Sasakian 3-metric as a generalized Ricci-Yamabe soliton

Abstract

In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if (g, V, λ, α, β, γ) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold M with potential function f , then M is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field V is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field ξ. Further, if h is the Hodge-de Rham potential for V, then, upto a constant, f = h.