Hyperbolic space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Hyperbolic_space


00:00:59 1 Formal definition
00:01:54 2 Models of hyperbolic space
00:03:42 2.1 The hyperboloid model
00:09:10 2.2 The Klein model
00:11:38 2.3 The Poincaré ball model
00:11:56 2.4 The Poincaré half-space model
00:12:28 3 Hyperbolic manifolds
00:13:47 3.1 Riemann surfaces
00:14:30 4 See also
00:15:24 5 References
00:16:38 See also



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SUMMARY
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In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.
When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.