Complete gradient Einstein-type Sasakian manifolds with α = 0

Catino et al. have launched an ambitious programme to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants α, β, μ and ρ. In this paper, we show that when α=0, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. Moreover, we obtain an explicit expression for the potential function.