A characterization of Einstein manifolds. (Q1873567)

Let \((M, g)\) be a compact Riemannian manifold of dimension \(n \geq 3\), with sectional curvature bounded below by a positive constant \(k_0 > 0\). The author proves that if \(M\) satisfies the following inequality \[ n | | Q | |^2 - S^2 \geq {{n - 4} \over{4 n k_0}} | | \text{grad}\, S | |^2 + {{1} \over {2 k_0}} | | F | |^2 \] (where \(Q\) is the Ricci operator, \(S\) the scalar curvature and \(F\) is the divergence of the curvature tensor), then \(M\) is Einstein.