# Conformally Flat metric

There is a theorem in D'Inverno Introducing Einstein's relativity which is as follows.

"Any two dimensional Riemannian manifold is conformally flat".

What does this mean? Does is mean that any two dimensional spacetime metric can be written as some function multiplied by the flat metric? Or does it mean that two dimensional spacetime is always flat and never curved?

• A conformally flat space is not necessarily flat, as you can check the conformal transformations in Appendix D of Wald's GR. – Drake Marquis Sep 3 '17 at 8:30
• – Qmechanic Oct 8 '19 at 18:21

It means exactly that any metric $g_{\mu \nu}(x)$ in $2D$ can be written as $$g_{\mu \nu}(x) = f(x) \delta_{\mu \nu}(x)$$ for some function $f > 0$ (assuming that $g_{\mu \nu}$ has signature $++$). It doesn't mean that space is actually flat. For instance, the Riemann sphere can be mapped to the complex plane with the metric $$ds^2 = \left(\frac{2}{1+z\bar{z}}\right)^2 dz d\bar{z}$$ and has scalar curvature 1 at every point $z$.