= Sectional curvatures from Ricci curvature in dimension three
{title2=$K_{ij}=\tfrac12(\operatorname{Ric}_{ii}+\operatorname{Ric}_{jj}-\operatorname{Ric}_{kk})$}
For an orthonormal triple and distinct $i,j,k$, tracing <sectional curvature> gives $\operatorname{Ric}_{ii}=K_{ij}+K_{ik}$. Solving the three equations gives the displayed formula. Equivalently $K_{ij}=\operatorname{Ric}_{ii}+\operatorname{Ric}_{jj}-\operatorname{Scal}/2$. This pointwise assertion does not require the triple to diagonalize the <Ricci curvature>.
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