [5, 7, 12, 17, 23, 25]). If (M2n, ϕ) is an almost paracomplex manifold
with Norden metric g, we say that (M2n, ϕ, g) is an almost para-Norden manifold. If ϕ is integrable,
we say that (M2n, ϕ, g) is a para-Norden manifold.
1.2. Paraholomorphic (almost paraholomorphic) tensor fields. Let
∗
t be a paracomplex tensor
field on Mn(R(j)). The real model of such a tensor field is a tensor field on M2n 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., [12, 18, 19, 23 – 25, 27]). In particular, being applied to a (0, q)-tensor field ω, the
purity means that for any X1, . . . , Xq ∈ =1
0(M2n), the following conditions should hold:
ω(ϕX1, X2, . . . , Xq) = ω(X1, ϕX2, . . . , Xq) = . . . = ω(X1, X2, . . . , ϕXq).
We define an operator
Φϕ : =0
q(M2n)→ =0
q+1(M2n)
applied to the pure tensor field ω by (see [27])
(Φϕω)(X,Y1, Y2, . . . , Yq) = (ϕX)(ω(Y1, Y2, . . . , Yq))−X(ω(ϕY1, Y2, . . . , Yq))+
+ω((LY1ϕ)X,Y2, . . . , Yq) + . . .+ ω(Y1, Y2, . . . , (LYqϕ)X),
where LY denotes the Lie differentiation with respect to Y.
When ϕ is a paracomplex structure on M2n and the tensor field Φϕω vanishes, the paracomplex
tensor field
∗
ω on Mn(R(j)) is said to be paraholomorphic (see [12, 23, 27]). Thus a paraholomorphic
tensor field
∗
ω on Mn(R(j)) is realized on M2n in the form of a pure tensor field ω, such that
(Φϕω)(X,Y1, Y2, . . . , Yq) = 0
for any X,Y1, . . . , Yq ∈ =1
0(M2n). Therefore such a tensor field ω on M2n is also called paraholo-
morphic tensor field. When ϕ is an almost paracomplex structure on M2n, a tensor field ω satisfying
Φϕω = 0 is said to be almost paraholomorphic.
1.3. Paraholomorphic Norden (para-Kähler – Norden) metrics. In a para-Norden manifold a
para-Norden metric g is called a paraholomorphic if
(Φϕg)(X,Y, Z) = 0 (1)
for any X,Y, Z ∈ =1
0(M2n).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
34 M. İŞCAN, M. ÖZKAN
By setting X = ∂k, Y = ∂i, Z = ∂j in the equation (1), we see that the components (Φϕg)kij
of Φϕg with respect to a local coordinate system x1, . . . , xn may be expressed as follows:
(Φϕg)kij = ϕm
k ∂mgij − ϕm
i ∂kgmj + gmj (∂iϕ
m
k − ∂kϕm
i ) + gim∂jϕ
m
k .
If (M2n, ϕ, g) is a para-Norden manifold with paraholomorphic Norden metric g, we say that
(M2n, ϕ, g) is a paraholomorphic Norden manifold.
In some aspects, paraholomorphic Norden manifolds are similar to Kähler manifolds. The
following theorem is analogue to the next known result: An almost Hermitian manifold is Kähler if
and only if the almost complex structure is parallel with respect to the Levi – Civita connection.
Theorem 1 [21] (for complex version see [10]). For an almost paracomplex manifold with para-
Norden metric g, the condition Φϕg = 0 is equivalent to ∇ϕ = 0, where ∇ is the Levi – Civita
connection of g.
A para-Kähler – Norden manifold can be defined as a triple (M2n, ϕ, g) which consists of a
manifold M2n endowed with an almost paracomplex 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
para-Nordenian. Therefore, there exist a one-to-one correspondence between para-Kähler – Norden
manifolds and para-Norden manifolds with a paraholomorphic metric. Recall that in such a manifold,
the Riemannian curvature tensor is pure and paraholomorphic, also the curvature scalar is locally
paraholomorphic function (see [10, 17]).
Remark 1. We know that the integrability of the almost paracomplex 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 para-Norden manifold with conditions Φϕg = 0 and Nϕ 6= 0,
i.e., almost paraholomorphic Norden manifolds (analogues of the almost para-Kähler manifolds with
closed para-Kähler form) does not exist.
2. Walker metrics in dimension four. A neutral metric g on a 4-manifold M4 is said to be
Walker metric if there exists a 2-dimensional null distribution D on M4, which is parallel with respect
to g. For such metrics a canonical form has been obtained by Walker [26], showing the existence of
suitable coordinates (x1, x2, x3, x4) around any point of M4 where the metric expresses as
g = (gij) =
0 0 1 0
0 0 0 1
1 0 a c
0 1 c b
,
for some functions a, b and c depending on the coordinates (x1, x2, x3, x4). Note that D =
= span {∂1, ∂2}
(
∂i =
∂
∂xi
)
. For an application of such a 4-dimensional Walker metric (see [9]).
Since the observation of the existence of almost paracomplex structures on Walker 4-manifolds in a
paper [20], the Walker 4-manifolds have been intensively studied, e.g., [1, 3, 4, 8, 13, 14, 16, 20, 22].
As in a resent paper [15], we shall study throughout this paper the following Walker metrics of
restricted type (c = 0):
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES WITH NORDEN METRICS 35
g = (gij) =
0 0 1 0
0 0 0 1
1 0 a 0
0 1 0 b
. (2)
3. Almost paracomplex structure ϕ in the case of c = 0. A natural way to construct of an
almost paracomplex structure ϕ on a neutral 4-manifold is as follows: choose a local orthonormal
basis {ei} , i = 1, . . . , 4, so that with respect to the basis the neutral metric becomes the standard
form
g = (g(ei, ej)) =
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
,
and then define ϕ by
ϕe1 = e2, ϕe2 = e1, ϕe3 = e4, ϕe4 = e3. (3)
We consider the Walker metrics with c = 0 as follows:
g = (gij) =
0 0 1 0
0 0 0 1
1 0 a 0
0 1 0 b
, (4)
where a and b are functions of suitable coordinates (x1, x2, x3, x4) around any point of M4. In this
case, we find a local orthonormal basis {e1, e2, e3, e4} [14] ((14)), as follows:
e1 =
1
4
√
a2 + 4
{
1
2
(
√
a2 + 4− a)∂1 + ∂3
}
,
e2 =
1
4
√
b2 + 4
{
1
2
(
√
b2 + 4− b)∂2 + ∂4
}
,
e3 =
1
4
√
a2 + 4
{
−1
2
(
√
a2 + 4 + a)∂1 + ∂3
}
,
e4 =
1
4
√
b2 + 4
{
−1
2
(
√
b2 + 4 + b)∂2 + ∂4
}
.
(5)
For the Walker metric (2) with c = 0, the dual basis
{
e1, e2, e3, e4
}
of 1-forms to the basis (5) of
vectors is given by [14] ((19))
e1 =
1
4
√
a2 + 4
{
dx1 +
1
2
(
√
a2 + 4 + a)dx3
}
,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
36 M. İŞCAN, M. ÖZKAN
e2 =
1
4
√
b2 + 4
{
dx2 +
1
2
(
√
b2 + 4 + b)dx4
}
,
e3 = − 1
4
√
a2 + 4
{
dx1 − 1
2
(
√
a2 + 4− a)dx3
}
,
e4 = − 1
4
√
b2 + 4
{
dx2 − 1
2
(
√
b2 + 4− b)dx4
}
.
We now put K = 4
√
(b2 + 4)/(a2 + 4). The almost paracomplex structures defined by (3) is
written explicitly as follows:
ϕ = e1 ⊗ e2 + e2 ⊗ e1 + e3 ⊗ e4 + e4 ⊗ e3 =
=
0
1
K
0
1
2
(
b
K
− aK
)
K 0
1
2
(
aK − b
K
)
0
0 0 0 K
0 0
1
K
0
, (6)
where these matrices are written with respect to the coordinate basis. In this case, the triple (M4, ϕ, g)
is called almost para-Norden – Walker manifold.
4. ϕ-Integrability (para-Norden structures). If we write as ϕ∂i =
∑4
j=1
ϕj
i∂j , then from (6)
we can read off the nonzero components ϕj
i as follows:
ϕ2
1 = K, ϕ1
2 =
1
K
, ϕ2
3 =
1
2
(
aK − b
K
)
,
ϕ4
3 =
1
K
, ϕ1
4 =
1
2
(
b
K
− aK
)
, ϕ3
4 = K.
(7)
The almost paracomplex structure ϕ is integrable if and only if the torsion of ϕ (Nijenhuis tensor)
vanishes, or equivalently the following components:
(Nϕ)ijk = ϕm
j ∂mϕ
i
k − ϕm
k ∂mϕ
i
j − ϕi
m∂jϕ
m
k + ϕi
m∂kϕ
m
j
all vanish (cf. [12, p. 124]), where ϕj
i are given by (7). By explicit calculation, we find the ϕ-
integrability condition as follows.
Theorem 2. The almost paracomplex structure ϕ on almost para-Norden – Walker manifolds is
integrable if and only if the following PDE’s hold:
K1 = 0, K2 = 0, K2a1 − b1 − 2KK3 = 0, K2a2 − b2 −
2
K
K4 = 0. (8)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES WITH NORDEN METRICS 37
In this note, we will try to find the general solutions according to suitable local coordinates for
the PDE’s in above theorem.
Theorem 3. The almost paracomplex structure ϕ on almost para-Norden – Walker manifolds is
integrable if and only if a and b satisfy one of the following:
type ϕA : K =
{
(b2 + 4)/(a2 + 4)
}1/4
(= C) is constant, and
a = a
(
x3, x4
)
, b = b
(
x3, x4
)
,
type ϕB : a
(
x1, x2, x3, x4
)
= b
(
x1, x2, x3, x4
)
(necessarily K = 1),
and either a1 6= 0 or a2 6= 0,
type ϕC : K =
{
(b2 + 4)/(a2 + 4)
}1/4
is not constant, and
a
(
x1, x2, x3, x4
)
= ψ−2
(
ψL+ φ− ψ4 − 1
ψL+ φ
)
,
b
(
x1, x2, x3, x4
)
= −ψL− φ− ψ4 − 1
ψL+ φ
,
(9)
with
L = L
(
x1, x2, x3, x4
)
= ψ3x
1 + ψ−2ψ4x
2, (10)
for a and b functions defined according to suitable coordinates values of (x1, x2, x3, x4), ψ =
= ψ(x3, x4) and φ = φ(x3, x4) are smooth functions of x3 and x4 such that ψ(x3, x4) 6= 0 and
φ(x3, x4) 6= 0 for suitable points (x3, x4), and x1 6= − ψ4
ψ2ψ3
x2 − φ
ψψ3
must be satisfied for suitable
points (x1, x2). Moreover, the function K depends only on x3, x4, and coincides with ψ.
We shall prove this theorem in three steps.
Proof. The first step: From the former two equations in (8), we must note that K =
=
{
(b2 + 4)/(a2 + 4)
}1/4
does not depend on x1, x2, and further that the latter two can be written
as follows:
(K2a− b)1 = 2KK3, (K2a− b)2 =
2
K
K4. (11)
Integrating these equations with respect to x1 and x2, respectively, we have
K2a− b = 2KK3x
1 + pA(x2, x3, x4),
K2a− b =
2
K
K4x
2 + pB(x1, x3, x4),
(12)
where pA and pB are arbitrary functions of x2, x3, x4 and of x1, x3, x4, respectively. Differ-
entiating the former equation by x2, then we have (K2a − b)2 = pA2 (x2, x3, x4) =
2
K
K4, and
hence pA(x2, x3, x4) =
2
K
K4x
2 + qA(x3, x4). For pB, similarly we can write pB(x1, x3, x4) =
= 2KK3x
1 + qB(x3, x4). Using these pA and pB in (12), we see that qA and qB coincide with each
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
38 M. İŞCAN, M. ÖZKAN
other, and denote them by 2f(x3, x4). In fact, we obtain
K2a− b = 2KK3x
1 +
2
K
K4x
2 + 2f(x3, x4). (13)
From the relation K =
{
(b2 + 4)/(a2 + 4)
}1/4
, we get
(K4)1 = 0⇒ bb1(a
2 + 4) = aa1(b
2 + 4),
(K4)2 = 0⇒ bb2(a
2 + 4) = aa2(b
2 + 4),
and hence
K4 =
b2 + 4
a2 + 4
=
bb1
aa1
=
bb2
aa2
. (14)
(End of the first step.)
In the subsequent steps of the proof, we divide the situation into two cases as follows:
Case I: K is constant.
Case II: K depends only on x3 and x4.
The second step: We consider here the first Case I: K is constant, denoted by K = C, i.e.,
C4(a2 + 4) = b2 + 4. In this case, the equations (11) reduce to
(C2a− b)i = (C2a±
√
C4(a2 + 4)− 4)i = 0, i = 1, 2.
There are two types of solutions to these equations as follows:
i) C2a− b = 0, where a and b can be functions of x1, x2, x3 and x4, or
ii) a1 = a2 = b1 = b2 = 0.
For i), in fact, the relation C2a−b = 0 together with C4(a2+4) = b2+4 implies thatK = C = 1,
and that a
(
x1, x2, x3, x4
)
= b
(
x1, x2, x3, x4
)
which is of type ϕB.
It is easy to see that if C2a − b 6= 0, then there is another possibility of the second case
ii) a1 = a2 = b1 = b2 = 0. Therefore, if K is constant (K = C) (including K = 1), then
a = a
(
x3, x4
)
and b = b
(
x3, x4
)
are solutions to (8). Such solutions are of type ϕA. Here, we must
note that such a and b are subject to a relation C4(a2 + 4) = b2 + 4. ( End of the second step.)
The third step: In this final step, we consider the Case II: K is independent of x1 and x2. From
(14), we have K2aa1 =
1
K2
bb1, and add −ba1 both sides of it. Then, we have
(K2a− b)a1 = − b
K2
(K2a− b)1.
From the former equation in (11), we obtain
(K2a− b)a1 = −2K3
K
b =
2K3
K
{
−K2a+ (K2a− b)
}
.
Using (13), we get(
KK3x
1 +
1
K
K4x
2 + f
)
a1 +KK3a =
2K3
K
(
KK3x
1 +
1
K
K4x
2 + f
)
. (15)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES WITH NORDEN METRICS 39
From a similar calculation, we have an analogous equation for x2 as follows:(
KK3x
1 +
1
K
K4x
2 + f
)
a2 +
1
K
K4a =
2K4
K3
(
KK3x
1 +
1
K
K4x
2 + f
)
. (16)
At this stage, we recall that for a function y(t) of single argument t, an ODE of the form
(αt+ β)
dy(t)
dt
+ αy(t) = γt+ δ (α, β, γ, δ — constants)
has a solution y(t) =
1
2
γt2 + δt+ αC
αt+ β
(C — constant). If we regard the equation (15) as such an
ODE with respect to x1, with x2, x3, x4 as parameters, then we have its solution as follows:
a =
K2
3
(
x1
)2
+
2K3K4
K2
x1x2 +
2
K
K3fx
1 +KK3h
A(x2, x3, x4)
KK3x1 +
1
K
K4x2 + f
.
In a similar way, we can obtain a solution to (16) as follows:
a =
1
K4
K2
4
(
x2
)2
+
2K3K4
K2
x1x2 +
2
K3
K4fx
2 +
K4
K
hB(x1, x3, x4)
KK3x1 +
1
K
K4x2 + f
.
In the above two equations, hA and hB are arbitrary functions of x2, x3, x4 and of x1, x3, x4,
respectively. Comparing the above two solutions for a, we have
KK3h
A(x2, x3, x4) =
1
K4
K2
4
(
x2
)2
+
2
K3
K4fx
2 + h(x3, x4),
K4
K
hB(x1, x3, x4) =
2
K
K3fx
1 +K2
3
(
x1
)2
+ h(x3, x4),
where h(x3, x4) is an arbitrary function of x3, x4. Therefore, we see that a is written as
a =
K2
3
(
x1
)2
+
2K3K4
K2
x1x2 +
1
K4
K2
4
(
x2
)2
+
2
K
K3fx
1 +
2
K3
K4fx
2 + h
KK3x1 +
1
K
K4x2 + f
=
=
1
K2
KK3x
1 +
1
K
K4x
2 + f − f2 −K2h
KK3x1 +
1
K
K4x2 + f
.
From this expression for a, we can obtain, with (13), the explicit form of the function b as well as a:
b = K2a− 2KK3x
1 − 2
K
K4x
2 − 2f(x3, x4) =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
40 M. İŞCAN, M. ÖZKAN
=−KK3x
1 − 1
K
K4x
2 − f − f2 −K2h
KK3x1 +
1
K
K4x2 + f
.
These expressions for a and b contain two arbitrary functions f and h of x3 and x4. Taking into
account of K =
{
(b2 + 4)/(a2 + 4)
}1/4
for the above solutions a and b, we can see that there is a
relation among f, h and K as follows:
f2 −K2h = K4 − 1.
At the final stage of the proof, we will arrange the expressions for a, b so that they look simple.
Keeping the last expression in mind, we can regard K as one of arbitrary functions with arguments
x3, x4, instead of h. Then, we denote K(x3, x4) by a new symbol ψ = ψ(x3, x4), and also put
φ = φ(x3, x4) = f(x3, x4). If we write L = K3x
1 +
1
K2
K4x
2 = ψ3x
1 + ψ−2ψ4x
2 as in (10), we
have arrived at the desired expressions as in (9). Also, for a and b functions defined according to
suitable coordinates values of (x1, x2, x3, x4), ψ = ψ(x3, x4) and φ = φ(x3, x4) must be smooth
functions of x3 and x4 such that ψ(x3, x4) 6= 0 and φ(x3, x4) 6= 0 for suitable points (x3, x4), and
x1 6= − ψ4
ψ2ψ3
x2− φ
ψψ3
must be satisfied for suitable points (x1, x2). Such a case is classified as type
ϕC . (End of the third step.)
Theorem 3 is proved.
5. Paraholomorphic Norden – Walker (para-Kähler – Norden – Walker) metrics on
(M4, ϕ, g). Let (M4, ϕ, g) be an almost para-Norden – Walker manifold. If
(Φϕg)kij = ϕm
k ∂mgij − ϕm
i ∂kgmj + gmj(∂iϕ
m
k − ∂kϕm
i ) + gim∂jϕ
m
k = 0, (17)
then by virtue of Theorem 1 ϕ is integrable and the triple (M4, ϕ, g) is called a paraholomorphic
Norden – Walker or a para-Kähler – Norden – Walker manifold. Taking account of Remark 1, we see
that almost para-Kähler – Norden – Walker manifold with condition Φϕg = 0 and Nϕ 6= 0 does not
exist.
We will write (4) and (7) in (17). By explicit calculation, we have the following theorem.
Theorem 4. The triple (M4, ϕ, g) is para-Kähler – Norden – Walker if and only if the following
PDEs hold:
K1 = 0, K2 = 0, a2 = a4 = b1 = b3 = 0, Ka1 − 2K3 = 0, Kb2 + 2K4 = 0.
Remark 2. In a recent paper [20], a proper almost paracomplex structure on a Walker 4-manifold
is defined and analyzed. The almost paracomplex structure ϕ defined in (6) coincides with that defined
in [20] ((3)) in each of the cases (a) c = 0 and a = b, and case (b) c = 0 and a = −b. Note that in
the former case (a), ϕ is integrable (cf. [20], Theorem 2). In fact, such happens in the following two
situations:
i) a
(
x3, x4
)
= b
(
x3, x4
)
in type ϕA (in Theorem 3),
ii) a
(
x1, x2, x3, x4
)
= b
(
x1, x2, x3, x4
)
in type ϕB (in Theorem 3).
1. Bonome A., Castro R., Hervella L. M., Matsushita Y. Construction of Norden structures on neutral 4-manifolds // JP
J. Geom. Top. – 2005. – 5, № 2. – P. 121–140.
2. Cruceanu V., Fortuny P., Gadea P. M. A survey on paracomplex geometry // Rocky Mountain J. Math. – 1996. – 26,
№ 1. – P. 83 – 115.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ON FOUR DIMENSIONAL PARACOMPLEX STRUCTURES WITH NORDEN METRICS 41
3. Davidov J., Dı́az-Ramos J. C., Garcı́a-Rı́o E., Matsushita Y., Muškarov O., Vázquez-Lorenzo R. Almost Kähler Walker
4-manifolds // J. Geom. Phys. – 2007. – 57. – P. 1075 – 1088.
4. Davidov J., Dı́az-Ramos J. C., Garcı́a-Rı́o E., Matsushita Y., Muškarov O., Vázquez-Lorenzo R. Hermitian – Walker
4-manifolds // J. Geom. Phys. – 2008. – 58. – P. 307 – 323.
5. Etayo F., Santamaria R. (J2 = ±1)-metric manifolds // Publ. Math. Debrecen. – 2000. – 57, № 3-4. – P. 435 – 444.
6. Gadea P. M., Grifone J., Munoz Masque J. Manifolds modelled over free modules over the double numbers // Acta
Math. hung. – 2003. – 100, № 3. – P. 187 – 203.
7. Ganchev G. T., Borisov A. V. Note on the almost complex manifolds with a Norden metric // C. R. Acad. Bulg. Sci. –
1986. – 39, № 5. – P. 31 – 34.
8. Garcı́a-Rı́o E., Haze S., Katayama N., Matsushita Y. Symplectic, Hermitian and Kahler structures on Walker 4-
manifolds // J. Geom. – 2008. – 90. – P. 56 – 65.
9. Ghanam R., Thompson G. The holonomy Lie algebras of neutral metrics in dimension four // J. Math. Phys. – 2001. –
42. – P. 2266 – 2284.
10. Iscan M., Salimov A. A. On Kähler – Norden manifolds // Proc. Indian Acad. Sci. Math. Sci. – 2009. – 119, № 1. –
P. 71 – 80.
11. Kobayashi S., Nomizu K. Foundations of differential geometry II. – New York; London: John Wiley, 1969.
12. Kruchkovich G. I. Hypercomplex structure on a manifold, I // Tr. Sem. Vect. Tens. Anal., Moscow Univ. – 1972. –
16. – P. 174 – 201.
13. Matsushita Y. Four-dimensional Walker metrics and symplectic structure // J. Geom. Phys. – 2004. – 52. – P. 89–99;
Erratum, J. Geom. Phys. – 2007. – 57. – P. 729.
14. Matsushita Y. Walker 4-manifolds with proper almost complex structure // J. Geom. Phys. – 2005. – 55. – P. 385 – 398.
15. Norden A. P. On a certain class of four-dimensional A-spaces // Iz. Vuzov. – 1960. – 4. – P. 145 – 157.
16. Özkan M., İşcan M. Some properties of para-Kähler – Walker metrics // Ann. pol. math. – 2014. – 112. – P. 115 – 125.
17. Salimov A. A. Almost analyticity of a Riemannian metric and integrability of a structure (in Russian) // Trudy Geom.
Sem. Kazan. Univ. – 1983. – 15. – P. 72 – 78.
18. Salimov A. A. Generalized Yano – Ako operator and the complete lift of tensor fields // Tensor (N. S.). – 1994. – 55,
№ 2. – P. 142 – 146.
19. Salimov A. A. Lifts of poly-affinor structures on pure sections of a tensor bundle // Russian Math. (Iz. Vuzov). –
1996. – 40, № 10. – P. 52 – 59.
20. Salimov A. A., Iscan M., Akbulut K. Notes on para-Norden – Walker 4-manifolds // Int. J. Geom. Methods Mod. Phys. –
2010. – 7, № 8. – P. 1331 – 1347.
21. Salimov A. A., Iscan M., Etayo F. Paraholomorphic B-manifold and its properties // Top. Appl. – 2007. – 154. –
P. 925 – 933.
22. Salimov A. A., Iscan M. Some properties of Norden – Walker metrics // Kodai Math. J. – 2010. – 33, № 2. –
P. 283 – 293.
23. Tachibana S. Analytic tensor and its generalization // Tôhoku Math. J. – 1960. – 12, № 2. – P. 208 – 221.
24. Vishnevskii V. V., Shirokov A. P., Shurygin V. V. Spaces over algebras. – Kazan: Kazan Gos. Univ., 1985 (in Russian).
25. Vishnevskii V. V. Integrable affinor structures and their plural interpretations // J. Math. Sci. – 2002. – 108, № 2. –
P. 151 – 187.
26. Walker A. G. Canonical form for a Riemannian space with a parallel field of null planes // Quart. J. Math. Oxford. –
1950. – 1, № 2. – P. 69 – 79.
27. Yano K., Ako M. On certain operators associated with tensor fields // Kodai Math. Sem. Rep. – 1968. – 20. –
P. 414 – 436.
Received 21.11.12,
after revision — 22.04.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
|
| id | umjimathkievua-article-1961 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:16:02Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/2d/3ebf9cfac169491fd29ce15cae091e2d.pdf |
| spelling | umjimathkievua-article-19612019-12-05T09:47:39Z On Four-Dimensional Paracomplex Structures with Norden Metrics Про чотиривимірні паракомплексні структури з метриками Нордена İşcan, M. Özkan, M. Іскан, М. Озкан, М. We study almost paracomplex structures with Norden metric on Walker 4-manifolds and try to find general solutions for the integrability of these structures on suitable local coordinates. We also discuss para-Kähler (paraholomorphic) conditions for these structures. Вивчаються майже паракомплексні структури з метрикою Нордена на 4-многовидах Уолкера. Встановлено загальні розв'язки щодо інтегровності таких структур у відповідних локальних координатах. Також обговорюються пара-кєлєрові (параголоморфні) умови для таких структур. Institute of Mathematics, NAS of Ukraine 2015-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1961 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 1 (2015); 32-41 Український математичний журнал; Том 67 № 1 (2015); 32-41 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1961/945 https://umj.imath.kiev.ua/index.php/umj/article/view/1961/946 Copyright (c) 2015 İşcan M.; Özkan M. |
| spellingShingle | İşcan, M. Özkan, M. Іскан, М. Озкан, М. On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title | On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title_alt | Про чотиривимірні паракомплексні структури з метриками Нордена |
| title_full | On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title_fullStr | On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title_full_unstemmed | On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title_short | On Four-Dimensional Paracomplex Structures with Norden Metrics |
| title_sort | on four-dimensional paracomplex structures with norden metrics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1961 |
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