First, we know that if the Ricci tensor $S$ is of the form $S = ag$, $R_{ij} = ag_{ij}$ where $a$ is a constant, then $M$ is called an Einstein manifold. At the same time we know that if $M$ is a Riemannian manifold and $S = ag$ where $a$ is a function on $M$, then $a$ is necessarily a constant provided that $\dim M > 2$.
I wonder can we say that an Einstein manifold has constant curvature if $\dim M > 2$, due to its definition and the result stated above.
A Riemannian manifold is said to have constant curvature if it has constant sectional curvature. An Einstein manifold has constant Ricci curvature (i.e. $\operatorname{Ric}(v)$ has the same value for any $v$ of unit length, namely the Einstein constant $a$). In dimension three, Einstein implies constant curvature, but this is not true in higher dimensions. For example, $S^2\times S^2$ endowed with its usual metric is Einstein but it does not have constant sectional constant curvature.