Axiom of $\Phi$-holomorphic $(2r+1)$-planes for generalized kenmotsu manifolds

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.