Integrability Properties of Generalized Kenmotsu Manifolds

Abu-Saleem A. , Rustanov A. R. , Kharitonova S. V.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3.

Abstract:
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.

Keywords: generalized Kenmotsu manifold, Kenmotsu manifold, normal manifold, Nijenhuis tensor, integrable structure, nearly Kahler manifold.

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