Find how to construct this metric, find the distance under the metric between $(e^{-2\pi},0)$ and $(-e^{-\pi},0)$ This is a very interesting question, I have an idea ,construct Riemannian covering space from Upper Half plane. Is it right? There are several method to construct ,is the distance invariant regardless any construction? Help to solve this! This question is very important to me! Thanks!
The punctured unit disc has the complete riemannian metric with constant curvature -1
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1Henry it usually helps if you separate the actual question from other comments you want to make in asking it, and use proper punctuation and less exclamation :) – 2011-07-03