Sasakian lift of Kaehler manifold and $\alpha$-Sasakian Ricci solitons

In this paper we provide a local construction of a Sasakian manifold given a Kähler manifold. Obatined in this way manifold we call Sasakian lift of Kähler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its Kähler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in Kähler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the lift and its base. We study lifts of symmetries different kind: of complex structure, of Khler metric, and Kähler structure automorphisms. In connection with $\eta$-Ricci solitons we introduce more general class of manifolds called twisted $\eta$-Ricci solitons. As we show class of $\alpha$-Sasakian twisted $\eta$-Ricci solitons is invariant under naturally defined group of structure deformations. As corollary it is proved that orbit of Sasakian lift of steady or shrinking Ricci-Kähler soliton contains $\alpha$-Sasakian Ricci soliton. In case of expanding Ricci-Kähler soliton existence of $\alpha$-Sasakina Ricci solition is assured provided expansion coefficient is small enough.