Universal Covering Space
The half pseudosphere of curvature −1 is covered by the portion of the hyperbolic upper half-plane with y ≥ 1. The covering map is periodic in the x direction of period 2π, and takes the horocycles y = c to the meridians of the pseudosphere and the vertical geodesics x = c to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is
where is the arclength parametrization of the tractrix.
Read more about this topic: Pseudosphere
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