$\eta$-Einstein Sasakian immersions in non-compact Sasakian space forms

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into ${\mathbb{B}}^N\times \mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $\eta$-Einstein case.