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 Quote by lavinia The extended complex plane does not admits a flat metric - if by flat you mean zero Gauss curvature.
Did you see the quote from wikipedia. When it says: "The round metric is therefore not intrinsic to the Riemann sphere." I understand that an abstract conformal Riemann sphere (wich is the extended complex plane) admits not only the positive curvature metric but other constant curvature metrics (up to conformal equivalence) like the euclidean metric.