In my notes it is staded that if in $(M^m,g)$ the sectional curvature at $sec(p)=k_p$ e.g. independent of the $2$-plane $P<T_pM$ then $ric(p)=(m-1)k_pg$.
I understand that in local coordinates $ric=\sum R_{ij}d\phi_i\otimes d\phi_j$ where $R_{ij}=\sum_{k}R^k_{ikj}$, $R^a_{b,c,d}$ being the components of the $(1,3)$ curvature tensor. So it boils down to $R_{ij}=(m-1)k_pg_{ij}$. I haven't managed to prove this and I cannot really think of any approach. I was able to verify the result for an orthonormal frame though.
Any hints or directions will be appreciated. Let me note that I am aware that there is a proof in Petersen's book but we haven't introduce the object $\mathfrak{R}$ and so I would like a different approach.