$eta$-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}^Ntimes mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $eta$-Einstein case.