Integrability properties of generalized Kenmotsu manifolds

A. Abu-Saleem ; A. R. Rustanov ; S. V. Kharitonova

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 4-20

Résumé

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.