We provide a direct proof of the result “A closed Yamabe soliton of dimension \(\ge 3\) has constant scalar curvature” through the derivation of a divergence formula. Next, we show that, if the potential vector field of a Yamabe soliton of dimension \(\ge 3\) leaves the Ricci tensor invariant, then the scalar curvature is constant. Finally, we provide a classification of a complete non-trivial gradient Kähler Yamabe soliton, showing that it is isometric to either (i) a product of a complex Euclidean space and a Kähler manifold of non-zero constant scalar curvature, or (ii) a complex Euclidean space.
