Closed vacuum static spaces with zero radial Weyl curvature (Q2110310)

Let \((M^n,g)\) be an \(n\)-dimensional Riemannian manifold. We say that \((M^n,g,f)\) is a vacuum static space if \(f\) is a smooth function in \(\ker \mathcal{L}^*_g\) where \(\mathcal{L}^*_g\) is the \(L_2\)-formal adjoint of \(\mathcal{L}_g\), the linearization of the scalar curvature \(R_g\) at \(g\) which is given in terms of the trace of a symmetric \((0,2)\)-tensor on \(M^n\), the divergence operator, the Beltrami operator, and the Ricci tensor of \(g\). The purpose of the author is to obtain a Bochner-type formula for the Weyl tensor to vacuum static space. Also, he gets a pointwise identity and employs it to prove that each closed vacuum static space of dimension \(n\ge 5\) with scalar curvature \(n(n-1)\) and zero radial Weyl curvature is Bach flat.