Let $(M, g)$ be a Riemannian manifold. Assume that $$ \text{sec}_p(\pi) = f(p) \qquad\text{ for all } 2-\text{planes } \pi \subset T_pM \text{ and for all } p \in M. $$ Where sec$_p(\pi)$ is the sectional curvature at $p$ and $f$ is some smooth function on $M$.
I need to prove that there is a smooth function $g$ on $M$ such that $$\text{Ric}(v) = g(p) \cdot v \qquad \text{for all }v \in T_pM \text{ and for all } p \in M.$$
How can I show that?