Closed Vacuum Static Spaces with Zero Radial Weyl Curvature.

In this paper, we study vacuum static spaces. We firstly derive a Bochner-type formula for the Weyl tensor to vacuum static space. Based on a global argument, under the condition of zero radial Weyl curvature, we then obtain a pointwise identity and use it to prove that each closed vacuum static space of dimension n ≥ 5 with scalar curvature n (n - 1) and zero radial Weyl curvature is Bach flat. [ABSTRACT FROM AUTHOR]