A rigidity theorem for Ricci flat metrics (Q1392728)

For complete, noncompact Riemannian \(n\)-dimensional manifolds \(M\) with \(n \geq 5\) under the assumptions that there exists an isometric \(S^{1}\) action on \(M\) and \(M\) is asymptotically flat, a rigidity theorem for Ricci-flat metrics is proven. It is also shown that these manifolds are isometric to the Riemannian Schwarzschild metric. In the course of the demonstration, a structure lemma for \(n \geq 5\) dimensions is used and proven. Connections to some particular cases are shown.