Rigidity of generalized Bach-flat vacuum static spaces

In this paper, we study the structure of generalized Bach-flat vacuum static spaces. Generalized Bach-flat metrics are considered as extensions of both Einstein and Bach-flat metrics. First, we prove that a compact Riemannian n-manifold with n≥4 which is a generalized Bach-flat vacuum static space is Einstein. A generalized Bach-flat vacuum static space with the potential function f having compact level sets is either Ricci-flat or a warped product with zero scalar curvature when n≥5, and when n=4, it is Einstein if f has its minimum. Secondly, we consider critical metrics for another quadratic curvature functional involving the Ricci tensor, and prove similar results. Lastly, by applying the technique developed above, we prove Besse conjecture when the manifold is generalized Bach-flat.