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I know that for a 2D manifold there can allways be found a local chart such that the metric is conformally flat. Is it possible globally? If the manifold is simply connected, like a disk, does it make it possible then? I'll be happy to see how it can be prove or a counter example. Thanks very much!

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    $\begingroup$ This is the classical uniformization theorem of Koebe: Any simply-connected, non-compact, connected Riemann surface is conformally equivalent to either the whole complex plane or to the unit disk. $\endgroup$
    – Robert Bryant
    Jun 18, 2013 at 12:30
  • $\begingroup$ Great! It is exactly what I need. Thank you! $\endgroup$
    – Michael
    Jun 18, 2013 at 15:08

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