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International Journal of Geometric Methods in Modern PhysicsVol. 7, No. 8 (2010) 1331–1347c© World Scientific Publishing CompanyDOI: 10.1142/S021988781000483X

NOTES ON PARA-NORDEN–WALKER 4-MANIFOLDS∗

A. A. SALIMOV∗,†, M. ISCAN‡ and K. AKBULUT§

Department of MathematicsFaculty of Science, Ataturk University

Erzurum, Turkey†asalimov@atauni.edu.tr‡miscan@atauni.edu.tr§kakbulut@atauni.edu.tr

Received 27 December 2009Accepted 23 May 2010

A Walker 4-manifold is a pseudo-Riemannian manifold, (M4, g) of neutral signa-ture, which admits a field of parallel null 2-plane. The main purpose of the presentpaper is to study almost paracomplex structures on 4-dimensional Walker manifolds.We discuss sequently the problem of integrability, para-Kahler (paraholomorphic),quasi-para-Kahler and isotropic para-Kahler conditions for these structures. The curva-ture properties for para-Norden–Walker metrics with respect to the almost paracomplexstructure and some properties of para-Norden–Walker metrics in context of almost prod-uct Riemannian manifolds are also investigated. Also, we discuss the Einstein conditionsfor these structures.

Keywords: Walker 4-manifolds; almost paracomplex structure; Norden metrics; holo-morphic metrics.

Mathematics Subject Classification: 53C50, 53B30

1. Introduction

Let M2n be a Riemannian manifold with neutral metric, i.e. with pseudo-Riemannian metric g of signature (n, n). We denote by �p

q(M2n) the set of alltensor fields of type (p, q) on M2n. Manifolds, tensor fields and connections arealways 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 thetwo eigenvalues +1 and −1 of ϕ, respectively, have the same rank. Note thatthe dimension of an almost paracomplex manifold is necessarily even. Consideringthe paracomplex structure ϕ, we obtain the following set of affinors on M2n: {id, ϕ},

∗This paper is supported by The Scientific and Technological Research Council of Turkey withnumber TBAG (108T590).

1331

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1332 A. A. Salimov, M. Iscan & K. Akbulut

ϕ2 = id, which form a basis of a representation of the algebra of order 2 over thefield 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 isassociative, commutative and unitial, i.e. it admits principal unit 1. The canonicalbasis of this algebra has the form {1, j}.

Let (M2n, ϕ) be an almost paracomplex manifold with almost paracomplexstructure ϕ. For almost paracomplex structure the integrability is equivalent tothe 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 = (ϕij) is reduced to the

constant form in a certain holonomic natural frame in a neighborhood Ux of everypoint x ∈ M2n. On the other hand, in order that an almost paracomplex structurebe integrable, it is necessary and sufficient that we can introduce a torsion-freelinear connection such that ∇ϕ = 0. A paracomplex manifold is an almost para-complex manifold (M2n, ϕ) such that the G-structure defined by the affinor fieldϕ is integrable. We can give another equivalent definition of paracomplex mani-fold in terms of local homeomorphisms in the space Rn(j) = {(X1, . . . , Xn)/X i ∈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 areal realization of the paraholomorphic manifold Mn(R(j)) over the algebra R(j).

1.1. Norden metrics

A metric g is a Norden metric [19] if

g(ϕX, Y ) = g(X, ϕY )

for any X, Y ∈ �10(M2n). Note that a linear operator ϕ satisfying the Norde-

nian property is called a self-adjoint. Metrics of this type have also been studiedunder the other names: Pure metrics, anti-Hermitian metrics and B-metrics (see[5, 7, 12, 20, 26, 28]). If (M2n, ϕ) is an almost paracomplex manifold with Nordenmetric g, we say that (M2n, ϕ, g) is an almost para-Norden manifold. If ϕ is inte-grable, we say that (M2n, ϕ, g) is a para-Norden manifold. A para-Norden manifold(M2n, ϕ, g) is a realization of the paraholomorphic manifold (Mn(R(j)),

∗g), where

∗g = (

∗guv), u, v = 1, . . . , n is a paracomplex metric tensor field on Mn(R(j)), which

in general, is non-paraholomorphic [12].

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 irrespective of whether its vector orcovector arguments is subject to the action of the affinor structure ϕ. Such tensorfields are said to be pure with respect to ϕ. They were studied by many authors

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Notes on Para-Norden–Walker 4-Manifolds 1333

(see, e.g. [12, 21, 22, 26−28, 31]). In particular, for a (0, q)-tensor field ω, the puritymeans 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 : �0q(M2n) → �0

q+1(M2n)

applied to a pure tensor field ω by (see [31])

(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, 26, 31]). 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 ∈ �10(M2n). Therefore such a tensor field ω on M2n is also

called paraholomorphic tensor field. When ϕ is an almost paracomplex structureon M2n, a tensor field ω satisfying Φϕω = 0 is said to be almost paraholomorphic.

1.3. Paraholomorphic Norden (Para-Kahler–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 ∈ �10(M2n).

By setting X = ∂k, Y = ∂i, Z = ∂j in Eq. (1), we see that the compo-nents (Φϕg)kij of Φϕg with respect to a local coordinate system x1, . . . , xn maybe expressed as follows:

(Φϕg)kij

= ϕmk ∂mgij − ϕm

i ∂kgmj + gmj(∂iϕmk − ∂kϕm

i ) + gim∂jϕmk .

If (M2n, ϕ, g) is a para-Norden manifold with paraholomorphic Norden metric,we say that (M2n, ϕ, g) is a paraholomorphic Norden manifold.

In some aspects, paraholomorphic Norden manifolds are similar to Kahler man-ifolds. The following theorem is an analogue to the next known result: An almostHermitian manifold is Kahler if and only if the almost complex structure is parallelwith respect to the Levi-Civita connection.

Theorem 1 ([23]). (For a complex version see [11].) For an almost paracomplexmanifold with para-Norden metric g, the condition Φϕg = 0 is equivalent to ∇ϕ = 0,

where ∇ is the Levi-Civita connection of g.

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1334 A. A. Salimov, M. Iscan & K. Akbulut

A para-Kahler–Norden manifold can be defined as a triple (M2n, ϕ, g) whichconsists of a manifold M2n endowed with an almost paracomplex structure ϕ anda pseudo-Riemannian metric g such that ∇ϕ = 0, where ∇ is the Levi-Civitaconnection of g and the metric g is assumed to be a para-Nordenian. Therefore,there exist a one-to-one correspondence between para-Kahler–Norden manifolds andpara-Norden manifolds with paraholomorphic metric. Recall that the Riemanniancurvature tensor of such a manifold is pure and paraholomorphic, also the curvaturescalar is locally paraholomorphic function (see [11, 20]).

Remark 1. We know that the integrability of an almost paracomplex structureϕ is equivalent to the existence of a torsion-free affine connection with respect towhich the equation ∇ϕ = 0 holds. Since the Levi-Civita connection ∇ of g is atorsion-free affine connection, we have: If Φϕg = 0, then ϕ is integrable. Thus,almost para-Norden manifold with conditions Φϕg = 0 and Nϕ �= 0, i.e. almostparaholomorphic Norden manifolds (analogues of the almost para-Kahler manifoldswith closed para-Kahler form) do not exist.

1.4. Twin para-Norden metrics

Let (M2n, ϕ, g) be an almost para-Norden manifold. The associated para-Nordenmetric of almost para-Norden manifold is defined by

G(X, Y ) = (g ◦ ϕ)(X, Y )

for all vector fields X and Y on M2n. One can easily prove that G is a new para-Norden metric:

G(ϕX, Y ) = (g ◦ ϕ)(ϕX, Y ) = g(ϕ2X, Y ) = g(X, Y )

= g(ϕX, ϕY ) = (g ◦ ϕ)(X, ϕY ) = G(X, ϕY )

which is called the twin(or dual) para-Norden metric of g and it plays a role similarto the Kahler form in Hermitian geometry.

We denote by ∇g the covariant differentiation of the Levi-Civita connection ofpara-Norden metric g. Then, we have

∇gG = (∇gg) ◦ ϕ + g ◦ (∇gϕ) = g ◦ (∇gϕ),

which implies ∇gG = 0 by virtue of Theorem 1. Therefore we have [26]: The Levi-Civita connection of para-Kahler–Norden metric g coincides with the Levi-Civitaconnection of twin para-Norden metric G (i.e. nonuniqueness of the metric for theLevi-Civita connection in para-Kahler–Norden manifolds).

2. Para-Norden–Walker Metrics

In the present paper, we shall focus our attention to para-Norden manifolds ofdimension 4. Using a Walker metric we construct new para-Norden–Walker metricstogether with a proper almost paracomplex structure.

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2.1. Walker metric g

A neutral metric g on a 4-manifold M4 is said to be a Walker metric if there existsa 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 [29], showing theexistence of suitable coordinates (x, y, z, t) where the metric expresses as

g = (gij) =

0 0 1 00 0 0 11 0 a c

0 1 c b

, (2)

for some functions a, b and c depending on the coordinates (x, y, z, t).

2.2. Almost para-Norden–Walker manifolds

Let F be an almost paracomplex structure on a Walker manifold M4, which satisfies

(i) F 2 = I,

(ii) g(FX, Y ) = g(X, FY ) (Nordenian property),(iii) F∂x = ∂y, F∂y = ∂x.

We easily see that these three properties define F nonuniquely, i.e.

F∂x = ∂y,

F∂y = ∂x,

F∂z = −α∂x − 12(b − a)∂y + ∂t,

F∂t =12(b − a)∂x + α∂y + ∂z

and F has the local components

F = (F ij ) =

0 1 −α12(b − a)

1 0 −12(b − a) α

0 0 0 10 0 1 0

with respect to the natural frame {∂x, ∂y, ∂z, ∂t}, where α = α(x, y, z, t) is anarbitrary function.

Our proposal is to find a nontrivial para-Norden structure with Walker metric g

explicitly. If α = c, then we have explicit examples of Walker metrics with hyperbolicfunctions a, b and c (Theorem 3). Therefore, we put α = c. Then g defines a uniquealmost paracomplex structure

ϕ = (ϕij) =

0 1 −c12(b − a)

1 0 −12(b − a) c

0 0 0 10 0 1 0

. (3)

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1336 A. A. Salimov, M. Iscan & K. Akbulut

The triple (M4, ϕ, g) is called almost para-Norden–Walker manifold. In conformitywith the terminology of [1, 3, 4, 14–16, 24] we call ϕ the proper almost paracomplexstructure.

2.3. Integrability of ϕ

We consider the general case.The almost paracomplex structure ϕ of an almost para-Norden–Walker manifold

is integrable if and only if

(Nϕ)ijk = ϕm

j ∂mϕik − ϕm

k ∂mϕij − ϕi

m∂jϕmk + ϕi

m∂kϕmj = 0. (4)

By explicit calculation, the nonzero components of the Nijenhuis tensor are asfollows:

Nxxz = −Ny

xt = −Nyyz = Nx

yt =12(bx − ax) − cy,

Nxxt = −Ny

xz = −Nyyt = Nx

yz =12(by − ay) − cx,

Nxzt = −1

2c(bx − ax) +

12(b − a)cx − 1

4(b − a)(by − ay) + ccy,

Nyzt =

12c(by − ay) − 1

2(b − a)cy +

14(b − a)(bx − ax) − ccx.

(5)

From (5) we have:

Theorem 2. The proper almost paracomplex structure ϕ of an almost para-Norden–Walker manifold is integrable if and only if the following PDEs hold:{

ax − bx + 2cy = 0,

ay − by + 2cx = 0.(6)

From this theorem, we easily see that if a = b and c = 0, then ϕ is integrable.Let (M4, ϕ, g) be a para-Norden–Walker manifolds (Nϕ = 0) and a = −b. Then

Eq. (6) reduces to {ax = −cy,

ay = −cx,(7)

from which follows

axx − ayy = 0,

cxx − cyy = 0,(8)

e.g. the functions a and c are hyperbolic with respect to the arguments x and y.Thus we have:

Theorem 3. If the triple (M4, ϕ, g) is para-Norden–Walker and a = −b, then a, b

and c are all hyperbolic with respect to the arguments x, y.

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3. Paraholomorphic Norden–Walker(Para-Kahler–Norden–Walker) Metrics on (M4, ϕ, g)

Let (M4, ϕ, g) be an almost para-Norden–Walker manifold. If

(Φϕg)kij = ϕmk ∂mgij − ϕm

i ∂kgmj + gmj(∂iϕmk − ∂kϕm

i ) + gim∂jϕmk = 0, (9)

then by virtue of Theorem 1, ϕ is integrable and the triple is called a paraholo-morphic Norden–Walker or a para-Kahler–Norden–Walker manifold. Taking intoaccount Remark 1, we see that an almost para-Kahler–Norden–Walker manifoldwith conditions Φϕg = 0 and Nϕ �= 0 does not exist.

We will write (2) and (3) in (9). By explicit calculation, the nonzero componentsof the Φϕg tensor are as follows:

(Φϕg)xzz = ay, (Φϕg)xzt = cy − 12(ax + bx), (Φϕg)xtt = by − 2cx, (Φϕg)yzz = ax,

(Φϕg)yzt = cx − 12(ay + by), (Φϕg)ytt = bx − 2cy, (Φϕg)zxz = −(Φϕg)txt = −cx,

(Φϕg)zxt = −(Φϕg)txz =12(ax − bx), (Φϕg)zyz = −(Φϕg)tyt = −cy,

(Φϕg)zyt = −(Φϕg)tyz =12(ay − by), (Φϕg)zzz = −cax − 2cz + at − 1

2(b − a)ay,

(Φϕg)zzt = −ccx − bz − 12(b − a)cy, (Φϕg)ztt = cbx + at − 2cz − 1

2(b − a)by,

(Φϕg)tzz = cay + bz +12(b − a)ax, (Φϕg)tzt = ccy − at + 2cz +

12(b − a)cx,

(Φϕg)ttt = cby + bz +12(b − a)bx. (10)

From the above equations, we have:

Theorem 4. A triple (M4, ϕ, g) is a para-Kahler–Norden–Walker manifold if andonly if the following PDEs hold:

ax = ay = bx = by = bz = cx = cy = 0, at − 2cz = 0. (11)

Corollary 1. A triple (M4, ϕ, g) with metric

g = (gij) =

0 0 1 00 0 0 11 0 a(z) 00 1 0 b(t)

is always para-Kahler–Norden–Walker.

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1338 A. A. Salimov, M. Iscan & K. Akbulut

4. A New Equivalent Condition for Quasi-Para-Kahler–NordenMetrics

The basis class of non-integrable almost paracomplex manifolds with para-Nordenmetric is the class of the quasi-para-Kahler manifolds. An almost para-Nordenmanifold (M2n, ϕ, g) is called a quasi-para-Kahler–Norden [7, 13], if

σX,Y,Z

g((∇Xϕ)Y, Z) = 0,

where σ is the cyclic sum by three arguments.If we add (Φϕg)(X, Y, Z) and (Φϕg)(Z, Y, X), then by virtue of g(Z, (∇Y ϕ)X) =

g((∇Y ϕ)Z, X), we find

(Φϕg)(X, Y, Z) + (Φϕg)(Z, Y, X) = 2g((∇Y ϕ)Z, X).

Since (Φϕg)(X, Y, Z) = (Φϕg)(X, Z, Y ), from last equation we have

(Φϕg)(X, Y, Z) + (Φϕg)(Y, Z, X) + (Φϕg)(Z, X, Y ) = σX,Y,Z

g((∇Xϕ)Y, Z).

Thus we have:

Theorem 5. Let (M2n, ϕ, g) be an almost para-Norden manifold. Then the para-Norden metric g is a quasi-para-Kahler–Norden if and only if

(Φϕg)(X, Y, Z) + (Φϕg)(Y, Z, X) + (Φϕg)(Z, X, Y ) = 0

for any X, Y, Z ∈ �10(M2n).

Remark 2. The equation in Theorem 5 can also be written in the form

(Φϕg)(X, Y, Z) + 2g((∇Xϕ)Y, Z) = 0.

Let (M4, ϕ, g) be an almost para-Norden–Walker manifold. A para-Norden–Walker manifold (M4, ϕ, g) satisfying the condition Φkgij + 2∇kGij to be zero iscalled a quasi-para-Kahler manifold, where G is defined by Gij = ϕm

i gmj .

Remark 3. From (2) and (3) we easily see that, the twin para-Norden metric G

is non-Walker.

Let now (M4, ϕ, g) be a quasi-para-Kahler–Norden–Walker manifold, which bydefinition is non-para-Kahler–Norden (∇ϕ �= 0):

(∇iϕmj )gmk + (∇jϕ

mk )gmi + (∇kϕm

i )gmj = 0.

The inverse of the metric tensor (2), g−1 = (gij), given by

g−1 =

−a −c 1 0−c −b 0 1

1 0 0 00 1 0 0

.

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Notes on Para-Norden–Walker 4-Manifolds 1339

The Levi-Civita connection of a Walker metric (2) is given by

∇∂x∂z =12ax∂x +

12cx∂y, ∇∂x∂t =

12cx∂x +

12bx∂y,

∇∂y∂z =12ay∂x +

12cy∂y, ∇∂y∂t =

12cy∂x +

12by∂y,

∇∂z∂z =12(aax + cay + az)∂x +

12(cax + bay − at + 2cz)∂y − 1

2ax∂z − 1

2ay∂t,

∇∂z∂t =12(at + acx + ccy)∂x +

12(bz + ccx + bcy)∂y − 1

2cx∂z − 1

2cy∂t,

∇∂t∂t =12(abx + cby − bz + 2ct)∂x +

12(cbx + bby + bt)∂y − 1

2bx∂z − 1

2by∂t.

For the covariant derivative ∇G of the twin para-Norden metric G put(∇G)ijk = ∇iGjk = (∇iϕ

mj )gmk. Then, after some calculations we obtain

∇xGzz = −∇xGtt = −cx, ∇yGzz = −∇yGtt = −cy,

∇zGxz = ∇zGzx = −∇zGyt = −∇zGty =12(ay − cx),

∇zGxt = ∇zGtx = −∇zGyz = −∇zGzy =12(cy − ax),

∇zGzz = −2cz + at +12(a − b)ay − cax,

∇zGzt = ∇zGtz =12cay − 1

2ccx − 1

4ax(a − b) +

14cy(a − b),

∇zGtt = 2cz − at − 12(a − b)cx + ccy,

∇tGxz = ∇tGzx = −∇tGyt = −∇tGty = −12(bx − cy),

∇tGxt = ∇tGtx = −∇tGyz = −∇tGzy =12(by − cx),

∇tGzz = −bz +12(a − b)cy − ccx,

∇tGzt = ∇tGtz =12ccy − 1

2cbx − 1

4cx(a − b) +

14by(a − b),

∇tGtt = bz + cby − 12bx(a − b).

(12)

From (10) and (12), we have:

Theorem 6. A triple (M4, ϕ, g) is a quasi-para-Kahler–Norden–Walker manifoldif and only if the following PDEs hold:

bx = by = bz = 0, ay−2cx = 0, ax−2cy = 0, cax−at+2cz−(a−b)cx = 0. (13)

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1340 A. A. Salimov, M. Iscan & K. Akbulut

5. A New Equivalent Condition for Isotropic Para-Kahler–NordenMetrics

It is well known that the inner product in the vector space can be extended to aninner product in the tensor space. In fact, if t

1and t

2are tensors of type (r, s) with

components t1

i1···ir

j1···jsand t

2

k1···kr

l1···ls , then

g(t1, t

2) = gi1k1 · · · girkrg

j1l1 · · · gjsls t1

i1···ir

j1···jst2

k1···kr

l1···ls .

If t1

= t2

= ∇ϕ ∈ �12(M2n), then the square norm ‖∇ϕ‖2 of ∇ϕ is defined by

‖∇ϕ‖2 = gijgklgms(∇ϕ)mik(∇ϕ)s

jl .

An almost paracomplex manifold with para-Norden metric satisfying thecondition

‖∇ϕ‖2 = 0, ∇ϕ �= 0

is called an isotropic para-Kahler–Norden [7, 8, 13].Using

∇jϕsi =

12gst[(Φϕg)ijt + (Φϕg)tji]

we find

‖∇ϕ‖2 =14gijgklgtr[(Φϕg)kir + (Φϕg)rik][(Φϕg)ljt + (Φϕg)tjl].

Thus we have:

Theorem 7. Let (M2n, ϕ, g) be an almost para-Norden manifold. Then the para-Norden metric g is an isotropic para-Kahler–Norden if and only if

gijgklgtr[(Φϕg)kir + (Φϕg)rik][(Φϕg)ljt + (Φϕg)tjl] = 0.

Taking account of g−1 and (10), on para-Norden–Walker manifold M4 we obtain

gijgklgtr[(Φϕg)kir + (Φϕg)rik][(Φϕg)ljt + (Φϕg)tjl] = 0, i, j, . . . = 1, . . . , 4,

from which by virtue of Theorem 7, we have:

Theorem 8. If (M4, ϕ, g) is an almost para-Norden–Walker manifold, then it isisotropic para-Kahlerian.

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Notes on Para-Norden–Walker 4-Manifolds 1341

6. Curvature Properties of Para-Norden–Walker Manifolds

If R and r are, respectively, the curvature and the scalar curvature of the Walkermetric, then the components of R and r have, respectively, expressions (see [15],Appendices A and C)

Rxzxz = −12axx, Rxzxt = −1

2cxx, Rxzyz = −1

2axy, Rxzyt = −1

2cxy,

Rxzyz =12axt − 1

2cxz − 1

4aybx +

14cxcy, Rxtxt = −1

2bxx, Rxtyz = −1

2cxy,

Rxtyt = −12bxy, Rxtzt =

12cxt − 1

2bxz − 1

4(cx)2 +

14axbx − 1

4bxcy +

14bycx,

Ryzyz = −12ayy, Ryzyt = −1

2cyy,

Ryzzt =12ayt − 1

2cyz − 1

4axcy +

14aycx − 1

4ayby +

14(cy)2, Rytyt = −1

2byy,

Rytzt =12cyt − 1

2byz − 1

4cxcy +

14aybx,

Rztzt = czt − 12att − 1

2bzz − 1

4a(cx)2 +

14aaxbx +

14caxby − 1

2ccxcy

− 12atcx +

12axct − 1

4axbz +

14caybx +

14bayby − 1

4b(cy)2

− 12bzcy +

14aybt +

14azbx +

12bycz − 1

4atby (14)

and

r = axx + 2cxy + byy. (15)

Suppose that the triple (M4, ϕ, g) is para-Kahler–Norden–Walker. Then fromthe last equation in (11) and (14), we see that

Rztzt = czt − 12att = −1

2(at − 2cz)t = 0.

From (11) we easily see that the other components of R in (14) directly all vanish.Thus we have:

Theorem 9. If a para-Norden–Walker manifold (M4, ϕ, g) is para-Kahler–Norden–Walker, then M4 is flat.

Remark 4. We note that a para-Kahler–Norden manifold is non-flat, in such man-ifold curvature tensor pure and paraholomorphic [23].

Let (M4, ϕ, g) be a para-Norden–Walker manifold with the integrable properstructure ϕ, i.e. Nϕ = 0. If a = −b, then from proof of the Theorem 3 we see thatEq. (7) hold. If c = c(y, z, t) and c = c(x, z, t), then cxy = (cx)y = (cy)x = 0. Inthese cases, by virtue of (7) we find a = a(x, z, t) and a = (y, z, t) respectively.

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1342 A. A. Salimov, M. Iscan & K. Akbulut

Using of cxy = 0 and axx + byy = 0 (see (8)), from (15) we obtain r = 0. Thus wehave:

Theorem 10. If (M4, ϕ, g) is a para-Norden–Walker non-para–Kahler manifoldwith metrics

g =

0 0 1 00 0 0 11 0 a(x, z, t) c(y, z, t)0 1 c(y, z, t) a(x, z, t)

, g =

0 0 1 00 0 0 11 0 a(y, z, t) c(x, z, t)0 1 c(x, z, t) a(y, z, t)

then M4 is scalar flat.

7. Para-Norden–Walker–Einstein Metrics

We now turn our attention to the Einstein conditions for the para-Norden–Walkermetric g in (2).

Let Rij and S denote the Ricci curvature and the scalar curvature of the metricg in (2). The Einstein tensor is defined by Gij = Rij − 1

4Sgij and has nonzerocomponents as follows (see [15], Appendix D):

Gxz =14axx − 1

4byy, Gxt =

12cxx +

12bxy,

Gyz =12axy +

12cyy, Gyt =

14byy − 1

4axx,

Gzz =14aaxx + caxy +

12bayy − ayt + cyz − 1

2aycx +

12axcy

+12ayby − 1

2(cy)2 − 1

2acxy − 1

4abyy,

Gzt =12acxx +

12ccxy +

12axt − 1

2cxz − 1

2aybx +

12cxcy +

12bcyy

− 12cyt +

12byz − 1

4caxx − 1

4cbyy,

Gtt =12abxx + cbxy + cxt − bxz − 1

2(cx)2 +

12axbx − 1

2bxcy +

12bycx

+14bbyy − 1

4baxx − 1

2bcxy.

(16)

The metric g in (2) is almost para-Norden–Walker–Einstein if all the above com-ponents Gij vanish (Gij = 0).

Theorem 11. Let (M4, ϕ, g) be a para-Norden–Walker manifold. If

ax = ay = bx = by = 0 (17)

g is para-Norden–Walker–Einstein.

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Notes on Para-Norden–Walker 4-Manifolds 1343

Proof. Suppose that the triple (M4, ϕ, g) be a para-Norden–Walker manifold. Thenfrom (6) and (17), we see that

cx = cy = 0. (18)

Substituting (17) and (18) into (16) we have Gij = 0.

Corollary 2. The triple (M4, ϕ, g) with metric

g = (gij) =

0 0 1 00 0 0 11 0 a(z, t) c(z, t)0 1 c(z, t) b(z, t)

is always para-Norden–Walker–Einstein.

8. Applications

We note that the almost para-Norden–Walker structure is a specialized almost prod-uct metric structure on a pseudo-Riemannian manifold (there is an extensive litera-ture on almost product metric structures, see, for examples [9, 10, 17, 18, 25, 30, 33]).In this context, we consider some properties of almost para-Norden–Walkermanifolds.

8.1.

An integrable almost product manifold is usually called a locally product mani-fold. Thus a para-Norden–Walker manifold (M4, ϕ, g) is a locally product pseudo-Riemannian manifold such that dim T+M4 = dim T−M4 = 2. From Theorem 2 wehave:

Corollary 3. An almost para-Norden–Walker manifold (M4, ϕ, g) is a locally prod-uct pseudo-Riemannian manifold if and only if ax − by + 2cy = 0 and ay − by +2cx = 0.

Let now (M4, ϕ, g) be a paraholomorphic Norden–Walker manifold (∇ϕ = 0).It is well known that a locally product (pseudo-)Riemannian manifold (M4, ϕ, g) isdecomposable if and only if ∇ϕ = 0 [30, p. 221], where ∇ is a Levi-Civita connectionof g. Then, from Theorems 1 and 4, we have:

Corollary 4. A para-Norden–Walker manifold (M4, ϕ, g) is locally decomposableif and only if ax = ay = bx = by = bz = cx = cy = 0 and at − 2cz = 0.

8.2.

Gil-Medrano and Naveira established in [10] that both distributions of the almostproduct structures on the (pseudo-)Riemannian manifold (M, P, g) are totally

January 14, 2011 14:40 WSPC/S0219-8878 IJGMMP-J043 00483

1344 A. A. Salimov, M. Iscan & K. Akbulut

geodesic if and only if

g((∇XP )Y, Z) + g((∇Y P )Z, X) + g((∇ZP )X, Y )

= σX,Y,Z

g((∇XP )Y, Z) = 0 (19)

for any X, Y, Z ∈ �10(M).

From Theorems 5 and 6 and (19), we have:

Corollary 5. Let (M4, ϕ, g) be an almost para-Norden–Walker manifold. Both dis-tributions of the manifold (M4, ϕ, g) are totally geodesic if and only if bx = by =bz = 0, ay − 2cx = 0, ax − 2cy = 0, cax − at + 2cz − (a − b)cx = 0.

8.3.

The minimal almost product structure on the (pseudo-)Riemannian manifold(M, P, g) is characterized by the equality of trgJ = (gjkJ i

jk) to zero on M

[17, 18, 25], where J is the Jordan tensor field, which in coordinate form can bewritten as

J ijk = P s

j (∇sPjk + ∇kP i

s) + P sk (∇sP

ij + ∇jP

is).

We can rewrite the condition of minimality as [25]

∇jPjk = 0. (20)

Now let (M4, ϕ, g) be an almost para-Norden–Walker manifold. Then using (20),we see that ∇1ϕ

1k + ∇2ϕ

2k + ∇3ϕ

3k + ∇4ϕ

4k = 0, k = 1, 2, 3, 4 if and only if{

ax − bx − 2cy = 0,

ay − by − 2cx = 0.(21)

Thus, we have:

Corollary 6. An almost para-Norden–Walker manifold (M4, ϕ, g) is a minimal, ifand only if g satisfies the condition (21).

8.4.

Let ∗T (Mn) denote the cotangent bundle of a manifold Mn and let π : ∗T (Mn) →Mn be the natural projection. A point ξ of the cotangent bundle is representedby an ordered pair (p, ω), where p = π(ξ) is a point on Mn and ω is a 1-form oncotangent spaces ∗Tp(Mn). For each coordinate neighborhood (U, xi) on Mn withp ∈ U , denote by xı, ı = n + 1, . . . , 2n the components of ω in the natural coframe{dxi}. Then, for any local coordinates (U, xi) on Mn, (xi, xı) are naturally inducedcoordinates in π−1(U) ⊂ ∗T (Mn).

For a given symmetric connection ∇ on Mn, the cotangent bundle ∗T (Mn) maybe equipped with a pseudo-Riemann metric ∇g of signature (n, n) [32, p. 268]: the

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Notes on Para-Norden–Walker 4-Manifolds 1345

Riemann extension of ∇, given by

∇g(V ω, CX) = V(ω(X)),∇g(V ω,V θ) = 0,∇g(CX,C Y ) = −γ(∇XY,∇Y X),

where Vω, Vθ are the vertical lifts to ∗T (Mn) of 1-forms ω, θ and CX,C Y denotethe complete lifts to ∗T (Mn) of vector fields X, Y on Mn. Moreover, for any vectorfield Z on Mn (Z = Zi∂i) γZ is the function on ∗T (Mn) defined by γZ =

∑i xıZi.

In a system of induced coordinates (xi, xı) on ∗T (Mn), the Riemann extension isexpressed by

∇g =

(−2∑

k xkΓkij δi

j

δji 0

)

with respect to {∂i, ∂ı}, where Γkij are the components of ∇. Since inverse (∇g)−1

of the matrix ∇g is given by

(∇g)−1 =

(0 δj

i

δij 2

∑k xkΓk

ij

),

the Walker metrics (2) can be viewed as Riemann extensions, i.e. they are locallyisometric to the cotangent bundle (∗T (

∑2),

∇g) of a surface (∑

2,∇) equipped withthe metric ∇g. Then we have:

Corollary 7. Let ∗T (∑

2) a cotangent bundle of a surface (∑

2, g). Then(∗T (

∑2),

∇g) is an almost para-Norden–Walker manifold with an almost para-complex structure (3), where a = 2(x3Γ1

11 + x4Γ211), b = 2(x3Γ1

22 + x4Γ222),

c = 2(x3Γ112 + x4Γ2

12).

Acknowledgments

We are grateful to Professor Yasuo Matsushita for valuable comments and usefuldiscussions.

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