Theorem on any 2-dimensional riemannian manifold conformally flat

Question

I found a theorem stating that "Any two-dimensional Riemannian manifold is conformally flat". Here the flat metric seems like the metric $g_{ab}=diag(-1,1)$. However, the normal Euclidean space $\mathbb{R}^2$ with the metric $g_{ab}=diag(1,1)$ cannot be conformally flat. So what is the exact meaning of this theorem? Could anyone please explain?

Answer 1

In the context of Riemannian (not pseudo-Riemannian or Lorentzian) geometry, flat means locally isometric to the Euclidean metric $\mathrm{diag}(1,1).$ The indefinite metric $\mathrm{diag}(-1,1)$ is not Riemannian, so of course no Riemannian metric will be locally isometric to it.