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I would like to understand the loose statement

"Every 2d manifold is conformally flat"

in the context of Pseudo-Riemannian manifolds.

It seems to me that there are two slightly different versions, which people refer to. The first one, is the one popping up in physics literature (cp. the stackexchange answer of se):

Theorem 1: Every 2D pseudo-Riemannian manifold $(M,g)$ is locally conformally flat, i.e. there locally exist isothermal coordinates.

So, for any metric $g$, we can (locally) choose coordinates, such that $g_{\mu \nu}=e^{2\rho} \eta_{\mu \nu}$ where $\eta = \operatorname{diag}(-1,1)$ for some function $\rho$.

The second one is related to a definition stated for example on Wikipedia:

Definition 1: $(M, g)$ is conformally flat if for each point $x$ in $M$, there exists a neighborhood $U$ of $x$ and a smooth function $f$ defined on $U$ such that $(U,e^{2f}g)$ is flat (i.e. the curvature of $e^{2f}g$ vanishes on $U$).

Now, I am confused, because the latter definition does not seem to involve any particular choice of coordinates at all, while Theorem 1 states that this is only true for a particular coordinate system. Furthermore, the function $\rho$ is not a scalar, since it's only defined for a particular coordinate system, while $f$ is a scalar by definition. So in some sense, using Definition 1 seems to give a stronger statement about "conformal flatness", than Theorem 1 does.

Can somebody clarify the relation between those two statements? Are they actually equivalent? In particular I want to understand if it is always possible on two-dimensional manifolds, given any metric $g$, to write (at least locally) $g_{\mu \nu}=e^{2f}g^0_{\mu \nu}$ for some Ricci-flat metric $g^0$ and some scalar $f$ (so without choosing a specific coordinate system).

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  • $\begingroup$ A scalar, at least to me, is a number, whereas $f$ here is a function which does not have to be constant. $\endgroup$ Dec 13, 2019 at 20:52
  • $\begingroup$ @MichaelAlbanese, thanks for the comment. I should've been more clear of what I meant by "scalar". I meant a function $f$ of coordinates $x$, that change under coordinate transformations as $f'(x')=f(x)$. Now, in face of the answer below, I realize that this concept seems meaningless for a Mathematician, since any smooth function on a manifold does that. $\endgroup$
    – user192049
    Dec 15, 2019 at 13:15

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It seems to me that you are trying to work out some difference which actually is not there. A local coordinate system on a manifold of dimesion $n$ is the same thing as a diffeomorphism $u:U\to V$ where $U\subset M$ and $V\subset\mathbb R^n$. This induces an isomorphism between smooth functions on $U$ and on $V$, so there is nothing like a subclass of "smooth functions defined independently of coordinates". The only issue is that you can ask whehther a smooth function on $U$ is the restriciton of a globally defined smooth function on $M$, but this is not relevant to any part of your question.

Gvien this fact it is clear that the condition in Theorem 1 implies the one in Definition one. For the other direction, you need the fact that flat metric on a Riemannian manifold of dimension $n$ is locally isometric to $\mathbb R^n$ (you only need $n=2$ here). Viewin a local diffeomorphism to $\mathbb R^n$ as a local coordinate system, the condition that it is an isometry exactly means that the metric in these coordinates is just given by $\eta_{\mu\nu}$.

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