Differential geometry of quasi-Sasakian manifolds

The full system of structure equations of a quasi-Sasakian structure is obtained. The structure of the main tensors on a quasi-Sasakian manifold (the Riemann–Christoffel tensor, the Ricci tensor, and other tensors) is studied on this basis. Interesting characterizations of quasi-Sasakian Einstein manifolds are obtained. Additional symmetry properties of the Riemann–Christoffel tensor are discovered and used for distinguishing a new class of $CR_1$ quasi-Sasakian manifolds. An exhaustive description of the local structure of manifolds in this class is given. A complete classification (up to the $\mathscr B$-transformation of the metric) is obtained for manifolds in this class having additional properties of the isotropy kind.