EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 671-681
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Locally Conformal Almost Cosymplectic Manifold of
Φ-holomorphic Sectional Conharmonic Curvature

Tensor

Habeeb M. Abood1, Farah Hassan J. Al-Hussaini1,∗

1 Department of Mathematics, College of Education for Pure Sciences, University of Basrah,
Basrah, Iraq

Abstract. The aim of the present paper is to study the geometry of locally conformal almost
cosymplectic manifold of Φ-holomorphic sectional conharmonic curvature tensor. In particular,
the necessary and sufficient conditions that locally conformal almost cosymplectic manifold is a
manifold of point constant Φ-holomorphic sectional conharmonic curvature tensor have been found.
The relation between the mentioned manifold and the Einstein manifold is determined.

2010 Mathematics Subject Classifications: 53C55, 53B35

Key Words and Phrases: Locally conformal almost cosymplectic manifold, conharmonic cur-
vature tensor, Φ-holomorphic sectional conharmonic curvature tensor, Einstein manifold.

1. Introduction

Sectional curvature provides a lot of information with regard to substance geometry
of Riemannian manifolds. Manifolds with constant sectional curvature are a great source
of study. Morever, contact geometry plays important roles in Physics, optics, differential
equations and phase spaces of a dynamical system. This stimulated the researchers to work
in the domain of constancy holomorphic sectional curvatures of locally conformal almost
cosymplectic manifold which is a motivating class of almost contact metric manifold..

The study of constant holomorphic sectional curvature of almost Hermitian manifolds
was started by Tanno [19] in 1973. He obtained an algebraic characterization for an almost
Hermitian manifold to constringe to a space of constant holomorphic sectional curvature,
which he later extended for Sasakian manifold. In 1988, Kim [7] studied total spaces of
constant Φ-holomorphic sectional curvature and in 1989, he studied [8] total spaces with
flat contact Bochner curvature tensor for fibred Sasakian spaces with conformal fibres.
In 1993, Takano [18] discuss fibred Sasakian spaces of constant Φ-holomorphic sectional

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3261

Email addresses: iraqsafwan2006@gmail.com (H. M. Abood),
farahalhussaini14@yahoo.com (F. H. J. Al-Hussaini)

http://www.ejpam.com 671 c© 2018 EJPAM All rights reserved.



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 672

and at the same time Nagaich [14] showed a generalized Tanno’s results for indefinite
almost Hermitian manifold. In 2009, Rani et al. [17] considered similar condition of [19]
to another distinct class of almost contact manifold known as (ε)-Sasakian manifold. In
2012, Kirichenko and Kharitonova [12] studied the constancy of Φ-holomorphic sectional
curvature of normal locally conformal almost cosymplectic Manifold.

2. Preliminaries

In this section, we will focus our efforts on the study of almost contact metric manifold.
In particular, we dedicate our study on the construction of the class of locally conformal
almost cosymplectic manifold in the G-adjoined structure space.

Definition 2.1. [1] Let M be 2n + 1 dimensional smooth manifold , η be differential 1-
form called a contact form, ξ be vector field called a characteristic, Φ be an endomorphism
of the module of the vector fields X(M) called a structure endomorphisim, then the triple
(η, ξ,Φ) is called an almost contact structure if the following conditions hold

(i) η(ξ) = 1 ;

(ii) Φ(ξ) = 0 ;

(iii) η ◦ Φ = 0 ;

(iv) Φ2 = −id+ η ⊗ ξ.

Morover, if there is a Riemannian metric g = 〈., .〉 on M such that 〈ΦX,ΦY 〉 = 〈X,Y 〉 −
η(X)η(Y ), X, Y ∈ X(M), then the tetrad of tensors (η, ξ,Φ, g) is called an almost contact
metric structure. In this case the manifold M equipped with this structure is called an
almost contact metric manifold.

Definition 2.2. [9] Let (M, η,Φ, g) be almost contat metric manifold (AC-manifold). In
the module X(M) we can determine two complementary projections m, `, where m = η⊗ξ
and ` = −Φ2; thus X(M) = L⊕ ℵ, where L =ImΦ = kerη and ℵ =Imm = kerΦ, where
` and m are the projections onto the submodules L and ℵ respectively.

Definition 2.3. [9] In the complexification module Lc of the module L define two endo-
morphisms σ and σ̄ as σ = 1

2(id −
√
−1Φ) and σ̄ = −1

2(id +
√
−1Φ). We can define two

projections by the forms:

Π = σ ◦ ` = −1

2
(Φ2 −

√
−1Φ) and Π̄ = σ̄ ◦ ` =

1

2
(Φ2 +

√
−1Φ),

where σ ◦Φ = Φ ◦ σ = iσ and σ̄ ◦Φ = Φ ◦ σ̄ = −iσ̄. Therefore, If we denote ImΠ = D
√
−1

Φ

and ImΠ̄ = D−
√
−1

Φ , then

Xc(M) = D
√
−1

Φ ⊕D−
√
−1

Φ ⊕D0
Φ,

where D
√
−1

Φ , D−
√
−1

Φ and D0
Φ are proper submodules.



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 673

Definition 2.4. [11] At each point p ∈ M2n+1, there is a frame in T cp (M) of the form
(p, ε0, ε1, ..., εn, ε1̂,
..., εn̂), where εa =

√
2σp(ep), εâ =

√
2σ̄(ep), â = a+n, ε0 = ξp, the mappings σp : Lp −→

D
√
−1

Φ , σ̄p : Lp −→ D−
√
−1

Φ are isomorphism and anti-isomorphism respectively, and ea are
orthonormal bases of Lp. The frame (p, ε0, ε1, ..., εn, ε1̂, ..., εn̂) is called an A-frame.

Lemma 2.1. [13] The matrices components of tensors Φp and gp in A-frame have the
following froms respectively:

(Φi
j) =

 0 0 0
0
√
−1In o

0 0 −
√
−1In

 , (gij) =

 1 0 0
0 0 −In
0 In 0

 ,

where In is the identity matrix of order n.

It is well known, that the set of such frames defines an G-structure on M with structure

group 1×U(n), represented by matrix of the form

 1 0 0
0 A 0
0 0 A

 , where A ∈ U(n). This

structure is called an G-adjoined structure.

Definition 2.5. [1] A skew-symmetric tensor Ω(X,Y ) = g(X,ΦY ) is called a fundamental
form of the AC-structure.

Definition 2.6. [4] An almost contact metric structure S = (η, ξ,Φ, g) is called an almost
cosymplectic structure ( AC∫ -structure) if

(i) dη = 0 ;

(ii) dΩ = 0 .

Definition 2.7. [15] A conformal transformation of an AC-structure S = (η, ξ,Φ, g) on
a manifold is the passage from S to an AC-structure S̃ = (η̃, ξ̃, Φ̃, g̃) such that

η̃ = e−ση, ξ̃ = eσξ, Φ̃ = Φ, g̃ = e−2σg

where σ is the determining function of the conformal transformation. If σ =const, then
the conformal transformation is said to be trivial.

Definition 2.8. [15] An AC-structure S on a manifold M is said to be locally conformal
almost cosymplectic (LCAC∫ -structure) if the restriction of this structure to some neigh-
borhood U of an arbitrary point p ∈ M admits a conformal transformation of an almost
cosymplectic structure.This transformation is called a locally conformal. A manifold M
equipped with an LCAC∫ -structure is called an LCAC∫ -manifold.

Lemma 2.2. [6] In the G-adjoined structure space, the collection of the structure equations
of LCAC∫ -manifold has the following forms:



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 674

(i) dωa = −ωab ∧ ωb +Bab
c ω

c ∧ ωb +Babcωb ∧ ωc +Ba
bω ∧ ωb +Babω ∧ ωb;

(ii) dωa = ωba ∧ ωb +Bc
abωc ∧ ωb +Babcω

b ∧ ωc +Bb
aω ∧ ωb +Babω ∧ ωb;

(iii) dω = Cbω ∧ ωb + Cbω ∧ ωb;

(iv) dωab = −ωac ∧ωcb +Aacdb ωc∧ωd+Aabcdω
c∧ωd+Aacbdω

d∧ωc+Aabc0ω∧ωc+Aac0b ω∧ωc;

where

(i) B[abc] = B[abc] = 0;

(ii) B[ab] = B[ab] = 0;

(iii) Ba
b = Bb

a = σ0δ
b
a;

(iv) Cab = Cab = 0;

(v) Bab
c = 2σ[aδ

b]
c , Bc

ab = 2σ[aδ
c
b];

(vi) Cb = −σb, Cb = −σb;

(vii) Aacdb = 2δ
[c
b σ

a]d − 2δ
[d
b σ

a]c +Bacd
b − 2σaδ

[d
b σ

c] − 2σeB
ae[dδ

c]
b + 2σbB

abc;

(viii) A
[acd]
b = σeB

e[daδ
c]
b ;

(ix) Aac[bd] = −2δ
[c
[bσ

a]
d] + 2σ[aδ

e]
b σ[eδ

c
d] − 2σ[aδ

e]
d σ[eδ

c
b] +

1

2
BaecBebd;

(x) Aac0b = −2δ
[c
b σ

a]0 +Dac
b − δabσc0 − 2BaecBeb − σaσ0δ

c
b + 2Bacσb −Baeσeδ

c
b;

(xi) A
[ac]0
b = σ

[c
0 δ

a]
b − σdδ

[a
b B

d]c + σ0σ
[cδ

a]
b +

1

2
BdcaBbd;

(xii) Ba[bcd] = −Ba[dbσC];

(xiii) Babc0 = −2Da[bc] −Badcσ0;

(xiv) σ[cd] = σbB
bcd.

Here Babc, Babc; B
ab, Bab; B

a
b , Bb

a; Cab, Cab; C
b, Cb; A

acd
b , Abacd; A

ac
bd; Aac0b , Abac0; Babci,

Babci; D
abi, Dabi and σij are smooth functions in the G-adjoined structure space.

The following lemma gives the expression for the nonzero components of Riemannian
curvature tensor of LCAC∫ -manifold in the G-adjoined structure space.

Lemma 2.3. [6] In the G-adjoined structure space, the components of Riemannian cur-
vature tensor of LCAC∫ -manifold have the following forms:

(i) Rabcd = 2(Aabcd + 4σ[aδ
h]
[cBd]hb − σ0Bb[dδ

a
c]);



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 675

(ii) Ra
b̂cd

= 2(2δ
[b
[cσ

a]
d] + 2BhabBhdc − δa[cδ

b
d]σ

2
0);

(iii) Ra
bcd̂

= Aadbc + 4σ[aδ
h]
c σ[hδ

d
b] − 4BdahBchb +BadBbc − δac δdbσ2

0;

(iv) Râbcd = 2(2B[c|ab|d] − 2σ[aBb]cd +Ba[cBd]b);

(v) Ra0cd = 2(σ0[cδ
a
d] +BabBbcd − 2σ[aδ

h]
[cBd]h);

(vi) Rabĉ0 = Aac0b + σbB
ac − δcbσ0σ

a;

(vii) Râbc0 = 2Bcab0 + 2Bcabσ0;

(viii) Ra0b0 = −δabσ00 − δabσ2
0 −BcbBac − σab − σaσb + 2σ[aδ

c]
b σc;

(ix) Ra
0b̂0

= 2σ0B
ab −Dab0 − σab − σaσb + 2Bbacσc.

and the other components are conjugate to the above components or can be obtained by the
property of symmetry for R or equal to zero.

Definition 2.9. [3] A Ricci tensor is a tensor of type (2,0) which is defined by

rij = −Rkijk

Lemma 2.4. In the G-adjoined structure space, the components of the Ricci tensor of
LCAC∫ -manifold are given by the following forms:

(i) rab = 2(−2Ac(ab)c − 4(σ[cδ
h]
[bBc]ha + σ[cδ

h]
[aBc]hb) + σ0Ba[cδ

c
b] + σ0Bb[cδ

c
a] + 2σ0Bab −

Dab0 − σab − σaσb + 2Bbahσ
h;

(ii) râb = −4(δ
[a
[bσ

c]
c] − σ[cδ

b
h]σ

[hδ
a]
c −

1

2
σ[aδ

h]
b σh + BhcaBhcb + BbchBcha) + (BcbBac −

BhbB
ah) +Acbac − δabσ00 − 2nσ2

0 − σab − σaσb;

(iii) ra0 = −Acac0 − σcBac + nσ0σa + 2(σ0[cδ
c
a] +BcbBbca − 2σ[cδ

h]
[cBa]h);

(iv) roo = −2n(σ00 + σ2
0)− 2BhcB

ch − 2(σcc + σcσc) + 4σ[cδ
h]
c σh.

and the other components can be found by taking the conjugate operator to the above
components.

Proof. The above components can obtained directly from the Definition 2.10 and
Lemma 2.5.

Definition 2.10. An LCAC∫ -manifold has Φ-invariant Ricci tensor, if Φ ◦ r = r ◦ Φ.



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 676

Lemma 2.5. An LCAC∫ -manifold has Φ-invariant Ricci tensor if and only if, in the
G-adjoined structure space, the following condition

râb = rab = 0

holds.

We conclude this section by remembering the main concept of our study which is a
conharmonic curvature tensor.

Definition 2.11. [5] Let M be an AC-manifold of dimension 2n+ 1. A tensor T of type
(4, 0) which is invariant under conharmonic transformation and defined by the form:

Tijkl = Rijkl −
1

2n− 1
(rilgjk − rjlgik + rjkgil − rikgjl)

is called a conharmonic tensor, where Tijkl = −Tjikl = −Tijlk = Tklij .

Theorem 2.1. In the G-adjoined structure space, the components of conharmonic curva-
ture tensor of LCAC∫ -manifold are given by the following forms:

(i) Tabcd = 2(2B[c|ab|d] − 2σ[aBb]cd +Ba[cBd]b);

(ii) Tâbcd = 2(Aabcd + 4σ[aδ
h]
[cBd]hb − σ0Bb[dδ

a
c])−

1
2n−1(rbcδ

a
d − rbdδac );

(iii) Tâbcd̂ = Aadbc + 4σ[aδ
h]
c σ[hδ

d
b] − 4BdahBchb +BadBbc − δac δdbσ2

0 − 1
2n−1(rdb δ

a
c + rac δ

d
b );

(iv) Tâb̂cd = 2(2δ
[b
[cσ

a]
d] + 2BhabBhdc − δa[cδ

b
d]σ

2
0)− 4

2n−1(r
[d
[aδ

c]
b]);

(v) Tâ0cd = 2(σ0[cδ
a
d] +BabBbcd − 2σ[aδ

h]
[cBd]h) + 1

2n−1(r0dδ
a
c − r0cδ

a
d);

(vi) Tâbĉ0 = Aac0b + σbB
ac − δcbσ0σ

a − 1
2n−1(ra0δ

c
b);

(vii) Tabc0 = 2Bcab0 + 2Bcabσ0;

(viii) Tâ0b0 = −δabσ00 − δabσ2
0 −BcbBac − σab − σaσb + 2σ[aδ

c]
b σc + 1

2n−1(r00δ
a
b + rab );

(ix) Tâ0b̂0 = 2σ0B
ab −Dab0 − σab − σaσb + 2Bbacσc + 1

2n−1(râb̂).

and the other components are conjugate to the above or can be obtained by the property of
symmetry for T or equal to zero.

Definition 2.12. [16] A Riemannian manifold is called an Einstein manifold, if the Ricci
tensor satisfies the equation rij = egij.

Definition 2.13. [10] Let M be an AC-manifold, an Φ-holomorphic sectional curvature
(ΦHS-curvature) of a manifold M in the direction X ∈ X(M); X 6= 0 is a function H(X)
which is defined as:

H(X) = 〈R(X,ΦX,X,ΦX, )〉‖X‖−4



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 677

Definition 2.14. [10] An AC-manifold is called a manifold of point constant ΦHS-curvature
if

〈R(X,ΦX,X,ΦX, )〉 = c‖X‖4

where c ∈ C∞(M); for all X ∈ X(M)

Lemma 2.6. [10] An AC-manifold is a manifold of point constant ΦHS-curvature c if and
only if, on the G-adjoined structure,

R
(a d)

(bc) =
c

2
δãdbc

where δãdbc = δab δ
d
c + δac δ

d
b is the symmetric second-order Kronecker delta.

Definition 2.15. Let M be an AC-manifold, an Φ-holomorphic sectional conharmonic
curvature (ΦHTS-curvature) of a manifold M in the direction X ∈ X(M); X 6= 0 is a
function H(X) which is defined as

H(X) = 〈T (X,ΦX,X,ΦX, )〉‖X‖−4

Definition 2.16. An AC-manifold is called a manifold of point constant ΦHST-curvature
if

〈T (X,ΦX,X,ΦX, )〉 = c‖X‖4

where c ∈ C∞(M); for all X ∈ X(M).

3. The main results

This section is devoted to study the theoretical application of LCAC∫ -manifold of
point constant Φ-holomorphic sectional conharmonic curvature. In particular, we found
the necessary and sufficient conditions in which the LCAC∫ -manifold of point constant
Φ-holomorphic sectional conharmonic curvature is an Eistein manifold.

The following theorems gives the necessary and sufficient condition in which an LCAC∫ -
manifold is a manifold of point constant ΦHS-curvature.

Theorem 3.1. An LCAC∫ -manifold is a manifold of point constant ΦHS-curvature c if

and only if, the relation A
(ad)
(bc) =

1

2
δãdbc (σ2

0 + c)− 4σ[aδ
h]
c σ[hδ

d
b] + 4B(da)hBchb−BadBbc holds

on the G-adjoined structure space.

Proof. According to the components of the Riemannian curvature tensor of LCAC∫ -
manifold, it follows that

Ra d
bc = Aadbc + 4σ[aδh]

c σ[hδ
d
b] − 4BdahBchb +BadBbc − δac δdbσ2

0

Symmetrizing with respect to the pair of upper and lower indices of the tensor Ra d
bc ,

we get

R
(a d)

(bc) = A
(ad)
(bc) + 4σ[aδh]

c σ[hδ
d
b] − 4B(da)hBchb +BadBbc −

1

2
δãdbc σ

2
0



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 678

By Lemma 2.6, the constancy condition on the ΦHS-curvature c for a LCAC∫ -manifold,
yields

A
(ad)
(bc) =

1

2
δãdbc (σ2

0 + c)− 4σ[aδh]
c σ[hδ

d
b] + 4B(da)hBchb −BadBbc

Theorem 3.2. Suppose that M is LCAC∫ -manifold. Then the necessary and suffcient
condition in which M is a manifold of point constant ΦHST-curvature C0 is

Aadbc = 4BdahBchb +BadBbc − 4σ[aδh]
c σ[hδ

d
b] + δac δ

d
bσ

2
0 − C0δ

a
b δ
d
c −

1

2n− 1
(rdb δ

a
c + rac δ

d
b )

Proof. Suppose that M is LCAC∫ -manifold of the point constant ΦHTS-curvature
tensor.
According to the Definition 2.16, we get

〈T (X,ΦX,X,ΦX, )〉 = C0‖X‖4

In the G-adjoined structure space, we have

TijklX
i(ΦX)jXk(ΦX)l = C0gijgklX

iXjXkX l

According to the property (ΦX)a =
√
−1Xa, (ΦX)â = −

√
−1X â and (ΦX)0 = 0 and

then using the properties of conharmonic tensor, we get

−4Tâbcd̂ = 4C0δ
a
b δ
d
c

Hence

Aadbc = 4BdahBchb +BadBbc − 4σ[aδh]
c σ[hδ

d
b] + δac δ

d
bσ

2
0 − C0δ

a
b δ
d
c −

1

2n− 1
(rdb δ

a
c + rac δ

d
b )

Theorem 3.3. If M is LCAC∫ -manifold of point constant ΦHST-curvature tensor with
flat holomorphic sectional curvature tensor and Φ-invariant Ricci tensor. Then M is an
Einstein manifold.

Proof. Suppose that M is a manifold of point constant ΦHST-curvature tensor.
According to Theorem 3.2, we have

Aadbc − 4BdahBchb −BadBbc + 4σ[aδh]
c σ[hδ

d
b] +

1

2n− 1
(rdb δ

a
c + rac δ

d
b ) = δac δ

d
bσ

2
0 − C0δ

a
b δ
d
c (3.1)

Symmetrizing and then antisymmetrizing (3.1) by the indices (a, h) and (a, d) respectively
and since M is a manifold with flat holomorphic sectional curvature tensor, then we have

1

2n− 1
(r

[d
b δ

a]
c + r[a

c δ
d]
b ) =

1

2
(δdb δ

a
c − δdc δab )(σ2

0 + C0) (3.2)



H. M. Abood, F. H. J. Al-Hussaini / Eur. J. Pure Appl. Math, 11 (3) (2018), 671-681 679

Contracting (3.2) by the indices (d, c), we deduce

− (n− 2)

2(2n− 1)
(rab + rddδ

a
b ) = −(n− 1)

2
δab (σ2

0 + C0) (3.3)

Symmetrizing and antisymmetrizing (3.3) by the indices (a, d), we obtain

rab = eδab

where e =
(2n− 1)(n− 1)

(n− 2)
(σ2

0 + C0)

Since the Ricci tensor is Φ-invariant
Therefore M is Einstein manifold.

Theorem 3.4. If M is LCAC∫ -manifold of point constant ΦHST-curvature tensor and Φ-
invariant Ricci tensor, then M is an Einstein manifold if and only if Aacbc = BacBbc + c1δ

a
b

.

Proof. Suppose that M is a manifold of point constant ΦHST-curvature tensor.
According to the Theorem 3.2, we have

Aadbc = 4BdahBchb +BadBbc − 4σ[aδh]
c σ[hδ

d
b] + δac δ

d
bσ

2
0 − C0δ

a
b δ
d
c −

1

(2n− 1)
(rdb δ

a
c + rac δ

d
b )(3.4)

Symmetrizing (3.4) by the indices (a, h), we get

Aadbc = BadBbc + δac δ
d
bσ

2
0 − C0δ

a
b δ
d
c −

1

(2n− 1)
(rdb δ

a
c + rac δ

d
b ) (3.5)

Contracting (3.5) by the indices (c, d), we deduce

Aacbc = BacBbc + (σ2
0 − nC0)δab −

2

(2n− 1)
rab (3.6)

Since M is an Einstein manifold, it follows that

Aacbc = BacBbc + c1δ
a
b

where c1 = σ2
0 − nC0 − 2e

(2n−1)

Conversely, by substituted Aacbc in equation (3.6), we get

rab = eδab

According to Φ-invariant of Ricci tensor, it follows that M is Einstein manifold.



REFERENCES 680

References

[1] D.E. Blair, The theory of quasi-Sasakian structures, J. Differential Geometry, N. 1,
P. 331-345, 1967.

[2] D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, in Progr.
Math. Birkhauser, Boston, MA, Vol. 203, 2002.

[3] E. Cartan, Riemannian Geometry in an Orthogonal Frame, From lectures Delivered
by E. Lie Cartan at the Sorbonne 1926-27, Izdat. Moskov. Univ., Moscow, 1960;
World Sci., Singapore, 2001.

[4] S.I. Goldberg and K. Yano, Integrabilty of almost cosymplectic structures, Pacific
Journal of Mathematics, 31, 373-382, 1969.

[5] Y. Ishi, On conharmonic transformation, Tensor N. S. 7, pp. 73-80, 1957.

[6] S. V. Kharitonova, On the geometry of locally conformal almost cosymplectic mani-
folds, Matematicheskie Zametki, Vol. 86, No. 1, pp. 126-138, 2009.

[7] B.H. Kim, Fibred Riemannian spaces with contact structure, Hiroshima Math.J.18,
493508, 1988.

[8] B. H. Kim, Fibred Sasakian spaces with vanishing contact Bochner curvature tensor,
ibid. 19, 181195, 1989.

[9] V. F. Kirichenko, The method of generalization of Hermitian geometry in the almost
Hermitian contact manifold, Problems of geometry VINITE ANSSR, V. 18, P. 25-71,
1986.

[10] V. F. Kirichenko, Generalized quasi-Kaehlerian manifolds and axioms of CR-
submanifolds in generalized Hermitian geometry I, Geo. Dedicata 51 (1), 75-104
(1994).

[11] V. F. Kirichenko and A. R. Rustanov, Differential Geometry of quasi Sasakian man-
ifolds, Mathematical Collection. 193(8): 71-100, 2002.

[12] V. F. Kirichenko and S. V. Kharitonova, On the geometry of normal locally conformal
almost cosymplectic manifolds, Matematicheskie Zametki, Vol. 91, No. 1, pp. 40-53,
2012.

[13] V. F. Kirichenko, Differential - Geometry structures on manifolds, Second edition,
expanded. Odessa, Printing House, 458 P., 2013.

[14] R. K. Nagaich, Constancy of holomorphic sectional curvature in indefinite almost
Hermitian manifolds, KodaiMath.J.16, 327, 1993.

[15] Z. Olszak, Locally conformal almost cosymplectic manifolds, Collq. Math. 57(1), 73-
87, 1989.



REFERENCES 681

[16] A. Z. Petrov, Einstein space, Phys-Math. Letr. Moscow, P. 463, 1961.

[17] R. Rani, R. Kumar and R. K. Nagaich,Constancy of Φ-Holomorphic Sectional Curva-
ture of (ε)-Sasakian Manifolds, ISSN 1995-0802, Lobachevskii Journal of Mathemat-
ics, Vol.30, No.1, pp.7680, 2009.

[18] K. Takano,On Fibred Sasakian Space with Vanishing Contact Bochner Curvature
Tensor,Colloquium Mathematicum, Vol. LXV, FASC.2, 1993.

[19] S. Tanno, Constancy of holomorphic sectional curvature in almost Hermitian mani-
folds, KodaiMath. Sem. Rep.25, 190, 1973.