Axiom of ?-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 ?-holomorphic (2r+1)-planes. After the preliminaries we give the definition of generalized Kenmotsu manifolds and the full structural equation group. Next, we define ?holomorphic generalized Kenmotsu manifolds and ?-paracontact generalized Kenmotsu manifold give a local characteristic of this subclasses. The ?-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 ?paracontact generalized Kenmotsu manifold is a special generalized Kenmotsu manifold of the second kind. An analytical expression is obtained for the tensor of ?-holomorphic sectional curvature of generalized Kenmotsu manifolds of the pointwise constant ?-holomorphic sectional curvature. Then we study the axiom of ?-holomorphic (2r+1)-planes for generalized Kenmotsu manifolds and propose a complete classification of simply connected generalized Kenmotsu manifolds satisfying the axiom of ?-holomorphic (2r+1)-planes. The main results are as follows. A simply connected GK-manifold of pointwise constant ?-holomorphic sectional curvature satisfying the axiom of ?-holomorphic (2r+1)-planes is a Kenmotsu manifold. A GK-manifold satisfies the axiom of ?-holomorphic (2r+1)-planes if and only if it is canonically concircular to one of the following manifolds: (1) CPn ?R; (2) Cn ?R; and (3) CHn ?R having the canonical cosymplectic structure. A