MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESeinspem.upm.edu.my/journal/fullpaper/vol9no2/2....
Transcript of MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESeinspem.upm.edu.my/journal/fullpaper/vol9no2/2....
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 βπ(Ξ¦)π = β¨Ξ¦π, πβ©π β π(π)Ξ¦π; π, π β π³(π). 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 (β1). Such structures are obtained by
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Abu-Saleem A. & Rustanov A. R
188 Malaysian Journal of Mathematical Sciences
the construction of the warped product πΉ Γπ πͺπ in the sense of Bishop and
O'Neill, 1969 complex Euclidean space and the real line, where π(π‘) = πππ‘
(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
Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind
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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 β smooth manifold of dimension 2π + 1, π³(π) β πΆβ-module
of smooth vector fields on M. In what follows, all manifolds, tensor fields,
etc. objects are assumed to be smooth of class πΆβ.
Definition 2.1 (Kirichenko, 2013): Almost contact structure on a manifold M
is a triple (π, π, Ξ¦) of tensor fields on the manifold, where Ξ· β differential 1-
form, called the contact form of the structure, ΞΎ β vector field, called the
characteristic, Π€ β endomorphism of π³(π) called the structure
endomorphism.
In this
1) π(π) = 1; 2) π β Ξ¦ = 0; 3) Ξ¦(π) = 0; 4) Ξ¦2 = βππ + π β π. (1)
If, moreover, M is fixed Riemannian structure π =<β ,β >, such that
β¨Ξ¦π, Ξ¦πβ© = β¨X, Yβ© β π(π)π(π), π, π β π³(π), (2)
four (π, π, Ξ¦, π =<β,β>) 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©(π, π) = β¨π, Ξ¦πβ©, π, π β π³(π) called the
fundamental form of AC-structure (Kirichenko, 2013).
Let (π, π, Ξ¦, π) β almost contact metric structure on the manifold
π2π+1 . In the module π³(π) internally defined two complementary
projectors π = π β π and β = ππ β π = βΞ¦2 (Kirichenko and Rustanov,
2002); thus π³(π) = β β β³ , where β = πΌπ(Ξ¦) = ππππ β the so-called
contact distribution, πππβ = 2π, β³ = πΌππ = πππ(Ξ¦) = πΏ(π) β linear hull
of structural vector (wherein β and π are the projections onto the
submodules β, β³, respectively). Clearly, the distribution of the β and β³ are
invariant with respect to Π€ and mutually orthogonal. It is also obvious that
Ξ¦Μ2 = βππ, β¨Ξ¦Μπ, Ξ¦Μπβ© = β¨π, πβ©, π, π β π³(π), where Ξ¦Μ = Ξ¦|β.
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190 Malaysian Journal of Mathematical Sciences
Consequently, {Ξ¦Μπ, ππ|β} β Hermitian structure on the space βπ .
Complexification π³(π)π module π³(π) is the direct sum π³(π)π = π·Ξ¦ββ1 β
π·Ξ¦βββ1 β π·Ξ¦
0 eigensubspaces of structural endomorphism Π€, corresponding
to the eigenvalues ββ1, βββ1 and 0, respectively. Projectors on the terms of
this direct sum will be, respectively, the endomorphisms Gray, 1959 π = π β
β = β1
2(Ξ¦2 + ββ1Ξ¦), οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β β =
1
2(βΞ¦2 + ββ1Ξ¦), π = ππ + Ξ¦2
where π =1
2(ππ β ββ1Ξ¦), οΏ½Μ οΏ½ =
1
2(ππ + ββ1Ξ¦).
The mappings ππ: βπ βΆ π·Ξ¦ββ1 and οΏ½Μ οΏ½π: βπ βΆ π·Ξ¦
βββ1 are
isomorphism and anti-isomorphism, respectively, Hermitian spaces.
Therefore, each point π β π2π+1 may be connected space frames family
ππ(π)πΆ of the form (π, ν0, ν1, β¦ , νπ, ν1Μ, β¦ , νοΏ½ΜοΏ½), where νπ = β2ππ(ππ), νοΏ½ΜοΏ½ =
β2οΏ½Μ οΏ½π(ππ), ν0 = ππ; where {ππ} β orthonormal basis of Hermitian space βπ.
Such a frame called A-frame (Kirichenko and Rustanov, 2002). It is easy to
see that the matrix components of tensors Ξ¦π and ππ in an A-frame have the
form, respectively:
(Ξ¦ππ) = (
0 0 0
0 ββ1πΌπ 0
0 0 βββ1πΌπ
) , (πππ) = (1 0 00 0 πΌπ
0 πΌπ 0), (3)
where πΌπ β 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(π) , represented by matrices of the form (1 0 00 π΄ 00 0 π΄
) , where
π΄ β π(π) . This structure is called a G-connected (Kirichenko, 2013);
Kirichenko and Rustanov, 2002).
3. SPECIALLY GENERALIZED MANIFOLDS KENMOTSU
TYPE II
Let (π2π+1, Ξ¦, π, π, π = β¨β,ββ©) β 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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βπ(Ξ¦)π + βπ(Ξ¦)π = βπ(π)Ξ¦π β π(π)Ξ¦π; π, π β π³(π), (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. ππ = 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 πΉππ = πΉππ = 0
are called SGK-manifolds of type II.
Let M β 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) ππ = 0; 2) πππ = βπππ β§ ππ + πΆπππππ β§ ππ + πΏπ
ππ β§ ππ; 3) πππ =ππ
π β§ ππ + πΆπππππ β§ ππ + πΏπππ β§ ππ,
(5)
where
πΆπππ =ββ1
2Ξ¦οΏ½ΜοΏ½,πΜ
π ; πΆπππ = βββ1
2Ξ¦π,π
οΏ½ΜοΏ½ ; πΆ[πππ] = πΆπππ; πΆ[πππ] =
πΆπππ; πΆπππΜ Μ Μ Μ Μ Μ = πΆπππ. (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) ππ = 0; 2) πππ = βπππ β§ ππ + πΆπππππ β§ ππ + πΏπ
ππ β§ππ; 3) πππ = ππ
π β§ ππ + πΆπππππ β§ ππ + πΏπππ β§ ππ; 4) πππ
π =
βπππ β§ ππ
π + (π΄ππππ β 2πΆππβπΆβππ)ππ β§ ππ; 5) ππΆπππ + πΆπππππ
π +
πΆππππππ + πΆπππππ
π = πΆππππππ β πΆππππ; 6) ππΆπππ β πΆππππππ β
πΆππππππ β πΆπππππ
π = πΆππππππ β πΆππππ. (7)
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192 Malaysian Journal of Mathematical Sciences
where
π΄[ππ]ππ = π΄ππ
[ππ]= 0, πΆπ[πππ] = 0, πΆπ[πππ] = 0. (8)
Taking the exterior derivative of (7(4)), we obtain:
ππ΄ππππ + π΄ππ
βππβπ + π΄ππ
πβπβπ β π΄βπ
ππππβ β π΄πβ
ππππβ = π΄ππβ
ππ πβ + π΄ππππβπβ β
2π΄πππππ, (9)
where
1) π΄π[πβ]ππ = 0; 2) π΄ππ
π[πβ]= 0; 3) (π΄π[π
ππβ 2πΆππππΆππ[π) πΆ|π|πβ] =
0; 4) (π΄πππ[π
β 2πΆπ[π|π|πΆπππ) πΆ|π|πβ] = 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 οΏ½ΜοΏ½πππ = 2πΆππβπΆβππ β 2πΏ[π
π πΏπ]π ; 2) π οΏ½ΜοΏ½πΜοΏ½ΜοΏ½
π = β2πΆππ[ππ]; 3) π 00ππ =
πΏππ; 4) π πποΏ½ΜοΏ½
π = π΄ππππ β πΆππβπΆβππ β πΏπ
ππΏππ. (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) π00 = β2π; 2) πποΏ½ΜοΏ½ = ποΏ½ΜοΏ½π = π΄ππππ β 3πΆππππΆπππ β πΏπ
π, (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 = ππππππ = 2π΄ππππ β 6πΆππππΆπππ β 2π. (13)
4. STRUCTURAL TENSORS SGK-MANIFOLDS OF TYPE II
Consider the family of functions πΆ = {πΆπππ}; πΆπ
οΏ½ΜοΏ½πΜ = πΆπππ; πΆοΏ½ΜοΏ½ππ =
πΆπππ; all other components of the family C β 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 π»Ξ¦ β tensor of type (2,1), by the fundamental
Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind
Malaysian Journal of Mathematical Sciences 193
theorem of tensor analysis βΞ¦π,ππ = πΞ¦π,π
π + Ξ¦π,ππ ππ
π β Ξ¦π,ππ ππ
π β Ξ¦π,ππ ππ
π β‘
0 (πππ ππ). In particular, βΞ¦οΏ½ΜοΏ½,πΜπ = πΞ¦οΏ½ΜοΏ½,πΜ
π + Ξ¦οΏ½ΜοΏ½,πΜπ ππ
π β Ξ¦οΏ½ΜοΏ½,πΜπ ποΏ½ΜοΏ½
οΏ½ΜοΏ½ β Ξ¦οΏ½ΜοΏ½,οΏ½ΜοΏ½π ππΜ
οΏ½ΜοΏ½ β‘
0 (πππ ππ). In view of (6), these relations can be rewritten as βπΆποΏ½ΜοΏ½πΜ =
ππΆπππ + πΆππππππ + πΆπππππ
π + πΆππππππ β‘ 0 (πππ ππ) . Similarly, βπΆοΏ½ΜοΏ½
ππ =
ππΆπππ β πΆππππππ β πΆπππππ
π β πΆππππππ β‘ 0 (πππ ππ).
Consequently, βπΆπππ β‘ 0 (πππ ππ). 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
πΆ: π³(π) Γ π³(π) βΆ π³(π) , defined by the formula πΆ(π, π) =πΆπππππππνοΏ½ΜοΏ½ + πΆπππππππνπ , where {πΆπππ , πΆπππ} β components of the
structure tensor SGK-manifold of type II. This tensor of type (2,1) is called a
composition and determines the module π³(π) the structure of the Q-algebra
(Kirkchenko, 1983;Kirichenko, 1986). Map C has the following properties:
1) πΆ(π, π) = πΆ(π, π) = 0; 2) πΆ(π, π) = βπΆ(π, π); 3) πΆ(Ξ¦π, π) =πΆ(π, Ξ¦π) = βΞ¦ β πΆ(π, π); 4) π β πΆ(π, π) = 0; βπ, π, π β π³(π). (14)
In fact:
1) Since ππ = ππ = 0 , then πΆ(π, π) = πΆπππππππνοΏ½ΜοΏ½ + πΆπππππππνπ = 0 .
Similarly, πΆ(π, π) = πΆπππππππνοΏ½ΜοΏ½ + πΆπππππππνπ = 0.
2) In view of (6(3)), (6(4)) we have
πΆ(π, π) = πΆπππππππνοΏ½ΜοΏ½ + πΆπππππππνπ = πΆπππππππνοΏ½ΜοΏ½ β πΆπππππππνπ =βπΆ(π, π).
3) πΆ(Ξ¦π, π) = πΆπππ(Ξ¦π)πππνοΏ½ΜοΏ½ + πΆπππ(Ξ¦π)πππνπ = ββ1πΆπππππππνοΏ½ΜοΏ½ β
ββ1πΆπππππππνπ = πΆπππππ(ββ1π)πνοΏ½ΜοΏ½ + πΆπππππ(βββ1π)
πνπ =
πΆπππππ(Ξ¦π)πνοΏ½ΜοΏ½ + πΆπππππ(Ξ¦π)πνπ = πΆ(π, Ξ¦π) , and also Ξ¦ β πΆ(π, π) =
Ξ¦ β (πΆπππXπππνοΏ½ΜοΏ½ + πΆπππXπππνπ) = πΆπππππππΞ¦(νοΏ½ΜοΏ½) + πΆπππππππΞ¦(νπ) =
βββ1πΆπππππππνοΏ½ΜοΏ½ + ββ1πΆπππππππνπ = β (πΆπππ(ββ1π)π
ππνοΏ½ΜοΏ½ +
πΆπππ(βββ1π)π
ππνπ) = πΆ(Ξ¦π, π).
Using the properties of the tensor C, we find an explicit analytic
expression for this tensor. By definition, we have: [πΆ(νοΏ½ΜοΏ½ , νπΜ)]πνπ =
πΆποΏ½ΜοΏ½πΜνπ =
ββ1
2Ξ¦οΏ½ΜοΏ½,πΜ
π νπ =ββ1
2(βποΏ½ΜοΏ½
(Ξ¦)νοΏ½ΜοΏ½)π
νπ =1
2(Ξ¦ β βποΏ½ΜοΏ½
(Ξ¦)νοΏ½ΜοΏ½)π
νπ,
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i.e. [πΆ(νοΏ½ΜοΏ½ , νπΜ)]πνπ =1
2(Ξ¦ β βποΏ½ΜοΏ½
(Ξ¦)νοΏ½ΜοΏ½)π
νπ . Since the vectors {νπ} form a
basis of the subspace (π·Ξ¦ββ1)
π, and the vectors {νοΏ½ΜοΏ½} form a basis of the
subspace (π·Ξ¦βββ1)
π and projectors module π³(π)πΆ to submodules
π·Ξ¦ββ1, π·Ξ¦
βββ1 are endomorphisms π = π β β = β1
2(Ξ¦2 + ββ1Ξ¦), οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β
β =1
2(βΞ¦2 + ββ1Ξ¦), then recorded the above equation can be rewritten as
(Ξ¦2 + ββ1Ξ¦) β πΆ(βΞ¦2π + ββ1Ξ¦π, βΞ¦2π + ββ1Ξ¦π) =1
2(Ξ¦2 +
ββ1Ξ¦) β (Ξ¦ β ββΞ¦2π+ββ1Ξ¦π(Ξ¦)(βΞ¦2π + ββ1Ξ¦π)) ; βπ, π β π³(π) .
Opening the brackets and simplifying, we obtain
πΆ(π, π) = β1
8{βΞ¦ β βΞ¦2π(Ξ¦)Ξ¦2π + Ξ¦ β βΞ¦π(Ξ¦)Ξ¦π + Ξ¦2 β
βΞ¦2π(Ξ¦)Ξ¦π + Ξ¦2 β βΞ¦π(Ξ¦)Ξ¦2π}. (15)
For SGK-manifolds of type II have the equality Ξ¦οΏ½ΜοΏ½,ππ = 0 , i.e.
Ξ¦οΏ½ΜοΏ½,ππ νπ = (βππ
(Ξ¦)νοΏ½ΜοΏ½)π
νπ = 0 . Since the vectors {νπ} form a basis of the
subspace (π·Ξ¦ββ1)
π, and the vectors {νοΏ½ΜοΏ½} form a basis of the subspace
(π·Ξ¦βββ1)
π and projectors module π³(π)πΆ to submodules π·Ξ¦
ββ1, π·Ξ¦βββ1 are
endomorphisms π = π β β = β1
2(Ξ¦2 + ββ1Ξ¦), οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β β =
1
2(βΞ¦2 +
ββ1Ξ¦) , then recorded the above equation can be rewritten as (Ξ¦2 +
ββ1Ξ¦) β βΞ¦2π+ββ1Ξ¦π(Ξ¦)(βΞ¦2π + ββ1Ξ¦π) = 0; βπ, π β π³(π).
Opening the brackets and simplifying, we obtain
Ξ¦2 β βΞ¦2π(Ξ¦)Ξ¦2π + Ξ¦2 β βΞ¦π(Ξ¦)Ξ¦π β Ξ¦ β βΞ¦π(Ξ¦)Ξ¦2π + Ξ¦ ββΞ¦2π(Ξ¦)Ξ¦π = 0; βπ, π β π³(π). We apply the operator Π€ on both sides of
this equation, then we have
Ξ¦ β βΞ¦2π(Ξ¦)Ξ¦2π + Ξ¦ β βΞ¦π(Ξ¦)Ξ¦π + Ξ¦2 β βΞ¦π(Ξ¦)Ξ¦2π β Ξ¦2 ββΞ¦2π(Ξ¦)Ξ¦π = 0; βπ, π β π³(π). (16)
Given the resulting equation (15), (16) takes the form:
πΆ(π, π) =1
4{Ξ¦ β βΞ¦2π(Ξ¦)Ξ¦2π β Ξ¦2 β βΞ¦2π(Ξ¦)Ξ¦π} = β
1
4{Ξ¦ β
βΞ¦π(Ξ¦)Ξ¦π + Ξ¦2 β βΞ¦π(Ξ¦)Ξ¦2π}; βπ, π β π³(π). (17)
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Thus we have proved.
Theorem 4.1. The structure tensor SGK-manifold of type II is calculated by
the formula πΆ(π, π) =1
4{Ξ¦ β βΞ¦2π(Ξ¦)Ξ¦2π β Ξ¦2 β βΞ¦2π(Ξ¦)Ξ¦π} =
β1
4{Ξ¦ β βΞ¦π(Ξ¦)Ξ¦π + Ξ¦2 β βΞ¦π(Ξ¦)Ξ¦2π}; βπ, π β π³(π).
Since C is a tensor of type (2,1), then by the fundamental theorem of tensor
analysis, we have:
ππΆπππ + πΆπ
πππππ β πΆπ
πππππ β πΆπ
πππππ = πΆπ
ππ,πππ, (18)
where {πΆπππ,π} β 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) πΆππ,π0 = βπΆπππ; 2) πΆοΏ½ΜοΏ½οΏ½ΜοΏ½,πΜ
0 = βπΆπππ; 3) πΆ0οΏ½ΜοΏ½,πΜπ = πΆπππ; 4) πΆππ,οΏ½ΜοΏ½
π =
βπΆππβπΆβππ; 5) πΆππΜ,ππ = πΆππβπΆβππ; 6) πΆοΏ½ΜοΏ½0,πΜ
π = βπΆπππ; 7) πΆοΏ½ΜοΏ½π,ππ =
βπΆππβπΆβππ; 8) πΆοΏ½ΜοΏ½πΜ,οΏ½ΜοΏ½π = πΆππππ; 9) πΆοΏ½ΜοΏ½πΜ,0
π = πΆπππ; 10) πΆ0π,ποΏ½ΜοΏ½ =
πΆπππ; 11) πΆπ0,ποΏ½ΜοΏ½ = βπΆπππ; 12) πΆππ,π
οΏ½ΜοΏ½ = πΆππππ; 13) πΆππ,0οΏ½ΜοΏ½ =
βπΆπππ; 14) πΆππΜ,οΏ½ΜοΏ½οΏ½ΜοΏ½ = βπΆππβπΆβππ; 15) πΆοΏ½ΜοΏ½π,οΏ½ΜοΏ½
οΏ½ΜοΏ½ = πΆππβπΆβππ; 16) πΆοΏ½ΜοΏ½πΜ,ποΏ½ΜοΏ½ =
βπΆππβπΆβππ. (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) πΆππ,π0 = βπΆπππ; 2) πΆοΏ½ΜοΏ½οΏ½ΜοΏ½,πΜ
0 = βπΆπππ; 3) πΆ0οΏ½ΜοΏ½,πΜπ = πΆπππ; 4) πΆππ,οΏ½ΜοΏ½
π =
βπΆππβπΆβππ; 5) πΆππΜ,ππ = πΆππβπΆβππ; 6) πΆοΏ½ΜοΏ½0,πΜ
π = βπΆπππ; 7) πΆοΏ½ΜοΏ½π,ππ =
βπΆππβπΆβππ; 8) πΆοΏ½ΜοΏ½πΜ,οΏ½ΜοΏ½π = πΆππππ; 9) πΆοΏ½ΜοΏ½πΜ,0
π = πΆπππ; 10) πΆ0π,ποΏ½ΜοΏ½ =
πΆπππ; 11) πΆπ0,ποΏ½ΜοΏ½ = βπΆπππ; 12) πΆππ,π
οΏ½ΜοΏ½ = πΆππππ; 13) πΆππ,0οΏ½ΜοΏ½ =
βπΆπππ; 14) πΆππΜ,οΏ½ΜοΏ½οΏ½ΜοΏ½ = βπΆππβπΆβππ; 15) πΆοΏ½ΜοΏ½π,οΏ½ΜοΏ½
οΏ½ΜοΏ½ = πΆππβπΆβππ; 16) πΆοΏ½ΜοΏ½πΜ,ποΏ½ΜοΏ½ =
βπΆππβπΆβππ and the other components are zero.
Consider the map π: π³(π) Γ π³(π) Γ π³(π) β π³(π), given by the
formula
π(π, π, π) = πΆππβπΆβππππππππνπ + πΆππβπΆβππππππππνοΏ½ΜοΏ½. (20)
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Theorem 4.3. For SGK-manifold of type II have the following relations:
1) π(π1 + π2, π, π) = π(π1, π, π) + π(π2, π, π); 2) π(Ξ¦π, π, π) =π(π, Ξ¦π, π) = βπ(π, π, Ξ¦π) = Ξ¦ β π(π, π, π); 3) π(π, π, Ξ¦2π) =βπ(π, π, π); 4) π β π(π, π, π) = 0; 5) π(π, π, π) = π(π, π, π) =π(π, π, π) = 0; 6) π(π, π, π) = βπ(π, π, π); 7) β¨π(π, π, π), πβ© =ββ¨π(π, π, π), πβ©; 8) π(π, π, π1 + π2) =π(π, π, π1) + π(π, π, π1); βπ, π, π, π β π³(π). (21)
Proof. The proof is by direct calculation. For example,
π(π1 + π2, π, π) = πΆππβπΆβππ(π1 + π2)πππππνπ + πΆππβπΆβππ(π1 +π2)πππππνοΏ½ΜοΏ½ = πΆππβπΆβππ(π1)πππππνπ + πΆππβπΆβππ(π2)πππππνπ +πΆππβπΆβππ(π1)πππππνοΏ½ΜοΏ½ + πΆππβπΆβππ(π1)πππππνοΏ½ΜοΏ½ = π(π1, π, π) +
π(π2, π, π).
Similarly we prove the remaining properties.
We consider the endomorphism c, given by an A-frame matrix
(πΆππ) = (πΆππππΆπππ).
This endomorphism Hermitian symmetric, and hence diagonalizable in a
suitable A-frame, ie, in this frame
πΆππ = πΆππΏπ
π, (22)
where {πΆπ} β the eigenvalues of this endomorphism. Moreover, the
Hermitian form β(π, π) = πΆππππππ , corresponding to this endomorphism,
positive semi-definite, since β(π, π) = πΆππππππ = πΆππππΆπππππππ =
β |πΆπππππ|2π,π β₯ 0. Consequently, πΆπ β₯ 0, π = 1,2, β¦ , π.
Let us consider some properties of the tensor π»πΆ.
Using the restore identity (Kirichenko, 2013; Kirichenko and Rustanov,
2002) to the relations (19), we obtain the following theorem.
Theorem 4.4. π»πΆ tensor has the following properties:
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1)βπ(πΆ)(π, π) = ββπ(πΆ)(π, π) = 0; 2)βΞ¦2π(πΆ)(π, Ξ¦2π) =
ββΞ¦π(πΆ)(ΞΎ, Ξ¦π) = βπ(πΆ)(π, π) = πΆ(π, π); 3)βπ(πΆ)(π, π) β
βπ(πΆ)(Ξ¦π, Ξ¦π) = β2πΆ(π, π); 4)βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) β
βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) β βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) β βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) =4β¨π, πΆ(π, π)β©π; βπ, π, π β π³(π) . (23)
From the (9) by the fundamental theorem of tensor analysis follows
that the family of functions {π΄ππππ } in the space of the associated G-structure,
symmetric with respect to the upper and lower indices form a pure tensor on
π2π+1 called tensor Π€-holomorphic sectional curvature (Abu-Saleem and
Rustanov, 2015;Umnova, 2002). This tensor defines a mapping π΄: π³(π) Γπ³(π) Γ π³(π) βΆ π³(π), which is given by π΄(π, π, π) = π΄ππ
ππ ππππππνπ +
π΄οΏ½ΜοΏ½οΏ½ΜοΏ½πΜοΏ½ΜοΏ½ ππππππνπ . Direct calculation is easy to check that the tensor Π€-
holomorphic sectional curvature has the following properties:
1) π΄(Ξ¦π, π, π) = π΄(π, Ξ¦π, π) =βπ΄(π, π, Ξ¦π); 2) π΄(π, π, Ξ¦2π) = βπ΄(π, π, π); 3) π βπ΄(π, π, π) = 0; 4) π΄(ΞΎ, π, π) = π΄(π, ΞΎ, π) = π΄(π, π, ΞΎ) =0; 5) π΄(π, π, π) = π΄(π, π, π); 6) β¨π΄(π, π, π), πβ© =β¨π΄(π, π, π), πβ©; βπ, π, π, π β π³(π). (24)
In fact,
π΄(Ξ¦π, π, π) = π΄ππππ (Ξ¦π)πππππνπ + π΄οΏ½ΜοΏ½οΏ½ΜοΏ½
πΜοΏ½ΜοΏ½ (Ξ¦π)πππππνπ =
ββ1π΄ππππ ππππππνπ β ββ1π΄οΏ½ΜοΏ½οΏ½ΜοΏ½
πΜοΏ½ΜοΏ½ ππππππνπ = π΄ππππ ππ(Ξ¦π)πππνπ +
π΄οΏ½ΜοΏ½οΏ½ΜοΏ½πΜοΏ½ΜοΏ½ ππ(Ξ¦π)πππνπ = π΄(π, Ξ¦π, π).
Similarly,
π΄(Ξ¦π, π, π) = π΄ππππ (Ξ¦π)πππππνπ + π΄οΏ½ΜοΏ½οΏ½ΜοΏ½
πΜοΏ½ΜοΏ½ (Ξ¦π)πππππνπ =
ββ1π΄ππππ ππππππνπ β ββ1π΄οΏ½ΜοΏ½οΏ½ΜοΏ½
πΜοΏ½ΜοΏ½ ππππππνπ = βπ΄ππππ ππYπ(Ξ¦π)πνπ β
π΄οΏ½ΜοΏ½οΏ½ΜοΏ½πΜοΏ½ΜοΏ½ ππYπ(Ξ¦π)πνπ = βπ΄(π, π, Ξ¦π).
Property 5 follows from the symmetry of the tensor π΄ππππ 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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198 Malaysian Journal of Mathematical Sciences
1) π΄(Ξ¦π, π, π) = π΄(π, Ξ¦π, π) = βπ΄(π, π, Ξ¦π); 2) π΄(π, π, Ξ¦2π) =βπ΄(π, π, π); 3) π β π΄(π, π, π) = 0; 4) π΄(ΞΎ, π, π) = π΄(π, ΞΎ, π) =π΄(π, π, ΞΎ) = 0; 5) π΄(π, π, π) = π΄(π, π, π); 6) β¨π΄(π, π, π), πβ© =
β¨π΄(π, π, π), πβ©; βπ, π, π, π β π³(π).
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 π (π, Ξ¦2π)Ξ¦2π β π (π, Ξ¦π)Ξ¦π = 0; βπ, π β π³(π) .
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: π 00π0 = πΏπ
0 = 0; π 00ππ = πΏπ
π; π 00ποΏ½ΜοΏ½ = πΏπ
οΏ½ΜοΏ½ =
0, i.e. to equality π 00ππ = πΏπ
π . In fixed point π β π last equality is equivalent
to the relation π (π, νπ)π = νπ . Since ππ = ν0 , and the vectors {νπ} form a
basis of the subspace (π·Ξ¦ββ1)
π, given that the projectors module π³(π)πΆ to
submodules π·Ξ¦ββ1 and π·Ξ¦
0 are endomorphisms π = π β β = β1
2(Ξ¦2 +
ββ1Ξ¦), π = ππ + Ξ¦2, identity π (π, νπ)π = νπ can be rewritten in the form
π (π, Ξ¦2π + ββ1Ξ¦π)π = Ξ¦2π + ββ1Ξ¦π; βπ β π³(π).
Expanding this relation is linear and separating the real and
imaginary parts of the resulting equation, we obtain an equivalent identity:
π (π, Ξ¦2π)π = Ξ¦2π; βπ β π³(π). (25)
We say that the identity (25), the first additional identity curvature
SGK-manifold of type II.
Since π (π, π)π = 0 and Ξ¦2 = βππ + π β π , then (25) takes the
form:
π (π, π)π = βπ + π(π)π; βπ β π³(π). (26)
Definition 5.1. We call the AC-manifold manifold of class π 1, if its curvature
tensor satisfies
π (π, π)π = 0; βπ β π³(π). (27)
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Let M β SGK-manifold of type II, which is a manifold of class π 1.
Then its Riemann curvature tensor satisfies (27). On the space of the
associated G-structure relation (27) can be written as: π 00ππ ππ = 0. In view of
(11), the last equation can be written as: π 00ππ ππ + π 00οΏ½ΜοΏ½
οΏ½ΜοΏ½ ποΏ½ΜοΏ½ = 0 . This
equality holds if and only if π 00ππ = π 00οΏ½ΜοΏ½
οΏ½ΜοΏ½ = 0. But since π 00ππ = πΏπ
π β 0,
then SGK-manifold of type II is not a manifold class π 1.
Thus we have proved.
Theorem 5.1. SGK-manifold of type II is not a manifold class π 1.
Since π 0ππ0 = 0, π 0ππ
π = 0, π 0πππΜ = 0, i.e. π 0ππ
π = 0. In fixed point π β π it
is obviously equivalent to the relation π (νπ , νπ)π = 0. Since ππ = ν0, and the
vectors {νπ} form a basis of the subspace (π·Ξ¦ββ1)
π, and projectors module
π³(π)πΆ to submodules π·Ξ¦ββ1 and π·Ξ¦
0 will endomorphisms π = π β β =
β1
2(Ξ¦2 + ββ1Ξ¦), π = ππ + Ξ¦2 , identity π (νπ , νπ)π = 0 can be rewritten
in the form π (Ξ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π)π = 0; βπ, π β π³(π).
Expanding this relation is linear and separating the real and imaginary parts
of the resulting equation, we obtain an equivalent identity:
π (Ξ¦2π, Ξ¦2π)π β π (Ξ¦π, Ξ¦π)π = 0; βπ, π β π³(π). (28)
Taking into account the equality Ξ¦2 = βππ + π β π last equation
can be written as
π (π, π)π β π(π)π (π, π)π β π(π)π (π, π)π + π(π)π(π)π (π, π)π βπ (Ξ¦π, Ξ¦π)π = π (π, π)π β π(π)π (π, π)π + π(π)π (π, π)π βπ (Ξ¦π, Ξ¦π)π = 0,
which in view of (26) takes the form:
π (Ξ¦π, Ξ¦π)π β π (π, π)π = βπ(π)π + π(π)π; βπ, π β π³(π). (29)
Similarly, applying the restore identity to the equalities π 0ποΏ½ΜοΏ½0 = 0,
π 0ποΏ½ΜοΏ½π = 0, π 0ποΏ½ΜοΏ½
πΜ = 0, i.e. π 0ποΏ½ΜοΏ½π = 0, we obtain the identity:
π (Ξ¦2π, Ξ¦2π)π + π (Ξ¦π, Ξ¦π)π = 0; βπ, π β π³(π). (30)
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200 Malaysian Journal of Mathematical Sciences
From the (28) and (30), we have:
π (Ξ¦2π, Ξ¦2π)π = π (Ξ¦π, Ξ¦π)π = 0; βπ, π β π³(π). (31)
Taking into account the equality Ξ¦2 = βππ + π β π and identity
(26), the identity of π (Ξ¦2π, Ξ¦2π)π = 0 takes the form:
π (π, π)π = βπ(π)π + π(π)π; βπ, π β π³(π). (32)
Using the restore identity to the relations π π0π0 = 0 , π π0π
π = 0 ,
π π0ππΜ = 0, i.e. π π0π
π = 0, we obtain the identity:
π (π, Ξ¦2π)Ξ¦2π β π (ΞΎ, Ξ¦π)Ξ¦π = 0; βπ, π β π³(π). (33)
In view of (26) and the relations π (π, π)π = 0, Ξ¦2 = βππ + π β π
last identity can be rewritten as:
π (π, π)π β π (ΞΎ, Ξ¦π)Ξ¦π = βπ(π)π + π(π)π(π)π; βπ, π β π³(π). (34)
Now apply recovery procedure identity to the equalities π π0οΏ½ΜοΏ½0 =
βπΏπππ0, π π0οΏ½ΜοΏ½
π = βπΏππππ = 0, π π0οΏ½ΜοΏ½
πΜ = βπΏππππΜ = 0, i.e. π π0οΏ½ΜοΏ½
π = βπΏππππ. In fixed
point π β π last equality is equivalent to the relation π (π, νοΏ½ΜοΏ½)νπ = βπΏπππ =
ββ¨νπ , νοΏ½ΜοΏ½β©π . Since ππ = ν0 , vectors {νπ} form a basis of the subspace
(π·Ξ¦ββ1)
π, and the vectors {νοΏ½ΜοΏ½} form a basis of the subspace (π·Ξ¦
βββ1)π
and
projectors of module π³(π)πΆ to submodules π·Ξ¦ββ1, π·Ξ¦
βββ1 and π·Ξ¦0 will
endomorphisms π = π β β = β1
2(Ξ¦2 + ββ1Ξ¦), οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β β =
1
2(βΞ¦2 +
ββ1Ξ¦), π = ππ + Ξ¦2 , the identity of π (π, νοΏ½ΜοΏ½)νπ = ββ¨νπ , νοΏ½ΜοΏ½β©π can be
rewritten in the form of π (π, βΞ¦2π + ββ1Ξ¦π)(Ξ¦2π + ββ1Ξ¦π) =
ββ¨βΞ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦πβ©π; βπ, π β π³(π) . Expanding this
relation is linear and separating the real and imaginary parts of the resulting
equation, we obtain an equivalent identity:
π (π, Ξ¦2π)(Ξ¦2π) + π (π, Ξ¦π)Ξ¦π = β2β¨Ξ¦π, Ξ¦πβ©π = β2β¨π, πβ©π +2π(π)π(π)π; βπ, π β π³(π). (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:
π (π, Ξ¦2π)(Ξ¦2π) = π (π, Ξ¦π)Ξ¦π = ββ¨Ξ¦π, Ξ¦πβ©π = ββ¨π, πβ©π +π(π)π(π)π; βπ, π β π³(π). (36)
Note 1. The identity of π (π, Ξ¦2π)(Ξ¦2π) = ββ¨π, πβ©π + π(π)π(π)π, taking
into account (15) and the equation Ξ¦2 = βππ + π β π, can be written as:
π (π, π)π = π(π)Ξ¦2π β β¨Ξ¦π, Ξ¦πβ©π = ββ¨π, πβ©π β π(π)π +2π(π)π(π)π; βπ, π β π³(π). (37)
Definition 5.2. We call the AC-manifold manifold of class π 2, if its curvature
tensor satisfies the equation:
π (π, Ξ¦2π)(Ξ¦2π) + π (π, Ξ¦π)Ξ¦π = 0; βπ, π β π³(π). (38)
Let M β SGK-manifold of type II, which is a manifold of class π 2.
Then its Riemann curvature tensor satisfies the condition (38). In view of
(33) the identity (38) can be written as: π (π, Ξ¦2π)(Ξ¦2π) = π (π, Ξ¦π)Ξ¦π =0; βπ, π β π³(π) . On the space of the associated G-structure relation
π (π, Ξ¦π)Ξ¦π = 0; βπ, π β π³(π) can be written as: π π0ππ (Ξ¦π)π(Ξ¦π)π = 0.
In view of (11) and the form of the matrix Π€, the last equation can be written
as: π π0οΏ½ΜοΏ½0 ποΏ½ΜοΏ½ππ + π οΏ½ΜοΏ½0π
0 ππποΏ½ΜοΏ½ = 0. This equality holds if and only if π π0οΏ½ΜοΏ½0 =
π οΏ½ΜοΏ½0π0 = 0. But since π π0οΏ½ΜοΏ½
0 = βπΏππ β 0, then the SGK-manifold of type II is
not a manifold class π 2.
Thus we have proved.
Theorem 5.2. SGK-manifold of type II is not a manifold class π 2.
Since π πππ0 = β2πΆ0π[ππ] = 0; π πππ
π = β2πΆοΏ½ΜοΏ½π[ππ] = 0; π ππποΏ½ΜοΏ½ = β2πΆππ[ππ] ,
i.e. π ππππ = β2πΆππ[ππ]. In fixed point π β π, taking into account (19), it is
obviously equivalent to the relation π (νπ , νπ)νπ = βππ(πΆ)(νπ, νπ) β
βππ(πΆ)(νπ , νπ). Since the vectors {νπ} form a basis of the subspace (π·Ξ¦
ββ1)π
,
and projectors of module π³(π)πΆ on the submodule π·Ξ¦ββ1 is the
endomorphism π = π β β = β1
2(Ξ¦2 + ββ1Ξ¦) , then the identity of the
π (νπ , νπ)νπ = βππ(πΆ)(νπ , νπ) β βππ
(πΆ)(νπ , νπ) can be rewritten in the form
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π (Ξ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π)(Ξ¦2π + ββ1Ξ¦π) =
βΞ¦2π+ββ1Ξ¦π(πΆ)(Ξ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π) β
βΞ¦2π+ββ1Ξ¦π(πΆ)(Ξ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π); βπ, π, π β π³(π).
Expanding this relation is linear and separating the real and imaginary parts
of the resulting equation, we obtain an equivalent identity:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π β π (Ξ¦2π, Ξ¦π)Ξ¦π β π (Ξ¦π, Ξ¦2π)Ξ¦π βπ (Ξ¦π, Ξ¦π)Ξ¦2π = βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) β βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) ββΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) β βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) β βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) +βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) + βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) +βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π); βπ, π, π β π³(π). (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 π 3, if its curvature
tensor satisfies the equation:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π β π (Ξ¦2π, Ξ¦π)Ξ¦π β π (Ξ¦π, Ξ¦2π)Ξ¦π βπ (Ξ¦π, Ξ¦π)Ξ¦2π = 0; βπ, π, π β π³(π). (40)
From the definition 5.3 and (39).
Theorem 5.3. SGK-manifold is a manifold of type II class π 3 if and only if
βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) β βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) β βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) ββΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) β βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) + βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) +
βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) + βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) = 0; βπ, π, π β π³(π).
Let M β SGK-manifold of type II, which is a manifold of class π 3.
Then its Riemann curvature tensor satisfies the condition (40), which in the
space of the associated G-structure can be written as:
π ππππ (Ξ¦2π)π(Ξ¦2π)π(Ξ¦2π)π β π πππ
π (Ξ¦2π)π(Ξ¦π)π(Ξ¦π)π β
π ππππ (Ξ¦π)π(Ξ¦2π)π(Ξ¦π)π β π πππ
π (Ξ¦π)π(Ξ¦π)π(Ξ¦2π)π = 0.
In view of (11) and the form of the matrix Π€, the last equation can be
written as: π ππποΏ½ΜοΏ½ ππππππ + π οΏ½ΜοΏ½οΏ½ΜοΏ½πΜ
π ππππππ = 0. This equality holds if and only
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Malaysian Journal of Mathematical Sciences 203
if π ππποΏ½ΜοΏ½ = π οΏ½ΜοΏ½οΏ½ΜοΏ½πΜ
π = 0. According to (2.8), πΆππ[ππ] = 0. From this equality and
(8) we have πΆππππ = 0.
Thus we have proved.
Theorem 5.4. SGK-manifold of type II is a manifold class π 3 if and only if
on the space of the associated G-structure πΆππππ = πΆππππ = 0.
Since,
π πππΜ0 = π΄ππ
0π β πΆ0πβπΆβππ β πΏπ0πΏπ
π = 0, π πππΜπ = π΄ππ
ππ β πΆππβπΆβππ β
πΏπππΏπ
π, π πππΜοΏ½ΜοΏ½ = π΄ππ
οΏ½ΜοΏ½π β πΆοΏ½ΜοΏ½πβπΆβππ β πΏποΏ½ΜοΏ½πΏπ
π = 0,
i.e. π πππΜπ = π΄ππ
ππ β πΆππβπΆβππ β πΏπππΏπ
π .
In fixed point π β π it is obviously equivalent to the relation π (νπ , νπΜ)νπ =π΄(νπ , νπ , νπΜ) + βποΏ½ΜοΏ½
(πΆ)(νπ , νπ) β νπβ¨νπ, νπΜβ© . Since the vectors {νπ} form a
basis of the subspace (π·Ξ¦ββ1)
π, and the vectors {νοΏ½ΜοΏ½} form a basis of the
subspace (π·Ξ¦βββ1)
π and projectors of the module π³(π)πΆ to submodules
π·Ξ¦ββ1, π·Ξ¦
βββ1 are endomorphisms π = π β β = β1
2(Ξ¦2 + ββ1Ξ¦),
οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β β =1
2(βΞ¦2 + ββ1Ξ¦) identity π (νπ , νπΜ)νπ = π΄(νπ , νπ , νπΜ) +
βποΏ½ΜοΏ½(πΆ)(νπ , νπ) β νπβ¨νπ , νπΜβ© can be rewritten in the form
π (Ξ¦2π + ββ1Ξ¦π, βΞ¦2π + ββ1Ξ¦π)(Ξ¦2π + ββ1Ξ¦π) = π΄(Ξ¦2π +
ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π, βΞ¦2π + ββ1Ξ¦π) + ββΞ¦2π+ββ1Ξ¦π(πΆ)(Ξ¦2π +
ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π) β (Ξ¦2π + ββ1Ξ¦π)β¨βΞ¦2π + ββ1Ξ¦π, Ξ¦2π +
ββ1Ξ¦πβ©; βπ, π, π β π³(π).
Expanding this relation by linearity, and separating the real and imaginary
parts of the resulting equation and using (24), we obtain an equivalent
identity:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π + π (Ξ¦2π, Ξ¦π)Ξ¦π β π (Ξ¦π, Ξ¦2π)Ξ¦π +π (Ξ¦π, Ξ¦π)Ξ¦2π = β4π΄(π, π, π) + βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) ββΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) + βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) + βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) β2Ξ¦2πβ¨Ξ¦π, Ξ¦πβ© β 2Ξ¦πβ¨π, Ξ¦πβ©; βπ, π, π β π³(π). (41)
Abu-Saleem A. & Rustanov A. R
204 Malaysian Journal of Mathematical Sciences
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 π 4, if its curvature
tensor satisfies the equation:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π + π (Ξ¦2π, Ξ¦π)Ξ¦π β π (Ξ¦π, Ξ¦2π)Ξ¦π +π (Ξ¦π, Ξ¦π)Ξ¦2π = 0; βπ, π, π β π³(π). (42)
Let M β SGK-manifold of type II, which is a manifold of class π 4.
Then its Riemann curvature tensor satisfies the condition (42), which in the
space of the associated G-structure can be written as:
π ππππ (Ξ¦2π)π(Ξ¦2π)π(Ξ¦2π)π + π πππ
π (Ξ¦2π)π(Ξ¦π)π(Ξ¦π)π β
π ππππ (Ξ¦π)π(Ξ¦2π)π(Ξ¦π)π + π πππ
π (Ξ¦π)π(Ξ¦π)π(Ξ¦2π)π = 0.
In view of (11) and the form of the matrix Π€, the last equation can be
written as: π πππΜπ ππππππ + π οΏ½ΜοΏ½οΏ½ΜοΏ½π
οΏ½ΜοΏ½ ππππππ = 0. This equality holds if and only
if π πππΜπ = π οΏ½ΜοΏ½οΏ½ΜοΏ½π
οΏ½ΜοΏ½ = 0 . According to (11), π΄ππππ β πΆππβπΆβππ β πΏπ
ππΏππ = 0 .
Simmetriruya last equality first in the indices a and b, then the indices c and
d, we obtain π΄(ππ)(ππ)
= πΏπππΏπ
π =1
2πΏππ
ππ. Due to the symmetry of the tensor π΄ππππ
by the upper and lower pair of indices, the resulting identity can be rewritten
as: π΄ππππ =
1
2πΏππ
ππ . Then the equality π΄ππππ β πΆππβπΆβππ β πΏπ
ππΏππ = 0 can be
written as πΆππβπΆβππ = πΏππππ.
Thus proved the following.
Theorem 5.5. Let M β SGK-manifold of type II. Then the following
conditions are equivalent:
1) M is a manifold of class π 4;
2) on the space of the associated G-structure πΆππβπΆβππ = πΏππππ;
3) on the space of the associated G-structure π΄ππππ =
1
2πΏππ
ππ,
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 π 4 has the form:
Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind
Malaysian Journal of Mathematical Sciences 205
π΄(π, π, π) = β1
2{Ξ¦2πβ¨Ξ¦π, Ξ¦πβ© + Ξ¦πβ¨π, Ξ¦πβ©}; βπ, π, π β π³(π).
Consider the relations π ππΜοΏ½ΜοΏ½0 = 0, π ππΜοΏ½ΜοΏ½
π = 0, π ππΜοΏ½ΜοΏ½οΏ½ΜοΏ½ = 2πΆππβπΆβππ β
2πΏπ[ππΏπ
π], i.e. π ππΜοΏ½ΜοΏ½
π = 2πΆοΏ½ΜοΏ½πβπΆβππ β 2πΏοΏ½ΜοΏ½[ππΏπ
π]. The last equality can be written
as π (νπΜ , νοΏ½ΜοΏ½)νπ = β2βποΏ½ΜοΏ½(πΆ)(νπ , νοΏ½ΜοΏ½) + νοΏ½ΜοΏ½β¨νπ , νπΜβ© β νπΜβ¨νπ , νοΏ½ΜοΏ½β© . Since the
vectors {νπ} form a basis of the subspace (π·Ξ¦ββ1)
π, and the vectors {νοΏ½ΜοΏ½} form
a basis of the subspace (π·Ξ¦βββ1)
π and projectors of module π³(π)πΆ to
submodules π·Ξ¦ββ1, π·Ξ¦
βββ1 are endomorphisms π = π β β = β1
2(Ξ¦2 +
ββ1Ξ¦), οΏ½Μ οΏ½ = οΏ½Μ οΏ½ β β =1
2(βΞ¦2 + ββ1Ξ¦) , identity π (νπΜ , νοΏ½ΜοΏ½)νπ =
β2βποΏ½ΜοΏ½(πΆ)(νπ , νοΏ½ΜοΏ½) + νοΏ½ΜοΏ½β¨νπ , νπΜβ© β νπΜβ¨νπ , νοΏ½ΜοΏ½β© can be rewritten in the form
π (βΞ¦2π + ββ1Ξ¦π, βΞ¦2π + ββ1Ξ¦π)(Ξ¦2π + ββ1Ξ¦π) =
β2ββΞ¦2π+ββ1Ξ¦π(πΆ)(βΞ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦π) + (βΞ¦2π +
ββ1Ξ¦π)β¨βΞ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦πβ© β
(βΞ¦2π + ββ1Ξ¦π)β¨βΞ¦2π + ββ1Ξ¦π, Ξ¦2π + ββ1Ξ¦πβ©; βπ, π, π β π³(π).
Expanding this relation by linearity, and separating the real and
imaginary parts of the resulting equation, we obtain an equivalent identity:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π + π (Ξ¦2π, Ξ¦π)Ξ¦π + π (Ξ¦π, Ξ¦2π)Ξ¦π βπ (Ξ¦π, Ξ¦π)Ξ¦2π = β2βΞ¦2π(πΆ)(Ξ¦2π, Ξ¦2π) β 2βΞ¦2π(πΆ)(Ξ¦π, Ξ¦π) β2βΞ¦π(πΆ)(Ξ¦2π, Ξ¦π) + 2βΞ¦π(πΆ)(Ξ¦π, Ξ¦2π) + Ξ¦2πβ¨Ξ¦2π, Ξ¦2πβ© +Ξ¦2πβ¨Ξ¦π, Ξ¦πβ© + Ξ¦πβ¨Ξ¦2π, Ξ¦πβ© β Ξ¦πβ¨Ξ¦π, Ξ¦2πβ© β Ξ¦2πβ¨Ξ¦2π, Ξ¦2πβ© βΞ¦πβ¨Ξ¦2π, Ξ¦πβ© β Ξ¦2πβ¨Ξ¦π, Ξ¦πβ© + Ξ¦πβ¨Ξ¦π, Ξ¦2πβ©; βπ, π, π β π³(π). (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 π 5, if its curvature
tensor satisfies the equation:
π (Ξ¦2π, Ξ¦2π)Ξ¦2π + π (Ξ¦2π, Ξ¦π)Ξ¦π + π (Ξ¦π, Ξ¦2π)Ξ¦π βπ (Ξ¦π, Ξ¦π)Ξ¦2π = 0; βπ, π, π β π³(π). (44)
Abu-Saleem A. & Rustanov A. R
206 Malaysian Journal of Mathematical Sciences
Let M β SGK-manifold of type II, which is a manifold of class π 5.
Then its Riemann curvature tensor satisfies the condition (44), which in the
space of the associated G-structure can be written as:
π ππππ (Ξ¦2π)π(Ξ¦2π)π(Ξ¦2π)π + π πππ
π (Ξ¦2π)π(Ξ¦π)π(Ξ¦π)π +
π ππππ (Ξ¦π)π(Ξ¦2π)π(Ξ¦π)π β π πππ
π (Ξ¦π)π(Ξ¦π)π(Ξ¦2π)π = 0.
In view of (11) and the form of the matrix Π€, the last equation can be
written as: π οΏ½ΜοΏ½πππ ππππππ + π ποΏ½ΜοΏ½πΜ
οΏ½ΜοΏ½ ππππππ = 0. This equality holds if and only
if π οΏ½ΜοΏ½πππ = π ποΏ½ΜοΏ½πΜ
οΏ½ΜοΏ½ = 0. According to (11), πΆππβπΆβππ = πΏ[ππ πΏπ]
π .
Thus, we have proved.
Theorem 5.7. SGK-manifold is a manifold of type II class π 5 if and only if
the space of the associated G-structure πΆππβπΆβππ = πΏ[ππ πΏπ]
π .
Let's turn the equality πΆππβπΆβππ = πΏ[ππ πΏπ]
π first in the indices a and b,
then the indices c and d, we obtain β |πΆπππ|2π,π,π =
1
2π(π β 1). From the
resulting equation implies that for π = 1, we have πΆπππ = 0, i.e. manifold is
Kenmotsu.
Thus we have proved the following.
Theorem 5.8. SGK-manifold of type II class π 5 of 3-dimensional manifold is
Kenmotsu.
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