Tachibana's Theorem Revisited

We prove Tachibana-type rigidity theorems for closed manifolds under weakened partial-sum conditions on the eigenvalues of curvature operators. Specifically, an Einstein manifold of dimension $n\ge 4$ is locally symmetric if its curvature operator is $\frac{2(n-1)}{3}$-nonnegative. The corresponding thresholds are $\frac{2(n+1)}{3}$ for the primitive curvature operator of a Kähler-Einstein manifold of complex dimension $n\ge 2$, and $\frac{2(n+2)}{3}$ for the curvature operator of a quaternionic-Kähler manifold of real dimension $4n\ge 8$.