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5 papers
Integrability properties of generalized Kenmotsu manifolds
A. Abu-Saleema,
A. R. Rustanovb,
S. V. Kharitonovac a Al al-Bayt University, P.O.Box 130040, Mafraq
25113,
Jordan
b National Research University (MGSU), 26 Yaroslavskoye Shosse, Moscow 129337,
Russia
c Orenburg State University, 13 Pobedy av., Orenburg 460000,
Russia
Abstract:
The
article is devoted to generalized Kenmotsu manofolds, namely the
study of their integrability properties. The study is carried out by
the method of associated
$G$-structures; therefore, the space of
the associated
$G$-structure of almost contact metric manifolds is
constructed first. Next, we define the generalized Kenmotsu
manifolds (in short, the
$GK$-manifolds) and give the complete group
of structural equations of such manifolds. The first, second, and
third fundamental identities of
$GK$-structures are defined.
Definitions of special generalized Kenmotsu manifolds (
$SGK$-manifolds)
of the I and II kinds are given. We consider
$GK$-manifolds the first
fundamental distribution of which is completely integrable. It is shown
that the almost Hermitian structure induced on integral manifolds of maximal
dimension of the first distribution of a
$GK$-manifold is nearly
Kahler. The local structure of a
$GK$-manifold with a closed contact
form is obtained, and the expressions of the first and second
structural tensors are given. We also compute the components of the
Nijenhuis tensor of a
$GK$-manifold. Since the setting of the
Nijenhuis tensor is equivalent to the specification of four tensors
$N^{(1)}$,
$N^{(2)}$,
$N^{(3)}$,
$N^{(4)}$, the geometric meaning of the vanishing of
these tensors is investigated. The local structure of the integrable
and normal GK-structure is obtained. It is proved that the
characteristic vector of a GK-structure is not a Killing vector. The
main result is
Theorem:
Let $M$ be a $GK$-
manifold.
Then the following statements are equivalent:
$1)$ $GK$-
manifold has
a closed contact form;
$2)$ $F^{ab}=F_{ab}=0;$ $3)$
$N^{(2)}(X,Y)=0;$ $4)$ $N^{(3)} (X)=0;$ $5)$ $M$ —
is a
second-kind $SGK$ manifold;
$6)$ $M$ is locally canonically
concircular with the product of a nearly Kahler manifold and a real
line.
Key words:
generalized Kenmotsu manifold, Kenmotsu manifold, normal manifold,
Nijenhuis tensor, integrable structure, nearly Kahler manifold.
UDC:
514.76
MSC: 58A05 Received: 11.07.2017
DOI:
10.23671/VNC.2018.3.17829