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On the other hand the Nash embedding theorem implies that hyperbolic n-space can be isometrically embedded into some Euclidean space of larger dimension (5 for the hyperbolic plane by the Nash embedding theorem). When isometrically embedded to a Euclidean space every point of a hyperbolic space is a saddle point.
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The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
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The Poincaré metric provides a hyperbolic metric on the space. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
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It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane.