Classification of Three-Dimensional Contact Metric Manifolds with Almost-Generalized Ƶ-Solitons

This work investigates the classification of three-dimensional complete contact metric manifolds that are non-Sasakian and satisfy the relation Qξ=σξ, focusing on those that support an almost-generalized Ƶ-soliton. In the scenario where σ is constant, we prove that if a generalized Ƶ-soliton (Mn,g,δ,η,V,μ,Λ) satisfies the condition g(V,ξ)=0, then Mn must be either an Einstein manifold or locally isometric to the Lie group E(1,1). Comparable classifications are obtained for (κ,μ,ϑ)-contact metric manifolds. Furthermore, we explore situations in which the potential vector field aligns with the Reeb vector field. We then provide the corresponding structural characterizations.