Admissible hypercomplex structures on distributions of Sasakian manifolds

The notions of admissible (almost) hypercomplex structure and almost contact hyper-Kählerian structure are introduced. On a manifold $M$ with an almost contact metric structure $(M,\vec\xi,\eta,\varphi,D)$ an interior symmetric connection $\nabla$ is defined. In the case of a contact manifold of dimension bigger than or equal to five, it is proved that the curvature tensor of the connection $\nabla$ is zero if and only if there exist adapted coordinate charts with respect to that the coefficients of the interior connection are zero. On the distribution $D$ of an almost contact structure as on the total space of the vector bundle $(D,\pi,M)$, an admissible almost hypercomplex structure $(\tilde D,J,J_1,J_2,\vec u,\lambda=\eta\circ\pi_*,D)$ is defined. Under the condition that the admissible almost complex structure $\varphi$ is integrable, it is proved that the constructed almost hypercomplex structure is integrable if and only if the distribution $D$ is a distribution of zero curvature. In the case of a Sasakian structure $(M,\vec\xi,\eta,\varphi,g,D)$, the conditions that imply that the admissible hypercomplex structure $(\tilde D,J,J_1,J_2,\vec u,\lambda=\eta\circ\pi_*,\tilde g,D)$ is an almost contact hyper-Kählerian structure.