MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESeinspem.upm.edu.my/journal/fullpaper/vol9no2/2....

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Malaysian Journal of Mathematical Sciences 9(2): 187-207 (2015) Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind 1 Abu-Saleem A. and 2 Rustanov A. R 1 Department of Mathematics, Al al -Bayt University, Mafraq, Jordan 2 Moscow State Pedagogical University, Moscow, Russia E-mail: [email protected] and [email protected] *Corresponding author ABSTRACT In this paper, we obtain an analytical expression of the structural tensor, brought additional identities curvature of special generalized manifolds Kenmotsu type II and based on them are highlighted in some of the classes of this class of manifolds and obtain a local characterization of the selected classes. Keywords: Riemann curvature tensor, special generalized manifolds Kenmotsu type II, flat manifolds, curvature additional identity. 1. INTRODUCTION The notions of almost contact structures and almost contact metric manifolds introduced by Gray, 1959. A careful analysis is subjected to special classes of almost contact metric manifolds. In 1972 Kenmotsu, 1972 introduced the class of almost contact metric structures, characterized by the identity βˆ‡ (Ξ¦) = 〈Φ, βŒͺ βˆ’ ()Ξ¦; , ∈ (). Kenmotsu structure, for example, arise naturally in the classification Tanno, 1969 connected almost contact metric manifolds whose automorphism group has maximal dimension . They have a number of remarkable properties. For example, the structure Kenmotsu normality and integrability. They are not a contact structure, and hence Sasakian. There are examples of Kenmotsu structures on odd Lobachevsky spaces of curvature (βˆ’1). Such structures are obtained by MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal

Transcript of MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESeinspem.upm.edu.my/journal/fullpaper/vol9no2/2....

Page 1: MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESeinspem.upm.edu.my/journal/fullpaper/vol9no2/2. abu...Abu-Saleem A. & Rustanov A. R 188 Malaysian Journal of Mathematical Sciences the construction

Malaysian Journal of Mathematical Sciences 9(2): 187-207 (2015)

Curvature Identities Special Generalized Manifolds Kenmotsu

Second Kind

1Abu-Saleem A. and

2Rustanov A. R

1Department of Mathematics, Al al -Bayt University, Mafraq, Jordan

2Moscow State Pedagogical University, Moscow, Russia

E-mail: [email protected] and [email protected]

*Corresponding author

ABSTRACT

In this paper, we obtain an analytical expression of the structural tensor,

brought additional identities curvature of special generalized manifolds

Kenmotsu type II and based on them are highlighted in some of the classes of

this class of manifolds and obtain a local characterization of the selected

classes.

Keywords: Riemann curvature tensor, special generalized manifolds

Kenmotsu type II, flat manifolds, curvature additional identity.

1. INTRODUCTION

The notions of almost contact structures and almost contact metric

manifolds introduced by Gray, 1959. A careful analysis is subjected to

special classes of almost contact metric manifolds. In 1972 Kenmotsu, 1972

introduced the class of almost contact metric structures, characterized by the

identity βˆ‡π‘‹(Ξ¦)π‘Œ = βŸ¨Ξ¦π‘‹, π‘ŒβŸ©πœ‰ βˆ’ πœ‚(π‘Œ)Φ𝑋; 𝑋, π‘Œ ∈ 𝒳(𝑀). Kenmotsu structure,

for example, arise naturally in the classification Tanno, 1969 connected

almost contact metric manifolds whose automorphism group has maximal

dimension . They have a number of remarkable properties. For example, the

structure Kenmotsu normality and integrability. They are not a contact

structure, and hence Sasakian. There are examples of Kenmotsu structures on

odd Lobachevsky spaces of curvature (βˆ’1). Such structures are obtained by

MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES

Journal homepage: http://einspem.upm.edu.my/journal

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Abu-Saleem A. & Rustanov A. R

188 Malaysian Journal of Mathematical Sciences

the construction of the warped product 𝑹 ×𝑓 π‘ͺ𝑛 in the sense of Bishop and

O'Neill, 1969 complex Euclidean space and the real line, where 𝑓(𝑑) = 𝑐𝑒𝑑

(see Kenmotsu, 1972).

Kenmotsu varieties studied by many authors, such as Bagewadi and

Venkatesha, 2007; De and Pathak, 2004 and Pitis, 2007 and many others.

But we would like to acknowledge the work of Kirichenko, 2001. In this

paper it is proved that the class of manifolds Kenmotsu coincides with the

class of almost contact metric manifolds derived from cosymplectic

manifolds canonical transformation concircular cosymplectic structure.

Further Umnova, 2002 studied Kenmotsu manifolds and their

generalizations. The author identified class of almost contact metric

manifolds, which is a generalization of manifolds Kenmotsu and named class

of generalized (in short, GK-) Kenmotsu manifolds. In Behzad and Niloufar,

2013, this class of manifolds is called the class of nearly Kenmotsu

manifolds. The authors prove that the second-order symmetric closed

recurrent tensor recurrence covector which annihilates the characteristic

vector ΞΎ, a multiple of the metric tensor g. In addition, the authors examine

the Π€-recurrent nearly Kenmotsu manifolds. It is proved that Π€-recurrent

nearly Kenmotsu manifold is Einstein and locally Π€-recurrent nearly

Kenmotsu manifold is a manifold of constant curvature -1. In Umnova, 2002

identifies two subclasses of generalized manifolds Kenmotsu called special

generalized manifolds Kenmotsu (briefly, SGK-) I and type II. In Tshikuna-

Matamba, 2012 SGK-manifolds of type II are called nearly Kenmotsu

manifolds. In Umnova, 2002 it is proved that the generalized manifolds of

constant curvature Kenmotsu are Kenmotsu manifolds of constant curvature -

1. Moreover, it is proved that the class of SGK-manifolds of type II coincides

with the class of almost contact metric manifolds, obtained from the closely

cosymplectic manifolds canonical transformation closely cosymplectic

structure, and given the local structure of these manifolds of constant

curvature.

This paper is organized as follows. In Section 2 we present the

preliminary information needed in the sequel, we construct the space of the

associated G-structure. In Section 3 we give the definition of generalized

manifolds Kenmotsu and SGK-manifolds of type II, we give a complete

group of structure equations, the components of Riemann-Christoffel tensor,

the Ricci tensor and the scalar curvature in the space of the associated G-

structure SGK-manifolds of type II. In Section 4, we study the properties of

the structure tensors SGK-manifold of type II. In Section 5, we discuss some

identities satisfied by the Riemann curvature tensor SGK-manifolds of type

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Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind

Malaysian Journal of Mathematical Sciences 189

II. On the basis of identities allocated classes SGK-manifolds of type II and

obtained a local characterization of the classes. The results obtained in

sections 4 and 5 are the main results.

2. PRELIMINARIES

Let M – smooth manifold of dimension 2𝑛 + 1, 𝒳(𝑀) – 𝐢∞-module

of smooth vector fields on M. In what follows, all manifolds, tensor fields,

etc. objects are assumed to be smooth of class 𝐢∞.

Definition 2.1 (Kirichenko, 2013): Almost contact structure on a manifold M

is a triple (πœ‚, πœ‰, Ξ¦) of tensor fields on the manifold, where Ξ· – differential 1-

form, called the contact form of the structure, ΞΎ – vector field, called the

characteristic, Π€ – endomorphism of 𝒳(𝑀) called the structure

endomorphism.

In this

1) πœ‚(πœ‰) = 1; 2) πœ‚ ∘ Ξ¦ = 0; 3) Ξ¦(πœ‰) = 0; 4) Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰. (1)

If, moreover, M is fixed Riemannian structure 𝑔 =<β‹…,β‹…>, such that

βŸ¨Ξ¦π‘‹, Ξ¦π‘ŒβŸ© = ⟨X, Y⟩ βˆ’ πœ‚(𝑋)πœ‚(π‘Œ), 𝑋, π‘Œ ∈ 𝒳(𝑀), (2)

four (πœ‚, πœ‰, Ξ¦, 𝑔 =<βˆ™,βˆ™>) is called an almost contact metric (shorter, AC-)

structure. Manifold with a fixed almost contact (metric) structure is called an

almost contact (metric (shorter, AC-)) manifold.

Skew-symmetric tensor Ξ©(𝑋, π‘Œ) = βŸ¨π‘‹, Ξ¦π‘ŒβŸ©, 𝑋, π‘Œ ∈ 𝒳(𝑀) called the

fundamental form of AC-structure (Kirichenko, 2013).

Let (πœ‚, πœ‰, Ξ¦, 𝑔) – almost contact metric structure on the manifold

𝑀2𝑛+1 . In the module 𝒳(𝑀) internally defined two complementary

projectors 𝓂 = πœ‚ βŠ— πœ‰ and β„“ = 𝑖𝑑 βˆ’ 𝓂 = βˆ’Ξ¦2 (Kirichenko and Rustanov,

2002); thus 𝒳(𝑀) = β„’ βŠ• β„³ , where β„’ = πΌπ‘š(Ξ¦) = π‘˜π‘’π‘Ÿπœ‚ – the so-called

contact distribution, π‘‘π‘–π‘šβ„’ = 2𝑛, β„³ = πΌπ‘šπ“‚ = π‘˜π‘’π‘Ÿ(Ξ¦) = 𝐿(πœ‰) – linear hull

of structural vector (wherein β„“ and 𝓂 are the projections onto the

submodules β„’, β„³, respectively). Clearly, the distribution of the β„’ and β„³ are

invariant with respect to Π€ and mutually orthogonal. It is also obvious that

Ξ¦Μƒ2 = βˆ’π‘–π‘‘, βŸ¨Ξ¦Μƒπ‘‹, Ξ¦Μƒπ‘ŒβŸ© = βŸ¨π‘‹, π‘ŒβŸ©, 𝑋, π‘Œ ∈ 𝒳(𝑀), where Ξ¦Μƒ = Ξ¦|β„’.

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Consequently, {Φ̃𝑝, 𝑔𝑝|β„’} – Hermitian structure on the space ℒ𝑝 .

Complexification 𝒳(𝑀)𝐂 module 𝒳(𝑀) is the direct sum 𝒳(𝑀)𝐂 = π·Ξ¦βˆšβˆ’1 βŠ•

π·Ξ¦βˆ’βˆšβˆ’1 βŠ• 𝐷Φ

0 eigensubspaces of structural endomorphism Π€, corresponding

to the eigenvalues βˆšβˆ’1, βˆ’βˆšβˆ’1 and 0, respectively. Projectors on the terms of

this direct sum will be, respectively, the endomorphisms Gray, 1959 πœ‹ = 𝜎 ∘

β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘ β„“ =

1

2(βˆ’Ξ¦2 + βˆšβˆ’1Ξ¦), 𝓂 = 𝑖𝑑 + Ξ¦2

where 𝜎 =1

2(𝑖𝑑 βˆ’ βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ =

1

2(𝑖𝑑 + βˆšβˆ’1Ξ¦).

The mappings πœŽπ‘: ℒ𝑝 ⟢ π·Ξ¦βˆšβˆ’1 and �̅�𝑝: ℒ𝑝 ⟢ 𝐷Φ

βˆ’βˆšβˆ’1 are

isomorphism and anti-isomorphism, respectively, Hermitian spaces.

Therefore, each point 𝑝 ∈ 𝑀2𝑛+1 may be connected space frames family

𝑇𝑝(𝑀)𝐢 of the form (𝑝, νœ€0, νœ€1, … , νœ€π‘›, νœ€1Μ‚, … , νœ€οΏ½Μ‚οΏ½), where νœ€π‘Ž = √2πœŽπ‘(π‘’π‘Ž), νœ€οΏ½Μ‚οΏ½ =

√2�̅�𝑝(π‘’π‘Ž), νœ€0 = πœ‰π‘; where {π‘’π‘Ž} – orthonormal basis of Hermitian space ℒ𝑝.

Such a frame called A-frame (Kirichenko and Rustanov, 2002). It is easy to

see that the matrix components of tensors Φ𝑝 and 𝑔𝑝 in an A-frame have the

form, respectively:

(Φ𝑗𝑖) = (

0 0 0

0 βˆšβˆ’1𝐼𝑛 0

0 0 βˆ’βˆšβˆ’1𝐼𝑛

) , (𝑔𝑖𝑗) = (1 0 00 0 𝐼𝑛

0 𝐼𝑛 0), (3)

where 𝐼𝑛 – identity matrix of order n. It is well known (Kirichenko, 2013)

that the set of such frames defines a G-structure on M with structure group

{1} Γ— π‘ˆ(𝑛) , represented by matrices of the form (1 0 00 𝐴 00 0 𝐴

) , where

𝐴 ∈ π‘ˆ(𝑛) . This structure is called a G-connected (Kirichenko, 2013);

Kirichenko and Rustanov, 2002).

3. SPECIALLY GENERALIZED MANIFOLDS KENMOTSU

TYPE II

Let (𝑀2𝑛+1, Ξ¦, πœ‰, πœ‚, 𝑔 = βŸ¨βˆ™,βˆ™βŸ©) – almost contact metric manifold.

Definition 3.1. (Abu-Sleem and Rustanov, 2015;Umnova, 2002): Class of

almost contact metric manifolds, characterized by the identity

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βˆ‡π‘‹(Ξ¦)π‘Œ + βˆ‡π‘Œ(Ξ¦)𝑋 = βˆ’πœ‚(π‘Œ)Φ𝑋 βˆ’ πœ‚(𝑋)Ξ¦π‘Œ; 𝑋, π‘Œ ∈ 𝒳(𝑀), (4)

called generalized manifolds Kenmotsu (shorter, GK-manifolds).

Definition 3.2. (Abu-Sleem and Rustanov, 2015;Umnova, 2002):GK-

manifold with a closed contact form, i.e. π‘‘πœ‚ = 0 are called SGK-manifolds of

type II.

This class of manifolds in (Tshikuna-Matamba, 2012) is a class of

nearly Kenmotsu manifolds. We will refer to these manifolds, as in Umnova,

2002 special generalized manifolds Kenmotsu type II, and briefly SGK-

manifolds of type II.

In terms of the structure tensors Kirichenko, 2013, Definition 2 can

be summarized as follows.

Definition 3.3. (Umnova, 2002) : GK-manifolds for which πΉπ‘Žπ‘ = πΉπ‘Žπ‘ = 0

are called SGK-manifolds of type II.

Let M – SGK-manifolds of type II. Then the first group of structure

equations SGK-manifolds of type II takes the form (Abu-Sleem and

Rustanov, 2015;Umnova, 2002).

1) π‘‘πœ” = 0; 2) π‘‘πœ”π‘Ž = βˆ’πœƒπ‘π‘Ž ∧ πœ”π‘ + πΆπ‘Žπ‘π‘πœ”π‘ ∧ πœ”π‘ + 𝛿𝑏

π‘Žπœ” ∧ πœ”π‘; 3) π‘‘πœ”π‘Ž =πœƒπ‘Ž

𝑏 ∧ πœ”π‘ + πΆπ‘Žπ‘π‘πœ”π‘ ∧ πœ”π‘ + π›Ώπ‘Žπ‘πœ” ∧ πœ”π‘,

(5)

where

πΆπ‘Žπ‘π‘ =βˆšβˆ’1

2Ξ¦οΏ½Μ‚οΏ½,𝑐̂

π‘Ž ; πΆπ‘Žπ‘π‘ = βˆ’βˆšβˆ’1

2Φ𝑏,𝑐

οΏ½Μ‚οΏ½ ; 𝐢[π‘Žπ‘π‘] = πΆπ‘Žπ‘π‘; 𝐢[π‘Žπ‘π‘] =

πΆπ‘Žπ‘π‘; πΆπ‘Žπ‘π‘Μ…Μ… Μ…Μ… Μ…Μ… = πΆπ‘Žπ‘π‘. (6)

Theorem 2.1 in Abu-Sleem and Rustanov, 2015 takes the form:

Theorem 3.1. Full group of structure equations SGK-manifolds of type II in

the space of the associated G-structure has the form:

1) π‘‘πœ” = 0; 2) π‘‘πœ”π‘Ž = βˆ’πœƒπ‘π‘Ž ∧ πœ”π‘ + πΆπ‘Žπ‘π‘πœ”π‘ ∧ πœ”π‘ + 𝛿𝑏

π‘Žπœ” βˆ§πœ”π‘; 3) π‘‘πœ”π‘Ž = πœƒπ‘Ž

𝑏 ∧ πœ”π‘ + πΆπ‘Žπ‘π‘πœ”π‘ ∧ πœ”π‘ + π›Ώπ‘Žπ‘πœ” ∧ πœ”π‘; 4) π‘‘πœƒπ‘

π‘Ž =

βˆ’πœƒπ‘π‘Ž ∧ πœƒπ‘

𝑐 + (π΄π‘π‘π‘Žπ‘‘ βˆ’ 2πΆπ‘Žπ‘‘β„ŽπΆβ„Žπ‘π‘)πœ”π‘ ∧ πœ”π‘‘; 5) π‘‘πΆπ‘Žπ‘π‘ + πΆπ‘‘π‘π‘πœƒπ‘‘

π‘Ž +

πΆπ‘Žπ‘‘π‘πœƒπ‘‘π‘ + πΆπ‘Žπ‘π‘‘πœƒπ‘‘

𝑐 = πΆπ‘Žπ‘π‘π‘‘πœ”π‘‘ βˆ’ πΆπ‘Žπ‘π‘πœ”; 6) π‘‘πΆπ‘Žπ‘π‘ βˆ’ πΆπ‘‘π‘π‘πœƒπ‘Žπ‘‘ βˆ’

πΆπ‘Žπ‘‘π‘πœƒπ‘π‘‘ βˆ’ πΆπ‘Žπ‘π‘‘πœƒπ‘

𝑑 = πΆπ‘Žπ‘π‘π‘‘πœ”π‘‘ βˆ’ πΆπ‘Žπ‘π‘πœ”. (7)

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where

𝐴[𝑏𝑐]π‘Žπ‘‘ = 𝐴𝑏𝑐

[π‘Žπ‘‘]= 0, πΆπ‘Ž[𝑏𝑐𝑑] = 0, πΆπ‘Ž[𝑏𝑐𝑑] = 0. (8)

Taking the exterior derivative of (7(4)), we obtain:

π‘‘π΄π‘π‘π‘Žπ‘‘ + 𝐴𝑏𝑐

β„Žπ‘‘πœƒβ„Žπ‘Ž + 𝐴𝑏𝑐

π‘Žβ„Žπœƒβ„Žπ‘‘ βˆ’ π΄β„Žπ‘

π‘Žπ‘‘πœƒπ‘β„Ž βˆ’ π΄π‘β„Ž

π‘Žπ‘‘πœƒπ‘β„Ž = π΄π‘π‘β„Ž

π‘Žπ‘‘ πœ”β„Ž + π΄π‘π‘π‘Žπ‘‘β„Žπœ”β„Ž βˆ’

2π΄π‘π‘π‘Žπ‘‘πœ”, (9)

where

1) 𝐴𝑏[π‘β„Ž]π‘Žπ‘‘ = 0; 2) 𝐴𝑏𝑐

π‘Ž[π‘‘β„Ž]= 0; 3) (𝐴𝑏[𝑐

π‘Žπ‘”βˆ’ 2πΆπ‘Žπ‘”π‘“πΆπ‘“π‘[𝑐) 𝐢|𝑔|π‘‘β„Ž] =

0; 4) (π΄π‘π‘”π‘Ž[𝑐

βˆ’ 2πΆπ‘Ž[𝑐|𝑓|𝐢𝑓𝑏𝑔) 𝐢|𝑔|π‘‘β„Ž] = 0. (10)

Theorems 2.3, 2.4 and 2.5 of Abu-Sleem and Rustanov, 2015 take the form:

Theorem 3.2. Nonzero essential components of Riemann-Christoffel tensor

in the space of the associated G-structure are of the form:

1) π‘…οΏ½Μ‚οΏ½π‘π‘‘π‘Ž = 2πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘ βˆ’ 2𝛿[𝑐

π‘Ž 𝛿𝑑]𝑏 ; 2) 𝑅�̂�𝑐̂�̂�

π‘Ž = βˆ’2πΆπ‘Žπ‘[𝑐𝑑]; 3) 𝑅00π‘π‘Ž =

π›Ώπ‘π‘Ž; 4) 𝑅𝑏𝑐�̂�

π‘Ž = π΄π‘π‘π‘Žπ‘‘ βˆ’ πΆπ‘Žπ‘‘β„ŽπΆβ„Žπ‘π‘ βˆ’ 𝛿𝑐

π‘Žπ›Ώπ‘π‘‘. (11)

Theorem 3.3. Covariant components of the Ricci tensor SGK-manifolds II in

space of the associated G-structure are of the form:

1) 𝑆00 = βˆ’2𝑛; 2) π‘†π‘ŽοΏ½Μ‚οΏ½ = π‘†οΏ½Μ‚οΏ½π‘Ž = π΄π‘Žπ‘π‘π‘ βˆ’ 3πΆπ‘Žπ‘π‘‘πΆπ‘‘π‘π‘ βˆ’ π›Ώπ‘Ž

𝑏, (12)

the other components are zero.

Theorem 3.4. Scalar curvature Ο‡ of SGK-manifold of type II in the space of

the associated G-structure is calculated by the formula

πœ’ = 𝑔𝑖𝑗𝑆𝑖𝑗 = 2π΄π‘Žπ‘π‘Žπ‘ βˆ’ 6πΆπ‘Žπ‘π‘‘πΆπ‘‘π‘π‘ βˆ’ 2𝑛. (13)

4. STRUCTURAL TENSORS SGK-MANIFOLDS OF TYPE II

Consider the family of functions 𝐢 = {πΆπ‘–π‘—π‘˜}; πΆπ‘Ž

�̂�𝑐̂ = πΆπ‘Žπ‘π‘; 𝐢�̂�𝑏𝑐 =

πΆπ‘Žπ‘π‘; all other components of the family C – zero. We show that this system

functions on the space of the associated G-structure globally defined tensor of

type (2,1) on M. In fact, since 𝛻Φ – tensor of type (2,1), by the fundamental

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theorem of tensor analysis βˆ‡Ξ¦π‘—,π‘˜π‘– = 𝑑Φ𝑗,π‘˜

𝑖 + Φ𝑗,π‘˜π‘™ πœƒπ‘™

𝑖 βˆ’ Φ𝑙,π‘˜π‘– πœƒπ‘—

𝑙 βˆ’ Φ𝑗,𝑙𝑖 πœƒπ‘˜

𝑙 ≑

0 (π‘šπ‘œπ‘‘ πœ”π‘˜). In particular, βˆ‡Ξ¦οΏ½Μ‚οΏ½,π‘Μ‚π‘Ž = 𝑑Φ�̂�,𝑐̂

π‘Ž + Ξ¦οΏ½Μ‚οΏ½,𝑐̂𝑑 πœƒπ‘‘

π‘Ž βˆ’ Ξ¦οΏ½Μ‚οΏ½,π‘Μ‚π‘Ž πœƒοΏ½Μ‚οΏ½

οΏ½Μ‚οΏ½ βˆ’ Ξ¦οΏ½Μ‚οΏ½,οΏ½Μ‚οΏ½π‘Ž πœƒπ‘Μ‚

οΏ½Μ‚οΏ½ ≑

0 (π‘šπ‘œπ‘‘ πœ”π‘˜). In view of (6), these relations can be rewritten as βˆ‡πΆπ‘ŽοΏ½Μ‚οΏ½π‘Μ‚ =

π‘‘πΆπ‘Žπ‘π‘ + πΆπ‘‘π‘π‘πœƒπ‘‘π‘Ž + πΆπ‘Žπ‘‘π‘πœƒπ‘‘

𝑏 + πΆπ‘Žπ‘π‘‘πœƒπ‘‘π‘ ≑ 0 (π‘šπ‘œπ‘‘ πœ”π‘˜) . Similarly, βˆ‡πΆοΏ½Μ‚οΏ½

𝑏𝑐 =

π‘‘πΆπ‘Žπ‘π‘ βˆ’ πΆπ‘‘π‘π‘πœƒπ‘Žπ‘‘ βˆ’ πΆπ‘Žπ‘‘π‘πœƒπ‘

𝑑 βˆ’ πΆπ‘Žπ‘π‘‘πœƒπ‘π‘‘ ≑ 0 (π‘šπ‘œπ‘‘ πœ”π‘˜).

Consequently, βˆ‡πΆπ‘–π‘—π‘˜ ≑ 0 (π‘šπ‘œπ‘‘ πœ”π‘˜). By the fundamental theorem of

tensor analysis family of C defines a real tensor field of type (2,1) on M,

which we denote by the same symbol. This tensor defines a mapping

𝐢: 𝒳(𝑀) Γ— 𝒳(𝑀) ⟢ 𝒳(𝑀) , defined by the formula 𝐢(𝑋, π‘Œ) =πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€π‘Ž , where {πΆπ‘Žπ‘π‘ , πΆπ‘Žπ‘π‘} – components of the

structure tensor SGK-manifold of type II. This tensor of type (2,1) is called a

composition and determines the module 𝒳(𝑀) the structure of the Q-algebra

(Kirkchenko, 1983;Kirichenko, 1986). Map C has the following properties:

1) 𝐢(πœ‰, 𝑋) = 𝐢(𝑋, πœ‰) = 0; 2) 𝐢(𝑋, π‘Œ) = βˆ’πΆ(π‘Œ, 𝑋); 3) 𝐢(Φ𝑋, π‘Œ) =𝐢(𝑋, Ξ¦π‘Œ) = βˆ’Ξ¦ ∘ 𝐢(𝑋, π‘Œ); 4) πœ‚ ∘ 𝐢(𝑋, π‘Œ) = 0; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (14)

In fact:

1) Since πœ‰π‘Ž = πœ‰π‘Ž = 0 , then 𝐢(πœ‰, 𝑋) = πΆπ‘Žπ‘π‘πœ‰π‘π‘‹π‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘πœ‰π‘π‘‹π‘νœ€π‘Ž = 0 .

Similarly, 𝐢(𝑋, πœ‰) = πΆπ‘Žπ‘π‘π‘‹π‘πœ‰π‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘π‘‹π‘πœ‰π‘νœ€π‘Ž = 0.

2) In view of (6(3)), (6(4)) we have

𝐢(𝑋, π‘Œ) = πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€π‘Ž = πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ βˆ’ πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€π‘Ž =βˆ’πΆ(π‘Œ, 𝑋).

3) 𝐢(Φ𝑋, π‘Œ) = πΆπ‘Žπ‘π‘(Φ𝑋)π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘(Φ𝑋)π‘π‘Œπ‘νœ€π‘Ž = βˆšβˆ’1πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ βˆ’

βˆšβˆ’1πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€π‘Ž = πΆπ‘Žπ‘π‘π‘‹π‘(βˆšβˆ’1π‘Œ)π‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘π‘‹π‘(βˆ’βˆšβˆ’1π‘Œ)

π‘νœ€π‘Ž =

πΆπ‘Žπ‘π‘π‘‹π‘(Ξ¦π‘Œ)π‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘π‘‹π‘(Ξ¦π‘Œ)π‘νœ€π‘Ž = 𝐢(𝑋, Ξ¦π‘Œ) , and also Ξ¦ ∘ 𝐢(𝑋, π‘Œ) =

Ξ¦ ∘ (πΆπ‘Žπ‘π‘Xπ‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘π‘Xπ‘π‘Œπ‘νœ€π‘Ž) = πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘Ξ¦(νœ€οΏ½Μ‚οΏ½) + πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘Ξ¦(νœ€π‘Ž) =

βˆ’βˆšβˆ’1πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€οΏ½Μ‚οΏ½ + βˆšβˆ’1πΆπ‘Žπ‘π‘π‘‹π‘π‘Œπ‘νœ€π‘Ž = βˆ’ (πΆπ‘Žπ‘π‘(βˆšβˆ’1𝑋)𝑏

π‘Œπ‘νœ€οΏ½Μ‚οΏ½ +

πΆπ‘Žπ‘π‘(βˆ’βˆšβˆ’1𝑋)𝑏

π‘Œπ‘νœ€π‘Ž) = 𝐢(Φ𝑋, π‘Œ).

Using the properties of the tensor C, we find an explicit analytic

expression for this tensor. By definition, we have: [𝐢(νœ€οΏ½Μ‚οΏ½ , νœ€π‘Μ‚)]π‘Žνœ€π‘Ž =

πΆπ‘ŽοΏ½Μ‚οΏ½π‘Μ‚νœ€π‘Ž =

βˆšβˆ’1

2Ξ¦οΏ½Μ‚οΏ½,𝑐̂

π‘Ž νœ€π‘Ž =βˆšβˆ’1

2(βˆ‡πœ€οΏ½Μ‚οΏ½

(Ξ¦)νœ€οΏ½Μ‚οΏ½)π‘Ž

νœ€π‘Ž =1

2(Ξ¦ ∘ βˆ‡πœ€οΏ½Μ‚οΏ½

(Ξ¦)νœ€οΏ½Μ‚οΏ½)π‘Ž

νœ€π‘Ž,

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Abu-Saleem A. & Rustanov A. R

194 Malaysian Journal of Mathematical Sciences

i.e. [𝐢(νœ€οΏ½Μ‚οΏ½ , νœ€π‘Μ‚)]π‘Žνœ€π‘Ž =1

2(Ξ¦ ∘ βˆ‡πœ€οΏ½Μ‚οΏ½

(Ξ¦)νœ€οΏ½Μ‚οΏ½)π‘Ž

νœ€π‘Ž . Since the vectors {νœ€π‘Ž} form a

basis of the subspace (π·Ξ¦βˆšβˆ’1)

𝑝, and the vectors {νœ€οΏ½Μ‚οΏ½} form a basis of the

subspace (π·Ξ¦βˆ’βˆšβˆ’1)

𝑝 and projectors module 𝒳(𝑀)𝐢 to submodules

π·Ξ¦βˆšβˆ’1, 𝐷Φ

βˆ’βˆšβˆ’1 are endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘

β„“ =1

2(βˆ’Ξ¦2 + βˆšβˆ’1Ξ¦), then recorded the above equation can be rewritten as

(Ξ¦2 + βˆšβˆ’1Ξ¦) ∘ 𝐢(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ) =1

2(Ξ¦2 +

βˆšβˆ’1Ξ¦) ∘ (Ξ¦ ∘ βˆ‡βˆ’Ξ¦2π‘Œ+βˆšβˆ’1Ξ¦π‘Œ(Ξ¦)(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋)) ; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀) .

Opening the brackets and simplifying, we obtain

𝐢(𝑋, π‘Œ) = βˆ’1

8{βˆ’Ξ¦ ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Ξ¦2𝑋 + Ξ¦ ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Φ𝑋 + Ξ¦2 ∘

βˆ‡Ξ¦2π‘Œ(Ξ¦)Φ𝑋 + Ξ¦2 ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Ξ¦2𝑋}. (15)

For SGK-manifolds of type II have the equality Ξ¦οΏ½Μ‚οΏ½,π‘π‘Ž = 0 , i.e.

Ξ¦οΏ½Μ‚οΏ½,π‘π‘Ž νœ€π‘Ž = (βˆ‡πœ€π‘

(Ξ¦)νœ€οΏ½Μ‚οΏ½)π‘Ž

νœ€π‘Ž = 0 . Since the vectors {νœ€π‘Ž} form a basis of the

subspace (π·Ξ¦βˆšβˆ’1)

𝑝, and the vectors {νœ€οΏ½Μ‚οΏ½} form a basis of the subspace

(π·Ξ¦βˆ’βˆšβˆ’1)

𝑝 and projectors module 𝒳(𝑀)𝐢 to submodules 𝐷Φ

βˆšβˆ’1, π·Ξ¦βˆ’βˆšβˆ’1 are

endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘ β„“ =

1

2(βˆ’Ξ¦2 +

βˆšβˆ’1Ξ¦) , then recorded the above equation can be rewritten as (Ξ¦2 +

βˆšβˆ’1Ξ¦) ∘ βˆ‡Ξ¦2π‘Œ+βˆšβˆ’1Ξ¦π‘Œ(Ξ¦)(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋) = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀).

Opening the brackets and simplifying, we obtain

Ξ¦2 ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Ξ¦2𝑋 + Ξ¦2 ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Φ𝑋 βˆ’ Ξ¦ ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Ξ¦2𝑋 + Ξ¦ βˆ˜βˆ‡Ξ¦2π‘Œ(Ξ¦)Φ𝑋 = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). We apply the operator Π€ on both sides of

this equation, then we have

Ξ¦ ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Ξ¦2𝑋 + Ξ¦ ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Φ𝑋 + Ξ¦2 ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Ξ¦2𝑋 βˆ’ Ξ¦2 βˆ˜βˆ‡Ξ¦2π‘Œ(Ξ¦)Φ𝑋 = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (16)

Given the resulting equation (15), (16) takes the form:

𝐢(𝑋, π‘Œ) =1

4{Ξ¦ ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Ξ¦2𝑋 βˆ’ Ξ¦2 ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Φ𝑋} = βˆ’

1

4{Φ ∘

βˆ‡Ξ¦π‘Œ(Ξ¦)Φ𝑋 + Ξ¦2 ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Ξ¦2𝑋}; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (17)

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Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind

Malaysian Journal of Mathematical Sciences 195

Thus we have proved.

Theorem 4.1. The structure tensor SGK-manifold of type II is calculated by

the formula 𝐢(𝑋, π‘Œ) =1

4{Ξ¦ ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Ξ¦2𝑋 βˆ’ Ξ¦2 ∘ βˆ‡Ξ¦2π‘Œ(Ξ¦)Φ𝑋} =

βˆ’1

4{Ξ¦ ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Φ𝑋 + Ξ¦2 ∘ βˆ‡Ξ¦π‘Œ(Ξ¦)Ξ¦2𝑋}; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀).

Since C is a tensor of type (2,1), then by the fundamental theorem of tensor

analysis, we have:

π‘‘πΆπ‘–π‘—π‘˜ + 𝐢𝑙

π‘—π‘˜πœƒπ‘™π‘– βˆ’ 𝐢𝑖

π‘™π‘˜πœƒπ‘—π‘™ βˆ’ 𝐢𝑖

π‘—π‘™πœƒπ‘˜π‘™ = 𝐢𝑖

π‘—π‘˜,π‘™πœ”π‘™, (18)

where {πΆπ‘–π‘—π‘˜,𝑙} – system functions, serving on the space of the bundle of all

frames components of a covariant differential structure tensor C. Rewriting

this equation in the space of the associated G-structure, we get:

1) πΆπ‘Žπ‘,𝑐0 = βˆ’πΆπ‘Žπ‘π‘; 2) 𝐢�̂��̂�,𝑐̂

0 = βˆ’πΆπ‘Žπ‘π‘; 3) 𝐢0οΏ½Μ‚οΏ½,π‘Μ‚π‘Ž = πΆπ‘Žπ‘π‘; 4) πΆπ‘Žπ‘,οΏ½Μ‚οΏ½

𝑐 =

βˆ’πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 5) 𝐢𝑏𝑐̂,π‘‘π‘Ž = πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘; 6) 𝐢�̂�0,𝑐̂

π‘Ž = βˆ’πΆπ‘Žπ‘π‘; 7) 𝐢�̂�𝑐,π‘‘π‘Ž =

βˆ’πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘; 8) 𝐢�̂�𝑐̂,οΏ½Μ‚οΏ½π‘Ž = πΆπ‘Žπ‘π‘π‘‘; 9) 𝐢�̂�𝑐̂,0

π‘Ž = πΆπ‘Žπ‘π‘; 10) 𝐢0𝑏,𝑐�̂� =

πΆπ‘Žπ‘π‘; 11) 𝐢𝑏0,𝑐�̂� = βˆ’πΆπ‘Žπ‘π‘; 12) 𝐢𝑏𝑐,𝑑

οΏ½Μ‚οΏ½ = πΆπ‘Žπ‘π‘π‘‘; 13) 𝐢𝑏𝑐,0οΏ½Μ‚οΏ½ =

βˆ’πΆπ‘Žπ‘π‘; 14) 𝐢𝑏𝑐̂,οΏ½Μ‚οΏ½οΏ½Μ‚οΏ½ = βˆ’πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 15) 𝐢�̂�𝑐,οΏ½Μ‚οΏ½

οΏ½Μ‚οΏ½ = πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 16) 𝐢�̂�𝑐̂,𝑑�̂� =

βˆ’πΆπ‘π‘β„ŽπΆβ„Žπ‘Žπ‘‘. (19)

And the other components are zero.

Thus we have proved the theorem.

Theorem 4.2. The components of the covariant differential structure tensor C

SGK-manifold of type II in the space of the associated G-structure are of the

form: 1) πΆπ‘Žπ‘,𝑐0 = βˆ’πΆπ‘Žπ‘π‘; 2) 𝐢�̂��̂�,𝑐̂

0 = βˆ’πΆπ‘Žπ‘π‘; 3) 𝐢0οΏ½Μ‚οΏ½,π‘Μ‚π‘Ž = πΆπ‘Žπ‘π‘; 4) πΆπ‘Žπ‘,οΏ½Μ‚οΏ½

𝑐 =

βˆ’πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 5) 𝐢𝑏𝑐̂,π‘‘π‘Ž = πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘; 6) 𝐢�̂�0,𝑐̂

π‘Ž = βˆ’πΆπ‘Žπ‘π‘; 7) 𝐢�̂�𝑐,π‘‘π‘Ž =

βˆ’πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘; 8) 𝐢�̂�𝑐̂,οΏ½Μ‚οΏ½π‘Ž = πΆπ‘Žπ‘π‘π‘‘; 9) 𝐢�̂�𝑐̂,0

π‘Ž = πΆπ‘Žπ‘π‘; 10) 𝐢0𝑏,𝑐�̂� =

πΆπ‘Žπ‘π‘; 11) 𝐢𝑏0,𝑐�̂� = βˆ’πΆπ‘Žπ‘π‘; 12) 𝐢𝑏𝑐,𝑑

οΏ½Μ‚οΏ½ = πΆπ‘Žπ‘π‘π‘‘; 13) 𝐢𝑏𝑐,0οΏ½Μ‚οΏ½ =

βˆ’πΆπ‘Žπ‘π‘; 14) 𝐢𝑏𝑐̂,οΏ½Μ‚οΏ½οΏ½Μ‚οΏ½ = βˆ’πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 15) 𝐢�̂�𝑐,οΏ½Μ‚οΏ½

οΏ½Μ‚οΏ½ = πΆπ‘π‘‘β„ŽπΆβ„Žπ‘Žπ‘; 16) 𝐢�̂�𝑐̂,𝑑�̂� =

βˆ’πΆπ‘π‘β„ŽπΆβ„Žπ‘Žπ‘‘ and the other components are zero.

Consider the map π’ž: 𝒳(𝑀) Γ— 𝒳(𝑀) Γ— 𝒳(𝑀) β†’ 𝒳(𝑀), given by the

formula

π’ž(𝑋, π‘Œ, 𝑍) = πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘π‘‹π‘π‘Œπ‘‘π‘π‘νœ€π‘Ž + πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘π‘‹π‘π‘Œπ‘‘π‘π‘Žνœ€οΏ½Μ‚οΏ½. (20)

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Abu-Saleem A. & Rustanov A. R

196 Malaysian Journal of Mathematical Sciences

Theorem 4.3. For SGK-manifold of type II have the following relations:

1) π’ž(𝑋1 + 𝑋2, π‘Œ, 𝑍) = π’ž(𝑋1, π‘Œ, 𝑍) + π’ž(𝑋2, π‘Œ, 𝑍); 2) π’ž(Φ𝑋, π‘Œ, 𝑍) =π’ž(𝑋, Ξ¦π‘Œ, 𝑍) = βˆ’π’ž(𝑋, π‘Œ, Φ𝑍) = Ξ¦ ∘ π’ž(𝑋, π‘Œ, 𝑍); 3) π’ž(𝑋, π‘Œ, Ξ¦2𝑍) =βˆ’π’ž(𝑋, π‘Œ, 𝑍); 4) πœ‚ ∘ π’ž(𝑋, π‘Œ, 𝑍) = 0; 5) π’ž(𝑋, π‘Œ, πœ‰) = π’ž(𝑋, πœ‰, π‘Œ) =π’ž(πœ‰, 𝑋, π‘Œ) = 0; 6) π’ž(𝑋, π‘Œ, 𝑍) = βˆ’π’ž(π‘Œ, 𝑋, 𝑍); 7) βŸ¨π’ž(𝑋, π‘Œ, 𝑍), π‘ŠβŸ© =βˆ’βŸ¨π’ž(𝑋, π‘Œ, π‘Š), π‘βŸ©; 8) π’ž(𝑋, π‘Œ, 𝑍1 + 𝑍2) =π’ž(𝑋, π‘Œ, 𝑍1) + π’ž(𝑋, π‘Œ, 𝑍1); βˆ€π‘‹, π‘Œ, 𝑍, π‘Š ∈ 𝒳(𝑀). (21)

Proof. The proof is by direct calculation. For example,

π’ž(𝑋1 + 𝑋2, π‘Œ, 𝑍) = πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋1 + 𝑋2)π‘π‘Œπ‘‘π‘π‘νœ€π‘Ž + πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋1 +𝑋2)π‘π‘Œπ‘‘π‘π‘Žνœ€οΏ½Μ‚οΏ½ = πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋1)π‘π‘Œπ‘‘π‘π‘νœ€π‘Ž + πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋2)π‘π‘Œπ‘‘π‘π‘νœ€π‘Ž +πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋1)π‘π‘Œπ‘‘π‘π‘Žνœ€οΏ½Μ‚οΏ½ + πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘(𝑋1)π‘π‘Œπ‘‘π‘π‘Žνœ€οΏ½Μ‚οΏ½ = π’ž(𝑋1, π‘Œ, 𝑍) +

π’ž(𝑋2, π‘Œ, 𝑍).

Similarly we prove the remaining properties.

We consider the endomorphism c, given by an A-frame matrix

(πΆπ‘π‘Ž) = (πΆπ‘Žπ‘π‘‘πΆπ‘‘π‘π‘).

This endomorphism Hermitian symmetric, and hence diagonalizable in a

suitable A-frame, ie, in this frame

πΆπ‘π‘Ž = 𝐢𝑏𝛿𝑏

π‘Ž, (22)

where {𝐢𝑏} – the eigenvalues of this endomorphism. Moreover, the

Hermitian form β„­(𝑋, π‘Œ) = πΆπ‘π‘Žπ‘‹π‘π‘Œπ‘Ž , corresponding to this endomorphism,

positive semi-definite, since β„­(𝑋, 𝑋) = πΆπ‘π‘Žπ‘‹π‘π‘‹π‘Ž = πΆπ‘Žπ‘π‘‘πΆπ‘‘π‘π‘π‘‹π‘π‘‹π‘Ž =

βˆ‘ |πΆπ‘π‘‘π‘Žπ‘‹π‘Ž|2𝑐,𝑑 β‰₯ 0. Consequently, πΆπ‘Ž β‰₯ 0, π‘Ž = 1,2, … , 𝑛.

Let us consider some properties of the tensor 𝛻𝐢.

Using the restore identity (Kirichenko, 2013; Kirichenko and Rustanov,

2002) to the relations (19), we obtain the following theorem.

Theorem 4.4. 𝛻𝐢 tensor has the following properties:

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1)βˆ‡πœ‰(𝐢)(πœ‰, 𝑋) = βˆ’βˆ‡πœ‰(𝐢)(𝑋, πœ‰) = 0; 2)βˆ‡Ξ¦2𝑋(𝐢)(πœ‰, Ξ¦2π‘Œ) =

βˆ’βˆ‡Ξ¦π‘‹(𝐢)(ΞΎ, Ξ¦π‘Œ) = βˆ‡π‘‹(𝐢)(πœ‰, π‘Œ) = 𝐢(𝑋, π‘Œ); 3)βˆ‡πœ‰(𝐢)(𝑋, π‘Œ) βˆ’

βˆ‡πœ‰(𝐢)(Φ𝑋, Ξ¦π‘Œ) = βˆ’2𝐢(𝑋, π‘Œ); 4)βˆ‡Ξ¦2𝑋(𝐢)(Ξ¦2π‘Œ, Ξ¦2𝑍) βˆ’

βˆ‡Ξ¦2𝑋(𝐢)(Ξ¦π‘Œ, Φ𝑍) βˆ’ βˆ‡Ξ¦π‘‹(𝐢)(Ξ¦2π‘Œ, Φ𝑍) βˆ’ βˆ‡Ξ¦π‘‹(𝐢)(Ξ¦π‘Œ, Ξ¦2𝑍) =4βŸ¨π‘‹, 𝐢(π‘Œ, 𝑍)βŸ©πœ‰; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀) . (23)

From the (9) by the fundamental theorem of tensor analysis follows

that the family of functions {π΄π‘Žπ‘π‘π‘‘ } in the space of the associated G-structure,

symmetric with respect to the upper and lower indices form a pure tensor on

𝑀2𝑛+1 called tensor Π€-holomorphic sectional curvature (Abu-Saleem and

Rustanov, 2015;Umnova, 2002). This tensor defines a mapping 𝐴: 𝒳(𝑀) ×𝒳(𝑀) Γ— 𝒳(𝑀) ⟢ 𝒳(𝑀), which is given by 𝐴(𝑋, π‘Œ, 𝑍) = π΄π‘Žπ‘

𝑐𝑑 π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ +

𝐴�̂��̂�𝑐̂�̂� π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ . Direct calculation is easy to check that the tensor Π€-

holomorphic sectional curvature has the following properties:

1) 𝐴(Φ𝑋, π‘Œ, 𝑍) = 𝐴(𝑋, Ξ¦π‘Œ, 𝑍) =βˆ’π΄(𝑋, π‘Œ, Φ𝑍); 2) 𝐴(𝑍, 𝑋, Ξ¦2π‘Œ) = βˆ’π΄(𝑍, 𝑋, π‘Œ); 3) πœ‚ ∘𝐴(𝑋, π‘Œ, 𝑍) = 0; 4) 𝐴(ΞΎ, π‘Œ, 𝑍) = 𝐴(𝑋, ΞΎ, 𝑍) = 𝐴(𝑋, π‘Œ, ΞΎ) =0; 5) 𝐴(𝑋, π‘Œ, 𝑍) = 𝐴(π‘Œ, 𝑋, 𝑍); 6) ⟨𝐴(𝑋, π‘Œ, 𝑍), π‘ŠβŸ© =⟨𝐴(𝑋, π‘Œ, π‘Š), π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍, π‘Š ∈ 𝒳(𝑀). (24)

In fact,

𝐴(Φ𝑋, π‘Œ, 𝑍) = π΄π‘Žπ‘π‘π‘‘ (Φ𝑋)π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ + 𝐴�̂��̂�

𝑐̂�̂� (Φ𝑋)π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ =

βˆšβˆ’1π΄π‘Žπ‘π‘π‘‘ π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ βˆ’ βˆšβˆ’1𝐴�̂��̂�

𝑐̂�̂� π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ = π΄π‘Žπ‘π‘π‘‘ π‘‹π‘Ž(Ξ¦π‘Œ)π‘π‘π‘‘νœ€π‘ +

𝐴�̂��̂�𝑐̂�̂� π‘‹π‘Ž(Ξ¦π‘Œ)π‘π‘π‘‘νœ€π‘ = 𝐴(𝑋, Ξ¦π‘Œ, 𝑍).

Similarly,

𝐴(Φ𝑋, π‘Œ, 𝑍) = π΄π‘Žπ‘π‘π‘‘ (Φ𝑋)π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ + 𝐴�̂��̂�

𝑐̂�̂� (Φ𝑋)π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ =

βˆšβˆ’1π΄π‘Žπ‘π‘π‘‘ π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ βˆ’ βˆšβˆ’1𝐴�̂��̂�

𝑐̂�̂� π‘‹π‘Žπ‘Œπ‘π‘π‘‘νœ€π‘ = βˆ’π΄π‘Žπ‘π‘π‘‘ π‘‹π‘ŽY𝑏(Φ𝑍)π‘‘νœ€π‘ βˆ’

𝐴�̂��̂�𝑐̂�̂� π‘‹π‘ŽY𝑏(Φ𝑍)π‘‘νœ€π‘ = βˆ’π΄(𝑋, π‘Œ, Φ𝑍).

Property 5 follows from the symmetry of the tensor π΄π‘Žπ‘π‘π‘‘ on the lower

pair of indices. The symmetry of the upper pair of indices implies property 6.

Thus, the theorem is proved.

Theorem 4.5. Tensor Π€-holomorphic sectional curvature SGK-manifold of

type II has the following properties:

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1) 𝐴(Φ𝑋, π‘Œ, 𝑍) = 𝐴(𝑋, Ξ¦π‘Œ, 𝑍) = βˆ’π΄(𝑋, π‘Œ, Φ𝑍); 2) 𝐴(𝑍, 𝑋, Ξ¦2π‘Œ) =βˆ’π΄(𝑍, 𝑋, π‘Œ); 3) πœ‚ ∘ 𝐴(𝑋, π‘Œ, 𝑍) = 0; 4) 𝐴(ΞΎ, π‘Œ, 𝑍) = 𝐴(𝑋, ΞΎ, 𝑍) =𝐴(𝑋, π‘Œ, ΞΎ) = 0; 5) 𝐴(𝑋, π‘Œ, 𝑍) = 𝐴(π‘Œ, 𝑋, 𝑍); 6) ⟨𝐴(𝑋, π‘Œ, 𝑍), π‘ŠβŸ© =

⟨𝐴(𝑋, π‘Œ, π‘Š), π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍, π‘Š ∈ 𝒳(𝑀).

5. CURVATURE IDENTITIES SGK-MANIFOLDS OF TYPE II

In Kirichenko, 2013; Kirichenko and Rustanov, 2002 allocated the

class of quasi-sasakian manifolds, the Riemann curvature tensor which

satisfies identity 𝑅(πœ‰, Ξ¦2𝑋)Ξ¦2π‘Œ βˆ’ 𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀) .

Following the idea described in these papers, we look at some of the identity

satisfied by the Riemann curvature tensor SGK-manifolds of type II.

We apply the restore identity (Kirichenko, 2013; Kirichenko and

Rustanov, 2002) to the equations: 𝑅00𝑏0 = 𝛿𝑏

0 = 0; 𝑅00π‘π‘Ž = 𝛿𝑏

π‘Ž; 𝑅00𝑏�̂� = 𝛿𝑏

οΏ½Μ‚οΏ½ =

0, i.e. to equality 𝑅00𝑏𝑖 = 𝛿𝑏

𝑖 . In fixed point 𝑝 ∈ 𝑀 last equality is equivalent

to the relation 𝑅(πœ‰, νœ€π‘)πœ‰ = νœ€π‘ . Since πœ‰π‘ = νœ€0 , and the vectors {νœ€π‘Ž} form a

basis of the subspace (π·Ξ¦βˆšβˆ’1)

𝑝, given that the projectors module 𝒳(𝑀)𝐢 to

submodules π·Ξ¦βˆšβˆ’1 and 𝐷Φ

0 are endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 +

βˆšβˆ’1Ξ¦), 𝓂 = 𝑖𝑑 + Ξ¦2, identity 𝑅(πœ‰, νœ€π‘)πœ‰ = νœ€π‘ can be rewritten in the form

𝑅(πœ‰, Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋)πœ‰ = Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋; βˆ€π‘‹ ∈ 𝒳(𝑀).

Expanding this relation is linear and separating the real and

imaginary parts of the resulting equation, we obtain an equivalent identity:

𝑅(πœ‰, Ξ¦2𝑋)πœ‰ = Ξ¦2𝑋; βˆ€π‘‹ ∈ 𝒳(𝑀). (25)

We say that the identity (25), the first additional identity curvature

SGK-manifold of type II.

Since 𝑅(πœ‰, πœ‰)πœ‰ = 0 and Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰ , then (25) takes the

form:

𝑅(πœ‰, 𝑋)πœ‰ = βˆ’π‘‹ + πœ‚(𝑋)πœ‰; βˆ€π‘‹ ∈ 𝒳(𝑀). (26)

Definition 5.1. We call the AC-manifold manifold of class 𝑅1, if its curvature

tensor satisfies

𝑅(πœ‰, 𝑋)πœ‰ = 0; βˆ€π‘‹ ∈ 𝒳(𝑀). (27)

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Malaysian Journal of Mathematical Sciences 199

Let M – SGK-manifold of type II, which is a manifold of class 𝑅1.

Then its Riemann curvature tensor satisfies (27). On the space of the

associated G-structure relation (27) can be written as: 𝑅00𝑗𝑖 𝑋𝑗 = 0. In view of

(11), the last equation can be written as: 𝑅00π‘π‘Ž 𝑋𝑏 + 𝑅00οΏ½Μ‚οΏ½

οΏ½Μ‚οΏ½ 𝑋�̂� = 0 . This

equality holds if and only if 𝑅00π‘π‘Ž = 𝑅00οΏ½Μ‚οΏ½

οΏ½Μ‚οΏ½ = 0. But since 𝑅00π‘π‘Ž = 𝛿𝑏

π‘Ž β‰  0,

then SGK-manifold of type II is not a manifold class 𝑅1.

Thus we have proved.

Theorem 5.1. SGK-manifold of type II is not a manifold class 𝑅1.

Since 𝑅0π‘Žπ‘0 = 0, 𝑅0π‘Žπ‘

𝑐 = 0, 𝑅0π‘Žπ‘π‘Μ‚ = 0, i.e. 𝑅0π‘Žπ‘

𝑖 = 0. In fixed point 𝑝 ∈ 𝑀 it

is obviously equivalent to the relation 𝑅(νœ€π‘Ž , νœ€π‘)πœ‰ = 0. Since πœ‰π‘ = νœ€0, and the

vectors {νœ€π‘Ž} form a basis of the subspace (π·Ξ¦βˆšβˆ’1)

𝑝, and projectors module

𝒳(𝑀)𝐢 to submodules π·Ξ¦βˆšβˆ’1 and 𝐷Φ

0 will endomorphisms πœ‹ = 𝜎 ∘ β„“ =

βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦), 𝓂 = 𝑖𝑑 + Ξ¦2 , identity 𝑅(νœ€π‘Ž , νœ€π‘)πœ‰ = 0 can be rewritten

in the form 𝑅(Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ)πœ‰ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀).

Expanding this relation is linear and separating the real and imaginary parts

of the resulting equation, we obtain an equivalent identity:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)πœ‰ βˆ’ 𝑅(Φ𝑋, Ξ¦π‘Œ)πœ‰ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (28)

Taking into account the equality Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰ last equation

can be written as

𝑅(𝑋, π‘Œ)πœ‰ βˆ’ πœ‚(𝑋)𝑅(πœ‰, π‘Œ)πœ‰ βˆ’ πœ‚(π‘Œ)𝑅(𝑋, πœ‰)πœ‰ + πœ‚(𝑋)πœ‚(π‘Œ)𝑅(πœ‰, πœ‰)πœ‰ βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)πœ‰ = 𝑅(𝑋, π‘Œ)πœ‰ βˆ’ πœ‚(𝑋)𝑅(πœ‰, π‘Œ)πœ‰ + πœ‚(π‘Œ)𝑅(πœ‰, 𝑋)πœ‰ βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)πœ‰ = 0,

which in view of (26) takes the form:

𝑅(Φ𝑋, Ξ¦π‘Œ)πœ‰ βˆ’ 𝑅(𝑋, π‘Œ)πœ‰ = βˆ’πœ‚(π‘Œ)𝑋 + πœ‚(𝑋)π‘Œ; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (29)

Similarly, applying the restore identity to the equalities 𝑅0π‘ŽοΏ½Μ‚οΏ½0 = 0,

𝑅0π‘ŽοΏ½Μ‚οΏ½π‘ = 0, 𝑅0π‘ŽοΏ½Μ‚οΏ½

𝑐̂ = 0, i.e. 𝑅0π‘ŽοΏ½Μ‚οΏ½π‘– = 0, we obtain the identity:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)πœ‰ + 𝑅(Φ𝑋, Ξ¦π‘Œ)πœ‰ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (30)

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From the (28) and (30), we have:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)πœ‰ = 𝑅(Φ𝑋, Ξ¦π‘Œ)πœ‰ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (31)

Taking into account the equality Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰ and identity

(26), the identity of 𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)πœ‰ = 0 takes the form:

𝑅(𝑋, π‘Œ)πœ‰ = βˆ’πœ‚(𝑋)π‘Œ + πœ‚(π‘Œ)𝑋; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (32)

Using the restore identity to the relations π‘…π‘Ž0𝑏0 = 0 , π‘…π‘Ž0𝑏

𝑐 = 0 ,

π‘…π‘Ž0𝑏𝑐̂ = 0, i.e. π‘…π‘Ž0𝑏

𝑖 = 0, we obtain the identity:

𝑅(πœ‰, Ξ¦2𝑋)Ξ¦2π‘Œ βˆ’ 𝑅(ΞΎ, Φ𝑋)Ξ¦π‘Œ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (33)

In view of (26) and the relations 𝑅(πœ‰, πœ‰)πœ‰ = 0, Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰

last identity can be rewritten as:

𝑅(πœ‰, 𝑋)π‘Œ βˆ’ 𝑅(ΞΎ, Φ𝑋)Ξ¦π‘Œ = βˆ’πœ‚(π‘Œ)𝑋 + πœ‚(𝑋)πœ‚(π‘Œ)πœ‰; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (34)

Now apply recovery procedure identity to the equalities π‘…π‘Ž0οΏ½Μ‚οΏ½0 =

βˆ’π›Ώπ‘Žπ‘πœ‰0, π‘…π‘Ž0οΏ½Μ‚οΏ½

𝑐 = βˆ’π›Ώπ‘Žπ‘πœ‰π‘ = 0, π‘…π‘Ž0οΏ½Μ‚οΏ½

𝑐̂ = βˆ’π›Ώπ‘Žπ‘πœ‰π‘Μ‚ = 0, i.e. π‘…π‘Ž0οΏ½Μ‚οΏ½

𝑖 = βˆ’π›Ώπ‘Žπ‘πœ‰π‘–. In fixed

point 𝑝 ∈ 𝑀 last equality is equivalent to the relation 𝑅(πœ‰, νœ€οΏ½Μ‚οΏ½)νœ€π‘Ž = βˆ’π›Ώπ‘Žπ‘πœ‰ =

βˆ’βŸ¨νœ€π‘Ž , νœ€οΏ½Μ‚οΏ½βŸ©πœ‰ . Since πœ‰π‘ = νœ€0 , vectors {νœ€π‘Ž} form a basis of the subspace

(π·Ξ¦βˆšβˆ’1)

𝑝, and the vectors {νœ€οΏ½Μ‚οΏ½} form a basis of the subspace (𝐷Φ

βˆ’βˆšβˆ’1)𝑝

and

projectors of module 𝒳(𝑀)𝐢 to submodules π·Ξ¦βˆšβˆ’1, 𝐷Φ

βˆ’βˆšβˆ’1 and 𝐷Φ0 will

endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘ β„“ =

1

2(βˆ’Ξ¦2 +

βˆšβˆ’1Ξ¦), 𝓂 = 𝑖𝑑 + Ξ¦2 , the identity of 𝑅(πœ‰, νœ€οΏ½Μ‚οΏ½)νœ€π‘Ž = βˆ’βŸ¨νœ€π‘Ž , νœ€οΏ½Μ‚οΏ½βŸ©πœ‰ can be

rewritten in the form of 𝑅(πœ‰, βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋)(Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ) =

βˆ’βŸ¨βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘ŒβŸ©πœ‰; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀) . Expanding this

relation is linear and separating the real and imaginary parts of the resulting

equation, we obtain an equivalent identity:

𝑅(πœ‰, Ξ¦2𝑋)(Ξ¦2π‘Œ) + 𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ = βˆ’2βŸ¨Ξ¦π‘‹, Ξ¦π‘ŒβŸ©πœ‰ = βˆ’2βŸ¨π‘‹, π‘ŒβŸ©πœ‰ +2πœ‚(𝑋)πœ‚(π‘Œ)πœ‰; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (35)

We say that the identity (35), the second additional identity curvature

SGK-manifold of type II.

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From (33) and (35) we have:

𝑅(πœ‰, Ξ¦2𝑋)(Ξ¦2π‘Œ) = 𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ = βˆ’βŸ¨Ξ¦π‘‹, Ξ¦π‘ŒβŸ©πœ‰ = βˆ’βŸ¨π‘‹, π‘ŒβŸ©πœ‰ +πœ‚(𝑋)πœ‚(π‘Œ)πœ‰; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (36)

Note 1. The identity of 𝑅(πœ‰, Ξ¦2𝑋)(Ξ¦2π‘Œ) = βˆ’βŸ¨π‘‹, π‘ŒβŸ©πœ‰ + πœ‚(𝑋)πœ‚(π‘Œ)πœ‰, taking

into account (15) and the equation Ξ¦2 = βˆ’π‘–π‘‘ + πœ‚ βŠ— πœ‰, can be written as:

𝑅(πœ‰, 𝑋)π‘Œ = πœ‚(π‘Œ)Ξ¦2𝑋 βˆ’ βŸ¨Ξ¦π‘‹, Ξ¦π‘ŒβŸ©πœ‰ = βˆ’βŸ¨π‘‹, π‘ŒβŸ©πœ‰ βˆ’ πœ‚(π‘Œ)𝑋 +2πœ‚(𝑋)πœ‚(π‘Œ)πœ‰; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (37)

Definition 5.2. We call the AC-manifold manifold of class 𝑅2, if its curvature

tensor satisfies the equation:

𝑅(πœ‰, Ξ¦2𝑋)(Ξ¦2π‘Œ) + 𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀). (38)

Let M – SGK-manifold of type II, which is a manifold of class 𝑅2.

Then its Riemann curvature tensor satisfies the condition (38). In view of

(33) the identity (38) can be written as: 𝑅(πœ‰, Ξ¦2𝑋)(Ξ¦2π‘Œ) = 𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ =0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀) . On the space of the associated G-structure relation

𝑅(πœ‰, Φ𝑋)Ξ¦π‘Œ = 0; βˆ€π‘‹, π‘Œ ∈ 𝒳(𝑀) can be written as: 𝑅𝑗0π‘˜π‘– (Φ𝑋)π‘˜(Ξ¦π‘Œ)𝑗 = 0.

In view of (11) and the form of the matrix Π€, the last equation can be written

as: π‘…π‘Ž0οΏ½Μ‚οΏ½0 π‘‹οΏ½Μ‚οΏ½π‘Œπ‘Ž + 𝑅�̂�0𝑏

0 π‘‹π‘π‘ŒοΏ½Μ‚οΏ½ = 0. This equality holds if and only if π‘…π‘Ž0οΏ½Μ‚οΏ½0 =

𝑅�̂�0𝑏0 = 0. But since π‘…π‘Ž0οΏ½Μ‚οΏ½

0 = βˆ’π›Ώπ‘Žπ‘ β‰  0, then the SGK-manifold of type II is

not a manifold class 𝑅2.

Thus we have proved.

Theorem 5.2. SGK-manifold of type II is not a manifold class 𝑅2.

Since π‘…π‘Žπ‘π‘0 = βˆ’2𝐢0π‘Ž[𝑏𝑐] = 0; π‘…π‘Žπ‘π‘

𝑑 = βˆ’2πΆοΏ½Μ‚οΏ½π‘Ž[𝑏𝑐] = 0; π‘…π‘Žπ‘π‘οΏ½Μ‚οΏ½ = βˆ’2πΆπ‘‘π‘Ž[𝑏𝑐] ,

i.e. π‘…π‘Žπ‘π‘π‘– = βˆ’2πΆπ‘–π‘Ž[𝑏𝑐]. In fixed point 𝑝 ∈ 𝑀, taking into account (19), it is

obviously equivalent to the relation 𝑅(νœ€π‘ , νœ€π‘)νœ€π‘Ž = βˆ‡πœ€π‘(𝐢)(νœ€π‘Ž, νœ€π‘) βˆ’

βˆ‡πœ€π‘(𝐢)(νœ€π‘Ž , νœ€π‘). Since the vectors {νœ€π‘Ž} form a basis of the subspace (𝐷Φ

βˆšβˆ’1)𝑝

,

and projectors of module 𝒳(𝑀)𝐢 on the submodule π·Ξ¦βˆšβˆ’1 is the

endomorphism πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦) , then the identity of the

𝑅(νœ€π‘ , νœ€π‘)νœ€π‘Ž = βˆ‡πœ€π‘(𝐢)(νœ€π‘Ž , νœ€π‘) βˆ’ βˆ‡πœ€π‘

(𝐢)(νœ€π‘Ž , νœ€π‘) can be rewritten in the form

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𝑅(Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ)(Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍) =

βˆ‡Ξ¦2𝑋+βˆšβˆ’1Φ𝑋(𝐢)(Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍, Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ) βˆ’

βˆ‡Ξ¦2π‘Œ+βˆšβˆ’1Ξ¦π‘Œ(𝐢)(Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍, Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋); βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀).

Expanding this relation is linear and separating the real and imaginary parts

of the resulting equation, we obtain an equivalent identity:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 βˆ’ 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 βˆ’ 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = βˆ‡Ξ¦2𝑋(𝐢)(Ξ¦2𝑍, Ξ¦2π‘Œ) βˆ’ βˆ‡Ξ¦2𝑋(𝐢)(Φ𝑍, Ξ¦π‘Œ) βˆ’βˆ‡Ξ¦π‘‹(𝐢)(Ξ¦2𝑍, Ξ¦π‘Œ) βˆ’ βˆ‡Ξ¦π‘‹(𝐢)(Φ𝑍, Ξ¦2π‘Œ) βˆ’ βˆ‡Ξ¦2π‘Œ(𝐢)(Ξ¦2𝑍, Ξ¦2𝑋) +βˆ‡Ξ¦2π‘Œ(𝐢)(Φ𝑍, Φ𝑋) + βˆ‡Ξ¦π‘Œ(𝐢)(Ξ¦2𝑍, Φ𝑋) +βˆ‡Ξ¦π‘Œ(𝐢)(Φ𝑍, Ξ¦2𝑋); βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (39)

We say that the identity (39) to the third additional identity curvature

SGK-manifold of type II.

Definition 5.3. We call the AC-manifold manifold of class 𝑅3, if its curvature

tensor satisfies the equation:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 βˆ’ 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 βˆ’ 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = 0; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (40)

From the definition 5.3 and (39).

Theorem 5.3. SGK-manifold is a manifold of type II class 𝑅3 if and only if

βˆ‡Ξ¦2𝑋(𝐢)(Ξ¦2𝑍, Ξ¦2π‘Œ) βˆ’ βˆ‡Ξ¦2𝑋(𝐢)(Φ𝑍, Ξ¦π‘Œ) βˆ’ βˆ‡Ξ¦π‘‹(𝐢)(Ξ¦2𝑍, Ξ¦π‘Œ) βˆ’βˆ‡Ξ¦π‘‹(𝐢)(Φ𝑍, Ξ¦2π‘Œ) βˆ’ βˆ‡Ξ¦2π‘Œ(𝐢)(Ξ¦2𝑍, Ξ¦2𝑋) + βˆ‡Ξ¦2π‘Œ(𝐢)(Φ𝑍, Φ𝑋) +

βˆ‡Ξ¦π‘Œ(𝐢)(Ξ¦2𝑍, Φ𝑋) + βˆ‡Ξ¦π‘Œ(𝐢)(Φ𝑍, Ξ¦2𝑋) = 0; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀).

Let M – SGK-manifold of type II, which is a manifold of class 𝑅3.

Then its Riemann curvature tensor satisfies the condition (40), which in the

space of the associated G-structure can be written as:

π‘…π‘—π‘˜π‘™π‘– (Ξ¦2𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Ξ¦2𝑍)𝑗 βˆ’ π‘…π‘—π‘˜π‘™

𝑖 (Ξ¦2𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Φ𝑍)𝑗 βˆ’

π‘…π‘—π‘˜π‘™π‘– (Φ𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Φ𝑍)𝑗 βˆ’ π‘…π‘—π‘˜π‘™

𝑖 (Φ𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Ξ¦2𝑍)𝑗 = 0.

In view of (11) and the form of the matrix Π€, the last equation can be

written as: π‘…π‘Žπ‘π‘οΏ½Μ‚οΏ½ π‘‹π‘π‘Œπ‘π‘π‘Ž + 𝑅�̂��̂�𝑐̂

𝑑 π‘‹π‘π‘Œπ‘π‘π‘Ž = 0. This equality holds if and only

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if π‘…π‘Žπ‘π‘οΏ½Μ‚οΏ½ = 𝑅�̂��̂�𝑐̂

𝑑 = 0. According to (2.8), πΆπ‘Žπ‘[𝑐𝑑] = 0. From this equality and

(8) we have πΆπ‘Žπ‘π‘π‘‘ = 0.

Thus we have proved.

Theorem 5.4. SGK-manifold of type II is a manifold class 𝑅3 if and only if

on the space of the associated G-structure πΆπ‘Žπ‘π‘π‘‘ = πΆπ‘Žπ‘π‘π‘‘ = 0.

Since,

π‘…π‘Žπ‘π‘Μ‚0 = π΄π‘Žπ‘

0𝑐 βˆ’ 𝐢0π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’ 𝛿𝑏0π›Ώπ‘Ž

𝑐 = 0, π‘…π‘Žπ‘π‘Μ‚π‘‘ = π΄π‘Žπ‘

𝑑𝑐 βˆ’ πΆπ‘‘π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’

π›Ώπ‘π‘π›Ώπ‘Ž

𝑑, π‘…π‘Žπ‘π‘Μ‚οΏ½Μ‚οΏ½ = π΄π‘Žπ‘

�̂�𝑐 βˆ’ πΆοΏ½Μ‚οΏ½π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’ π›Ώπ‘οΏ½Μ‚οΏ½π›Ώπ‘Ž

𝑐 = 0,

i.e. π‘…π‘Žπ‘π‘Μ‚π‘– = π΄π‘Žπ‘

𝑖𝑐 βˆ’ πΆπ‘–π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’ π›Ώπ‘π‘π›Ώπ‘Ž

𝑖 .

In fixed point 𝑝 ∈ 𝑀 it is obviously equivalent to the relation 𝑅(νœ€π‘ , νœ€π‘Μ‚)νœ€π‘Ž =𝐴(νœ€π‘Ž , νœ€π‘ , νœ€π‘Μ‚) + βˆ‡πœ€οΏ½Μ‚οΏ½

(𝐢)(νœ€π‘Ž , νœ€π‘) βˆ’ νœ€π‘βŸ¨νœ€π‘Ž, νœ€π‘Μ‚βŸ© . Since the vectors {νœ€π‘Ž} form a

basis of the subspace (π·Ξ¦βˆšβˆ’1)

𝑝, and the vectors {νœ€οΏ½Μ‚οΏ½} form a basis of the

subspace (π·Ξ¦βˆ’βˆšβˆ’1)

𝑝 and projectors of the module 𝒳(𝑀)𝐢 to submodules

π·Ξ¦βˆšβˆ’1, 𝐷Φ

βˆ’βˆšβˆ’1 are endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 + βˆšβˆ’1Ξ¦),

οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘ β„“ =1

2(βˆ’Ξ¦2 + βˆšβˆ’1Ξ¦) identity 𝑅(νœ€π‘ , νœ€π‘Μ‚)νœ€π‘Ž = 𝐴(νœ€π‘Ž , νœ€π‘ , νœ€π‘Μ‚) +

βˆ‡πœ€οΏ½Μ‚οΏ½(𝐢)(νœ€π‘Ž , νœ€π‘) βˆ’ νœ€π‘βŸ¨νœ€π‘Ž , νœ€π‘Μ‚βŸ© can be rewritten in the form

𝑅(Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ)(Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍) = 𝐴(Ξ¦2𝑍 +

βˆšβˆ’1Φ𝑍, Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ) + βˆ‡βˆ’Ξ¦2π‘Œ+βˆšβˆ’1Ξ¦π‘Œ(𝐢)(Ξ¦2𝑍 +

βˆšβˆ’1Φ𝑍, Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋) βˆ’ (Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋)βŸ¨βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ, Ξ¦2𝑍 +

βˆšβˆ’1Ξ¦π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀).

Expanding this relation by linearity, and separating the real and imaginary

parts of the resulting equation and using (24), we obtain an equivalent

identity:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 + 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 βˆ’ 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 +𝑅(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = βˆ’4𝐴(𝑍, 𝑋, π‘Œ) + βˆ‡Ξ¦2π‘Œ(𝐢)(Ξ¦2𝑍, Ξ¦2𝑋) βˆ’βˆ‡Ξ¦2π‘Œ(𝐢)(Φ𝑍, Φ𝑋) + βˆ‡Ξ¦π‘Œ(𝐢)(Ξ¦2𝑍, Φ𝑋) + βˆ‡Ξ¦π‘Œ(𝐢)(Φ𝑍, Ξ¦2𝑋) βˆ’2Ξ¦2π‘‹βŸ¨Ξ¦π‘Œ, Ξ¦π‘βŸ© βˆ’ 2Ξ¦π‘‹βŸ¨π‘Œ, Ξ¦π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (41)

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We say that the identity (41), fourth additional identity curvature

SGK-manifold of type II.

Definition 5.4. We call the AC-manifold manifold of class 𝑅4, if its curvature

tensor satisfies the equation:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 + 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 βˆ’ 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 +𝑅(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = 0; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (42)

Let M – SGK-manifold of type II, which is a manifold of class 𝑅4.

Then its Riemann curvature tensor satisfies the condition (42), which in the

space of the associated G-structure can be written as:

π‘…π‘—π‘˜π‘™π‘– (Ξ¦2𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Ξ¦2𝑍)𝑗 + π‘…π‘—π‘˜π‘™

𝑖 (Ξ¦2𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Φ𝑍)𝑗 βˆ’

π‘…π‘—π‘˜π‘™π‘– (Φ𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Φ𝑍)𝑗 + π‘…π‘—π‘˜π‘™

𝑖 (Φ𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Ξ¦2𝑍)𝑗 = 0.

In view of (11) and the form of the matrix Π€, the last equation can be

written as: π‘…π‘Žπ‘π‘Μ‚π‘‘ π‘‹π‘π‘Œπ‘π‘π‘Ž + 𝑅�̂��̂�𝑐

οΏ½Μ‚οΏ½ π‘‹π‘π‘Œπ‘π‘π‘Ž = 0. This equality holds if and only

if π‘…π‘Žπ‘π‘Μ‚π‘‘ = 𝑅�̂��̂�𝑐

οΏ½Μ‚οΏ½ = 0 . According to (11), π΄π‘Žπ‘π‘‘π‘ βˆ’ πΆπ‘‘π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’ 𝛿𝑏

π‘π›Ώπ‘Žπ‘‘ = 0 .

Simmetriruya last equality first in the indices a and b, then the indices c and

d, we obtain 𝐴(π‘Žπ‘)(𝑑𝑐)

= π›Ώπ‘π‘π›Ώπ‘Ž

𝑑 =1

2π›Ώπ‘Žπ‘

𝑐𝑑. Due to the symmetry of the tensor π΄π‘Žπ‘π‘π‘‘

by the upper and lower pair of indices, the resulting identity can be rewritten

as: π΄π‘Žπ‘π‘π‘‘ =

1

2π›Ώπ‘Žπ‘

𝑐𝑑 . Then the equality π΄π‘Žπ‘π‘‘π‘ βˆ’ πΆπ‘‘π‘β„ŽπΆβ„Žπ‘Žπ‘ βˆ’ 𝛿𝑏

π‘π›Ώπ‘Žπ‘‘ = 0 can be

written as πΆπ‘‘π‘β„ŽπΆβ„Žπ‘Žπ‘ = π›Ώπ‘Žπ‘π‘π‘‘.

Thus proved the following.

Theorem 5.5. Let M – SGK-manifold of type II. Then the following

conditions are equivalent:

1) M is a manifold of class 𝑅4;

2) on the space of the associated G-structure πΆπ‘‘π‘β„ŽπΆβ„Žπ‘Žπ‘ = π›Ώπ‘Žπ‘π‘π‘‘;

3) on the space of the associated G-structure π΄π‘Žπ‘π‘π‘‘ =

1

2π›Ώπ‘Žπ‘

𝑐𝑑,

and as an intermediate result can be formulated in the following theorem.

Theorem 5.6. Tensor Π€-holomorphic sectional curvature SGK-manifold of

type II class 𝑅4 has the form:

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Malaysian Journal of Mathematical Sciences 205

𝐴(𝑍, 𝑋, π‘Œ) = βˆ’1

2{Ξ¦2π‘‹βŸ¨Ξ¦π‘Œ, Ξ¦π‘βŸ© + Ξ¦π‘‹βŸ¨π‘Œ, Ξ¦π‘βŸ©}; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀).

Consider the relations 𝑅𝑏𝑐̂�̂�0 = 0, 𝑅𝑏𝑐̂�̂�

π‘Ž = 0, 𝑅𝑏𝑐̂�̂��̂� = 2πΆπ‘Žπ‘β„ŽπΆβ„Žπ‘π‘‘ βˆ’

2π›Ώπ‘Ž[𝑐𝛿𝑏

𝑑], i.e. 𝑅𝑏𝑐̂�̂�

𝑖 = 2πΆοΏ½Μ‚οΏ½π‘β„ŽπΆβ„Žπ‘π‘‘ βˆ’ 2𝛿�̂�[𝑐𝛿𝑏

𝑑]. The last equality can be written

as 𝑅(νœ€π‘Μ‚ , νœ€οΏ½Μ‚οΏ½)νœ€π‘ = βˆ’2βˆ‡πœ€οΏ½Μ‚οΏ½(𝐢)(νœ€π‘ , νœ€οΏ½Μ‚οΏ½) + νœ€οΏ½Μ‚οΏ½βŸ¨νœ€π‘ , νœ€π‘Μ‚βŸ© βˆ’ νœ€π‘Μ‚βŸ¨νœ€π‘ , νœ€οΏ½Μ‚οΏ½βŸ© . Since the

vectors {νœ€π‘Ž} form a basis of the subspace (π·Ξ¦βˆšβˆ’1)

𝑝, and the vectors {νœ€οΏ½Μ‚οΏ½} form

a basis of the subspace (π·Ξ¦βˆ’βˆšβˆ’1)

𝑝 and projectors of module 𝒳(𝑀)𝐢 to

submodules π·Ξ¦βˆšβˆ’1, 𝐷Φ

βˆ’βˆšβˆ’1 are endomorphisms πœ‹ = 𝜎 ∘ β„“ = βˆ’1

2(Ξ¦2 +

βˆšβˆ’1Ξ¦), οΏ½Μ…οΏ½ = οΏ½Μ…οΏ½ ∘ β„“ =1

2(βˆ’Ξ¦2 + βˆšβˆ’1Ξ¦) , identity 𝑅(νœ€π‘Μ‚ , νœ€οΏ½Μ‚οΏ½)νœ€π‘ =

βˆ’2βˆ‡πœ€οΏ½Μ‚οΏ½(𝐢)(νœ€π‘ , νœ€οΏ½Μ‚οΏ½) + νœ€οΏ½Μ‚οΏ½βŸ¨νœ€π‘ , νœ€π‘Μ‚βŸ© βˆ’ νœ€π‘Μ‚βŸ¨νœ€π‘ , νœ€οΏ½Μ‚οΏ½βŸ© can be rewritten in the form

𝑅(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ)(Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍) =

βˆ’2βˆ‡βˆ’Ξ¦2π‘Œ+βˆšβˆ’1Ξ¦π‘Œ(𝐢)(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, Ξ¦2𝑍 + βˆšβˆ’1Φ𝑍) + (βˆ’Ξ¦2π‘Œ +

βˆšβˆ’1Ξ¦π‘Œ)βŸ¨βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋, Ξ¦2𝑍 + βˆšβˆ’1Ξ¦π‘βŸ© βˆ’

(βˆ’Ξ¦2𝑋 + βˆšβˆ’1Φ𝑋)βŸ¨βˆ’Ξ¦2π‘Œ + βˆšβˆ’1Ξ¦π‘Œ, Ξ¦2𝑍 + βˆšβˆ’1Ξ¦π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀).

Expanding this relation by linearity, and separating the real and

imaginary parts of the resulting equation, we obtain an equivalent identity:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 + 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 + 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = βˆ’2βˆ‡Ξ¦2π‘Œ(𝐢)(Ξ¦2𝑋, Ξ¦2𝑍) βˆ’ 2βˆ‡Ξ¦2π‘Œ(𝐢)(Φ𝑋, Φ𝑍) βˆ’2βˆ‡Ξ¦π‘Œ(𝐢)(Ξ¦2𝑋, Φ𝑍) + 2βˆ‡Ξ¦π‘Œ(𝐢)(Φ𝑋, Ξ¦2𝑍) + Ξ¦2π‘ŒβŸ¨Ξ¦2𝑋, Ξ¦2π‘βŸ© +Ξ¦2π‘ŒβŸ¨Ξ¦π‘‹, Ξ¦π‘βŸ© + Ξ¦π‘ŒβŸ¨Ξ¦2𝑋, Ξ¦π‘βŸ© βˆ’ Ξ¦π‘ŒβŸ¨Ξ¦π‘‹, Ξ¦2π‘βŸ© βˆ’ Ξ¦2π‘‹βŸ¨Ξ¦2π‘Œ, Ξ¦2π‘βŸ© βˆ’Ξ¦π‘‹βŸ¨Ξ¦2π‘Œ, Ξ¦π‘βŸ© βˆ’ Ξ¦2π‘‹βŸ¨Ξ¦π‘Œ, Ξ¦π‘βŸ© + Ξ¦π‘‹βŸ¨Ξ¦π‘Œ, Ξ¦2π‘βŸ©; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (43)

We say that the identity (41), the fifth additional identity curvature

SGK-manifold of type II.

Definition 5.5. We call the AC-manifold manifold of class 𝑅5, if its curvature

tensor satisfies the equation:

𝑅(Ξ¦2𝑋, Ξ¦2π‘Œ)Ξ¦2𝑍 + 𝑅(Ξ¦2𝑋, Ξ¦π‘Œ)Φ𝑍 + 𝑅(Φ𝑋, Ξ¦2π‘Œ)Φ𝑍 βˆ’π‘…(Φ𝑋, Ξ¦π‘Œ)Ξ¦2𝑍 = 0; βˆ€π‘‹, π‘Œ, 𝑍 ∈ 𝒳(𝑀). (44)

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Abu-Saleem A. & Rustanov A. R

206 Malaysian Journal of Mathematical Sciences

Let M – SGK-manifold of type II, which is a manifold of class 𝑅5.

Then its Riemann curvature tensor satisfies the condition (44), which in the

space of the associated G-structure can be written as:

π‘…π‘—π‘˜π‘™π‘– (Ξ¦2𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Ξ¦2𝑍)𝑗 + π‘…π‘—π‘˜π‘™

𝑖 (Ξ¦2𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Φ𝑍)𝑗 +

π‘…π‘—π‘˜π‘™π‘– (Φ𝑋)π‘˜(Ξ¦2π‘Œ)𝑙(Φ𝑍)𝑗 βˆ’ π‘…π‘—π‘˜π‘™

𝑖 (Φ𝑋)π‘˜(Ξ¦π‘Œ)𝑙(Ξ¦2𝑍)𝑗 = 0.

In view of (11) and the form of the matrix Π€, the last equation can be

written as: 𝑅�̂�𝑏𝑐𝑑 π‘‹π‘π‘Œπ‘π‘π‘Ž + π‘…π‘ŽοΏ½Μ‚οΏ½π‘Μ‚

οΏ½Μ‚οΏ½ π‘‹π‘π‘Œπ‘π‘π‘Ž = 0. This equality holds if and only

if 𝑅�̂�𝑏𝑐𝑑 = π‘…π‘ŽοΏ½Μ‚οΏ½π‘Μ‚

οΏ½Μ‚οΏ½ = 0. According to (11), πΆπ‘Žπ‘‘β„ŽπΆβ„Žπ‘π‘ = 𝛿[π‘π‘Ž 𝛿𝑐]

𝑑 .

Thus, we have proved.

Theorem 5.7. SGK-manifold is a manifold of type II class 𝑅5 if and only if

the space of the associated G-structure πΆπ‘Žπ‘‘β„ŽπΆβ„Žπ‘π‘ = 𝛿[π‘π‘Ž 𝛿𝑐]

𝑑 .

Let's turn the equality πΆπ‘Žπ‘‘β„ŽπΆβ„Žπ‘π‘ = 𝛿[π‘π‘Ž 𝛿𝑐]

𝑑 first in the indices a and b,

then the indices c and d, we obtain βˆ‘ |πΆπ‘Žπ‘π‘|2π‘Ž,𝑏,𝑐 =

1

2𝑛(𝑛 βˆ’ 1). From the

resulting equation implies that for 𝑛 = 1, we have πΆπ‘Žπ‘π‘ = 0, i.e. manifold is

Kenmotsu.

Thus we have proved the following.

Theorem 5.8. SGK-manifold of type II class 𝑅5 of 3-dimensional manifold is

Kenmotsu.

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