= Poincare metric on a Riemann surface
{c}
{title2=$ds_{\mathbb D}=2|dz|/(1-|z|^2)$}
A connected <Riemann surface> whose <universal cover> is the <unit disc> inherits the complete curvature-minus-one metric displayed on its cover. <Möbius transformations> preserving the disc are isometries, so <deck transformations> preserve the metric and it descends to the surface. This uniformization notion must be distinguished from the potential-theoretic existence of a Green function. <Holomorphic maps> between such surfaces contract their metric by lifting to <Schwarz-Pick theorem> on the covers.
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