Integrability properties of generalized Kenmotsu manifolds

The article is devoted to generalized Kenmotsu manifolds, namely, the study of their integrability properties. The study is carried out by the method of adjoint G-structures; therefore, at first, the space of the adjoint G-structure of almost contact metric manifolds was constructed. Next, generalized Kenmotsu manifolds are defined (shorter than a GK-manifold), and a complete group of structural equations of such varieties is given. The first, second, and third fundamental identities of GK-structures are defined. The definitions of special generalized Kenmotsu manifolds (SGK-manifolds) of the first and second genera are formulated. In this paper, we study GK-manifolds whose first fundamental distribution is completely integrable. It is shown that an almost Hermitian structure induced on integral manifolds of maximal dimension of the first distribution of a GK-manifold is approximately Kähler. The local structure of a GK-manifold with a closed contact form is obtained, the expressions of the first and second structural tensors are given ...