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DOI: 10.23671/VNC.2018.3.17829

# Integrability Properties of Generalized Kenmotsu Manifolds

Abu-Saleem A. , Rustanov A. R. , Kharitonova S. V.
Vladikakazian 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.
Language: Russian Download the full text
For citation: Abu-Saleem A., Rustanov A. R.,  Kharitonova S. V. Integrability Properties of Generalized Kenmotsu Manifolds. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no.  2, pp.4-20 . DOI 10.23671/VNC.2018.3.17829

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