Extended structures on codistributions of contact metric manifolds

In the paper, the notion of an $AP$-manifold is introduced. Such a manifold is an almost contact metric manifold that is locally equivalent to the direct product of a contact metric manifold and an Hermitian manifold. A normal $AP$-manifold with a closed fundamental form is a quasi-Sasakian manifold. A quasi-Sasakian AP-manifold is called in the paper a special quasi-Sasakian manifold ($\mathrm{SQS}$-manifold). A $\mathrm{SQS}$-manifold is locally equivalent to the product of a Sasakian manifold and a Kählerian manifold. As a subsidiary result, a proposition is proved stating that a contact metric space with a zero curvature distribution is a K–contact metric space. The codistribution $D^*$ of a contact metric structure $(M, \vec{\xi}, \eta, \varphi, g, D)$ is defined as the subbundle of the cotangent bundle $T^*M$, consisting of all 1-forms annihilating the structure vector $\vec{\xi}$. On the codistribution $D^*$, the extended almost contact metric structure $(D^*,\vec{u}=\partial_n,\mu=\eta\circ \pi_{*},J,G,\tilde{D})$ is defined. Structural equations are introduced. These equations were used to prove the statement that the extended almost contact metric structure defines a structure of an $AP$-manifold if and only if the Schouten tensor of the contact metric manifold $M$ is equal to zero. Finally we prove the theorem stating that the extended almost contact metric structure is a SQS-structure if and only if the initial manifold is a Sasakian manifold with a zero curvature distribution.