On rigidity and scalar curvature of Einstein-type manifolds

Here, the scalar curvature of an Einstein-type manifold or equivalently an almost Yamabe quasi-soliton is explicitly determined in terms of its soliton function and some rigidity theorems are obtained. Among the others its shown; if the soliton function is negative, then every compact conformal Yamabe quasi-soliton is isometric to the standard Euclidean sphere.