Geometry of integral manifolds of contact distribution

In this paper, various classes of almost contact metric structures are considered under the assumption that their contact distribution is completely integrable. An analytical criterion for the completely integrability of the contact distribution of an almost contact metric manifold is obtained. It is found which almost Hermitian structures are induced on the integral manifolds of the contact distribution of some almost contact metric manifolds. In particular, it is proved that an almost Hermitian structure induced on integral submanifolds of maximum dimension of the first fundamental distribution of a Kenmotsu manifold is a Kähler structure. An almost Hermitian structure induced on integral manifolds of maximum dimension of a completely integrable first fundamental distribution of a normal manifold is a Hermitian structure. We show that a nearly cosymplectic structure with an involutive first fundamental distribution is the most closely cosymplectic one and approximately Kähler structure is induced on its integral submanifolds of the maximum dimension of a completely integrable contact distribution. It is also proved that the contact distribution of an inquasi-Sasakian manifold is integrable only in case of this manifold is cosymplectic. Kähler structure is induced on the maximal integral manifolds of the contact distribution of a cosymplectic manifold. If $M$ is a $lcQS$-manifold with an involutive first fundamental distribution, then the structure of the class $W_4$ of almost Hermitian structures in the Gray-Hervella classification is induced on integral manifolds of the maximum dimension of its contact distribution. It is Kähler if and only if $grad \ \sigma \subset M$, where $\sigma$ is an arbitrary smooth function on $M$ of corresponding conformal transformation.