RICCI–YAMABE SOLITONS ON THE LIE GROUP Sol3

We study Ricci–Yamabe solitons on the three-dimensional solvable Lie group Sol₃, one of Thurston's eight model geometries. After computing the Levi-Civita connection and the Ricci tensor of the canonical left-invariant metric, we derive necessary and sufficient conditions for the existence of such solitons and give an explicit classification of the associated vector fields. We further prove that Sol₃ admits no non-trivial gradient Ricci–Yamabe soliton, and we characterize the conditions under which the dual $1$-form of the soliton vector field defines a contact structure. As an application, we determine when the quintuple (Sol₃, g, X, μ₁, μ₂) constitutes a hyperbolic Ricci soliton.