Abstract

Let W be a simply connected two-dimensional Riemannian manifold with a Cysmooth metric ds. We can always introduce local isothermal coordinates near any point of W, that is, such coordinates ðx; yÞ that the metric is represented in the form dsðx; yÞ 1⁄4 oðx; yÞðdx þ dyÞ, for some positive weight function o. And since W is orientable as a two-dimensional simply-connected manifold, these local isothermal coordinates can serve as conformal charts for a complex structure on W (see [1], pp. 124–126). The Kœbe uniformization theorem then says that W is conformally equivalent to one of the three sets: the Riemann sphere S 1⁄4 CW fyg, the complex plane C, or the open unit disk D. This equivalence, together with the choice of isothermal coordinates, allows us to identify W with one of the above three sets W supplied with the isothermal Riemannian metric

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