= Constant scalar curvature does not imply an Einstein metric
{title2=$M=S^2\times S^1,\quad\operatorname{Scal}=2$}
The <product Riemannian metric> of the round unit two-sphere and a circle has <Ricci curvature> equal to $g_{S^2}$ on the sphere directions and zero on the circle direction. Its <scalar curvature> is constantly two, but no single scalar multiple of the whole metric equals that Ricci tensor. In two dimensions the identity $\operatorname{Ric}=\operatorname{Scal}\,g/2$ does make constant scalar curvature equivalent to an <Einstein metric>.
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