When is a Riemannian manifold a Riemann surface?

A Riemann surface is a manifold of one complex dimension. If we have an orientable, connected Riemannian manifold of two real dimensions, what further conditions (if any) are needed for it be equipped with a complex structure and therefore seen as a Riemann surface?

I've read something on Wikipedia about isothermal coordinates, which seem to be a way of locally constructing a complex structure from the metric. But when and how can these local structures be stitched together to form a global complex structure?

• Always. Hint: Conformal maps of open subsets in $R^2$ are holomorphic. – Moishe Kohan Nov 8 '17 at 3:29
• when it dimension is two it is called surface – Guy Fsone Nov 8 '17 at 3:29
• A conformal map can never be Holomorphic. but rather it can be anti-holomorphic – Guy Fsone Nov 8 '17 at 3:30
• @GuyFsone: It depends on your definition of a conformal map: Mine requires conformal maps to preserve oriented angles between tangent vectors. – Moishe Kohan Nov 9 '17 at 3:11

Thus we can promote the $\mathbb{R}^2$ coordinate domains to $\mathbb{C}$ with its usual complex structure, and now all the transition maps are holomorphic, so the manifold is a complex manifold. We can also pull back the complex structure from $\mathbb{C}$ to explicitly construct the complex structure on the manifold. This will be consistent across coordinate charts, since a complex structure is conformally invariant.
• Also, Chern's assumptions are not optimal, the theorem works even for measurable Riemannian metrics (under a certain $L_\infty$ condition). He even gives a reference to the more general result (Morrey). – Moishe Kohan Nov 9 '17 at 3:14