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3 papers
MATHEMATICS
Axiom of $\Phi$-holomorphic $(2r+1)$-planes for generalized kenmotsu manifolds
Ahmad Abu-Saleema,
A. R. Rustanovb,
S. V. Kharitonovac a Al al-Bayt University, Mafraq, Jordan
b National Research Moscow
State University of civil engineering, Institute of fundamental education, Russia
c Orenburg State University,
Russia
Abstract:
In this paper we study generalized Kenmotsu manifolds (shortly, a GK-manifold) that satisfy the axiom of
$\Phi$-holomorphic
$(2r+1)$-planes. After the preliminaries we give the definition of generalized Kenmotsu manifolds and the full structural equation group. Next, we define
$\Phi$-holomorphic generalized Kenmotsu manifolds and
$\Phi$-paracontact generalized Kenmotsu manifold give a local characteristic of this subclasses. The
$\Phi$-holomorphic generalized Kenmotsu manifold coincides with the class of almost contact metric manifolds obtained from closely cosymplectic manifolds by a canonical concircular transformation of nearly cosymplectic structure. A
$\Phi$-paracontact generalized Kenmotsu manifold is a special generalized Kenmotsu manifold of the second kind. An analytical expression is obtained for the tensor of
$\Phi$-holomorphic sectional curvature of generalized Kenmotsu manifolds of the pointwise constant
$\Phi$-holomorphic sectional curvature.
Then we study the axiom of
$\Phi$-holomorphic
$(2r+1)$-planes for generalized Kenmotsu manifolds and propose a complete classification of simply connected generalized Kenmotsu manifolds satisfying the axiom of
$\Phi$-holomorphic
$(2r+1)$-planes. The main results are as follows. A simply connected GK-manifold of pointwise constant
$\Phi$-holomorphic sectional curvature satisfying the axiom of
$\Phi$-holomorphic
$(2r+1)$-planes is a Kenmotsu manifold. A GK-manifold satisfies the axiom of
$\Phi$-holomorphic
$(2r+1)$-planes if and only if it is canonically concircular to one of the following manifolds: (1)
$\mathbf{CP^n}\times\mathbf{R}$; (2)
$\mathbf{C^n}\times\mathbf{R}$; and (3)
$\mathbf{CH^n}\times\mathbf{R}$ having the canonical cosymplectic structure.
Keywords:
almost contact metric structure, Kentmotsu structure, generalized Kentmotsu manifold, special generalized Kentmotsu manifold, axiom of $\Phi$-holomorphic planes, $\Phi$-quasiinvariant manifold, $\Phi$-paracontact manifold.
UDC:
514.76
MSC: 53C25,
53D15 Received: 04.10.2019
DOI:
10.17223/19988621/66/1