Integrability properties of generalized Kenmotsu manifolds

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.