Gap Theorems for Compact Quasi Sasaki–Ricci Solitons

Gradient quasi Sasaki–Ricci solitons are generalization of gradient Sasaki–Ricci solitons and Sasaki–Einstein manifolds. The main focus of this paper is to establish two gap results for the transverse Ricci curvature $${\rm Ric}^{T}$$ and the transverse scalar curvature $${\mathscr {S}}^{T}$$ , based on which we can derive necessary and sufficient conditions for gradient quasi Sasaki–Ricci solitons to be Sasaki–Einstein. Our results generalize some recent works on this direction.