On distributions with special quasi-sasakian structure

In the paper, the notion of an almost contact metric structure $(M, \vec{\xi}, \eta, \varphi, g, D)$ of the first order is introduced. On an almost contact metric manifold $M$ satisfying the condition $rk d\eta=2p,$ $2p<2m,$ $2p \neq 0,$ is defined the integrable distribution $K$ that is equal to the kernel of the form $\omega=d \eta.$ In this case, on the manifold $M$ appears the decomposition $TM=L\oplus L^{\bot}\oplus D^{\bot},$ where $L^{\bot}=K\cap D,$ and $L$ is its orthogonal distribution in $D.$ An almost contac metric structure is called in the paper the structure of the first order, if the distribution $L$ is invariant under the action of the endomorphism $\varphi.$ If an almost contact metric structure of the first order is a quasi-Sasakian structure and it holds $d \eta(\vec{x},\vec{y})=\Omega(\vec{x},\vec{y}),$ $\vec{x},\vec{y}\in \Gamma(L),$ where $\Omega(\vec{x},\vec{y})$ is the fundamental form of the structure, then such a structure is called a special quasi-Sasakian structure (SQS-structure), and the manifold $M$ is called a SQS-manifold. On the manifold $M,$ an interior connection $\nabla$ and the corresponding associated connection $\nabla^A$ are defined. The curvature tensor of the interior connection is called the Schouten tensor. The properties of the Schouten tensor are studied. In particular, it is shown that the Schouten tensor is zero if and only if there exists an atlas consisting of adapted charts with respect to that the Christoffel components of the interior connection are zero. The distribution of an almost contact metric structure with zero Schouten tensor is called in the paper the distribution of zero curvature. On the distribution $D$ of a manifold $M$ with a contact metric structure $(M, \vec{\xi}, \eta, \varphi, g, D),$ an almost contact metric structure $(D,J,\vec{u},\lambda=\eta\circ \pi_{*},\tilde{g},\tilde{D}),$ is defined, which is a structure of the first order, and it is called an extended almost contact metric structure. It is shown that an extended structure is a SQS-structure, if the initial manifold $M$ is a Sasakian manifold with a distribution of zero curvature.