Hyperbolic Space
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چکیده
Radial lines, suitably parameterized, are geodesics, but notice that the distance from the origin to the (Euclidean) unit sphere is infinite. This model makes it intuitively clear that the boundary at infinity of hyperbolic space is Sn−1. Hyperbolic space together with its boundary at infinity has the topology of a closed ball, and isometries of hyperbolic space extend uniquely to a homeomorphism of this ball. The Brouwer Fixed Point Theorem tells us that any such homeomorphism has a fixed point. Thus, any hyperbolic isometry either fixes a point in hyperbolic space or fixes a point at infinity.
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