Riemannian manifolds with harmonic curvature (Q2833644)

Let \(\mathcal{M}=(M^n,g)\) be a compact Riemannian manifold of dimension \(n\geq3\). Let \(R_{ijk}{}^l\) be the curvature tensor, \(\tau\) the scalar curvature, \(E\) the trace-free Ricci tensor, and \(W\) the Weyl curvature. Assume that the curvature tensor is \textit{harmonic}, i.e., that the divergence for the curvature tensor vanishes, \(R_{ijk}{}^l{}_{;l}=0\). The authors establish the inequality. NEWLINE\[NEWLINE\int_M\left(\tau-\sqrt{n(n-1)}\|E\|\right)\|E||^{(n-2)/n} \leq\sqrt{\frac{(n-1)(n-2)}2}\int_M\| W\|\cdot\|E||^{(n-2)/n}.NEWLINE\]NEWLINE They show that equality holds if and only if {\parindent=0.7cm\begin{itemize}\item[--] \((M,g)\) is Einstein. \item[--] \((M,g)\) is isometrically covered by \(S^1\times S^{n-1}\) with the product metric. \item[--] \((M,g)\) is isometrically covered by \(S^1\times S^{n-1}\) with the Derdzinski metric. NEWLINENEWLINE\end{itemize}}