On the geometry of nearly trans-Sasakian manifolds

Authors: ALIGADZHI R. RUSTANOV, TATIANA L. MELEKHINA, SVETLANA V. KHARITONOVA

Abstract: The geometry of nearly trans-Sasakian manifolds is researched in this paper. The complete group of structural equations and the components of the Lee vector on the space of the associated $G$-structure are obtained for such manifolds. Conditions are found under which a nearly trans-Sasakian structure is a trans-Sasakian, a cosymplectic, a closely cosymplectic, a Sasakian structure or a Kenmotsu structure. The conditions are obtained when the nearly trans-Sasakian structure is a special generalized Kenmotsu structure of the second kind. A complete classification of nearly trans-Sasakian manifolds is obtained, i.e. it is proved that a nearly trans-Sasakian manifold is either a trans-Sasakian manifold or has a closed contact form. It is proved that the nearly trans-Sasakian structure with a nonclosed contact form is homothetic to the Sasakian structure. The criterion of ownership of a nearly trans-Sasakian structure is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form, which are locally conformal to the closely cosymplectic manifolds. Examples of such manifolds are given. The necessary and sufficient conditions for the complete integrability of the first fundamental distribution of a nearly trans-Sasakian manifold are obtained. It is proved that a nearly K\"ahler structure on the leaves of the first fundamental distribution of a nearly trans-Sasakian manifold is induced.

Keywords: Trans-Sasakian structure, nearly trans-Sasakian manifold, closely cosymplectic structure, linear extension of almost Hermitian structure, Lee form

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