In the picture below, seemly, if the scalar curvature is positive , then ,the Ricci tensor is positive , because I can use norm coordinate make the $g_{ij}\ge 0$.
As I know ,$R=g^{ij}R_{ij}$ , I think the positive of $R$ can't make $R_{ij}$ is positive . Why the paper is not so ?
If $g$ is an Einstein metric on a smooth $n$-manifold, and if $R_{ij} = \lambda g_{ij}$ for some real $\lambda$, then $$ R = g^{ij} R_{ij} = n\lambda, $$ which has the same sign as $\lambda$.