In Relativity and Singularities, Natário states that
Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric).
Why is that so? It doesn't seem obvious to me, and I found no precise proof (from the mathematical point of view) around the web.
